Policy Transforms and Learning Optimal Policies

abstractWe study the problem of choosing optimal policy rules in uncertain environments using models that may be incomplete and/or partially identified. We consider a policymaker who wishes to choose a policy to maximize a particular counterfactual quantity called a policy transform. We characterize learnability of a set of policy options by the existence of a decision rule that closely approximates the maximin optimal value of the policy transform with high probability. Sufficient conditions are provided for the existence of such a rule. However, learnability of an optimal policy is an ex-ante notion (i.e. before observing a sample), and so ex-post (i.e. after observing a sample) theoretical guarantees for certain policy rules are also provided. Our entire approach is applicable when the distribution of unobservables is not parametrically specified, although we discuss how semiparametric restrictions can be used. Finally, we show possible applications of the procedure to a simultaneous discrete choice example and a program evaluation example.

comment\subsection{Relation to Previous Work} To study counterfactuals requires at least a basic understanding of their historical context within economics. There are at least two popular notions of counterfactuals within economics, and both are connected with two different notions of causality used in econometrics. The first notion of counterfactuals comes from the definition of structural models/structural equations. The analysis of counterfactuals (or `ex-ante policy prediction') using structural economic models has a long history, going at least as far back as haavelmo1943statistical in the early days of the Cowles commission. The contribution of haavelmo1943statistical was to view a structural econometric model as representing a set of hypothetical experiments---encoded by a system of equations---which could be used to provide quantitative answers to policy questions. Following haavelmo1943statistical, the papers of koopmans1947measurement, marshak1953measurements, and hurwicz1966structural are widely regarded as having had a profound impact in forming the epistemology of modern econometrics. Within this notion of causality/counterfactuals, differences arise depending on whether the underlying system is recursive or nonrecursive (see strotz1960recursive), with the latter usually being associated with an “equilibrium” interpretation of causality and counterfactuals. The notion of counterfactuals based on structural equations has been developed extensively in the computer science literature by Judea Pearl, with the seminal article pearl1995causal, and a complete introduction in Another notion of counterfactual is an extension of the potential outcome model of neyman1923applications and rubin1974estimating. Explications of this notion of causality in econometrics are given by heckman2007econometric and heckman2008econometric, who also discuss the differences between the Neyman-Rubin potential outcome model, inluding the role of explicit models of outcomes and choice, and the role of randomization. In general, this notion of causality relies on sources of exogenous variation to reveal causal relationships. \begin{itemize} • heckman2007econometric introduce forecasting of policies in new environments as one of three central policy evaluation questions (the other two being the evaluation of historical policy, and predicting the impact of new policies never experienced). • Incomplete models: \begin{itemize} • GH2011: In economics, coordination games and participation games are the most natural area of application, including models of labour force participation (including Bjorn and Vuong 1985) models of discrete choice with social interactions (in Brock and Durlauf 2001), oligopoly entry models (including Andrews, Berry and Jia, 2003; Ciliberto and Tamer, auction models (including Bajari, Hong and Ryan, 2010), bargaining models (including network effects (including Engers and stern 2002) and network effects (including sweeting 2009). • Epstein, Kaido and Seo (2016): These models include: entry games with multiple Nash equilibria (Bresnahan and Reiss (1990, 1991), Berry (1992), Jia (2008), Ciliberto and Tamer (2009);2 first-price auctions (Haile and Tamer (2003)); innovation and product variety (Eizenberg (2014)); sincere versus strategic voting (Kawai and Watanabe (2013)); municipal mergers (Weese (2015)); discrete-choice with social interactions (Soetevent and Kooreman (2007)); matching with externalities (Uetake andWanatabe (2012)); and friendship networks (Miyauchi (2014)). \end{itemize} • Other interesting quotes: \begin{itemize} • heckman2007econometric: “Counterfactuals are required to forecast the effects of policies that have been tried in one environment but are proposed to be applied in new environments and to forecast the effects of new policies.” • heckman2007econometric: “Models for counterfactuals are the basis for extending historically experienced policies to new environments and for forecasting the effects of policies never previously experienced.” \end{itemize} \end{itemize}

assumptionThere exists a fixed probability space $(\Omega,\mathfrak{A},P)$, and a random element $U: (\Omega,\mathfrak{A}) \to (\mathcal{U},\mathfrak{B}(\mathcal{U}))$ where $\mathcal{U}$ is a compact second-countable Hausdorff space. In addition, the parameter space $\Theta$ is a Polish space equipped with the $\sigma-$algebra $\mathfrak{B}(\Theta)$.

assumption[Factual Domain] The factual domain is represented by random vectors $Y: (\Omega,\mathfrak{A}) \to (\mathcal{Y},\mathfrak{B}(\mathcal{Y}))$ and $Z: (\Omega,\mathfrak{A}) \to (\mathcal{Z},\mathfrak{B}(\mathcal{Z}))$, where $\mathcal{Y}$ and $\mathcal{Z}$ are Polish spaces. There exists a (possibly multi-valued) map $\bm G^{-} : \mathcal{Y} \times\mathcal{Z}\times\Theta \to \mathcal{U}$ which is closed and Effros-measurable, and satisfies:\todolt{does conditionally matter?} \begin{align} P\left(U \in \bm G^{-}(Y,Z,\theta_{0})|Y=y,Z=z\right)=1, \end{align} $(y,z)-$a.s. for some $\theta_{0} \in \Theta$. Furthermore, \begin{align} \E_{P_{U|Y,Z} \times P_{Y,Z}} [m_{j}(Y,Z,U,\theta_{0})] \leq 0, \qquad j=1,\ldots,J, \end{align} for some measurable functions $m_{j}: \mathcal{Y}\times \mathcal{Z}\times \mathcal{U}\times \Theta \to \mathbb{R}$, for $j=1,\ldots,J$, bounded in absolute value for each $\theta \in \Theta$.

assumption[$\Gamma$-Counterfactual Domains] The $\Gamma$-counterfactual domains are represented by a stochastic process $\{Y^\star(\omega,\gamma) : \gamma \in \Gamma\}$ where $(\Gamma,\mathfrak{B}(\Gamma))$ is a measurable space with $\Gamma$ a Polish space, and where $Y_{\gamma}^\star:=Y^\star(\cdot,\gamma)$ is such that $Y^\star: (\Omega\times\Gamma,\mathfrak{A}\otimes \mathfrak{B}(\Gamma)) \to (\mathcal{Y}^\star,\mathfrak{B}(\mathcal{Y}^\star)$ is measurable, with $\mathcal{Y}^\star$ a Polish space. Furthermore, there exists a (possibly multi-valued) map $\bm G^{\star}: \mathcal{Y}\times\mathcal{Z}\times\mathcal{U}\times \Theta \times \Gamma \to \mathcal{Y}^{\star}$ which is closed and Effros measurable, and satisfies: \begin{align} P\left(Y_{\gamma}^\star \in \bm G^\star(Y,Z,U,\theta_{0},\gamma)|Y=y,Z=z,U=u\right)=1, \end{align} $(y,z,u)-$a.s. for the same $\theta_{0} \in \Theta$ from Assumption (ref), and for all $\gamma \in \Gamma$ .

remark[The “No Back-Tracking” Principle] From a purely mathematical standpoint there is no reason that the moment functions in Assumption (ref) cannot also be functions of $Y_{\gamma}^\star$ and/or $\gamma \in \Gamma$. However, we omit this extension for interpretive reasons and caution researchers interested in this approach. In particular, if the researcher is not judicious in her formulation of such moment functions, then it is possible to have environments where the counterfactual $\gamma \in \Gamma$ of interest has “identifying power” for the structural parameters $\theta \in \Theta$. Such environments are extremely puzzling since, intuitively, in these cases the counterfactual domain $\gamma \in \Gamma$ under consideration contains “information” on the values of the structural parameters $\theta \in \Theta$ existing in the factual domain. Environments that avoid such difficulties will be said to satisfy the “no back-tracking principle.”\footnote{This principle is named in honour of the philosopher David Lewis who argued against similar “back-tracking counterfactuals” in lewis1979counterfactual.} We will return to this idea at some point in our example on simultaneous discrete choice models.

example[Simultaneous Discrete Choice] Consider a simultaneous discrete choice problem. In particular, assume that a binary outcome vector $Y:=(Y_{1},\ldots,Y_{K})$ has generic element $Y_{k} \in \mathcal{Y}$ determined by the equation: \begin{align} Y_{k}&=\mathbbm{1}\{ \pi_{k} \left(Z_{k},Y_{-k};\theta \right) \geq U_{k}\}. \end{align} Here $Z_{k}$ is a vector of covariates, $U_{k}$ is an unobserved random variable, and $\theta$ is a vector of model parameters. We will define the vector $Z:=(Z_{1},\ldots,Z_{K})$ and $U:=(U_{1},\ldots,U_{K})$ where each variable $Z_{k}$ has support $\mathcal{Z}=\{z_{1},\ldots,z_{L}\}$, a finite subset of euclidean space, and each $U_{k}$ has support $\mathcal{U}= [-1,1]^{d_{u}}$.\footnote{Note that we could instead define $\mathcal{U}:=\overline{\mathbb{R}}^{d_{u}}$, but then: \begin{align*} \mathbbm{1}\{ \pi_{k} \left(Z_{k},Y_{-k};\theta \right) \geq U_{k}\} = \mathbbm{1}\{ \tilde{\pi}_{k} \left(Z_{k},Y_{-k};\theta \right) \geq \tilde{U}_{k}\}, \end{align*} where $\tilde{\pi}_{k}\left(Z_{k},Y_{-k};\theta \right) = \text{tanh}\left(\pi_{k} \left(Z_{k},Y_{-k};\theta \right)\right)$ and $\tilde{U}_{k} = \text{tanh}(U_{k})$. In other words, the case with $\mathcal{U}:=\overline{\mathbb{R}}^{d_{u}}$ is homeomorphic to the case $\mathcal{U}:=[-1,1]^{d_{u}}$. } For each $k$, we assume that $\pi_{k}$ is a known measurable function of $(Z_{k},Y_{-k},\theta)$, mapping to $[-1,1]$ that is linear in parameters $\theta$ and has a gradient (with respect to $\theta$) bounded away from zero for each $(z,y_{-k})$. We also assume that $\theta=(\theta_{1},\ldots,\theta_{K})$, and that each $\pi_{k}$ depends only on the subvector $\theta_{k}$. For simplicity we will assume that the parameter space $\Theta$ is a compact subset of $\mathbb{R}^{d_{\theta}}$, and that $U$ is continuously distributed. \todo{How will you enforce this?} To illustrate the use of semi-parametric restrictions, we will also assume that each coordinate of the vector $U$ is (i) median zero, and (ii) median independent of $(Z_{k},Y_{-k})$. Finally, we assume all random variables are supported on the same probability space $(\Omega,\mathfrak{A},P)$. Verification of Assumption (ref) under these conditions is presented in Appendix (ref). For the factual domain, we have the following multifunction: \begin{align} \bm G^{-} \left( Y,Z,\theta \right):= cl\left\{ u \in \mathcal{U} : Y_{k}=\mathbbm{1}\{ \pi_{k} \left(Z_{k},Y_{-k};\theta \right) \geq u_{k}\},\,\, k=1,\ldots,K. \right\}. \end{align} Note the closure is taken to ensure that $\bm G^{-}(\cdot,\theta)$ is a closed set for each $\theta$. However, this introduces no additional structure and serves merely as a technical simplification, since $\bm G^{-}(\cdot,\theta_{0})$ as defined above will be almost surely equal to the right hand side of (ref) without taking the closure, which follows from the assumption that $U$ is continuously distributed. To complete the description of the factual domain, we will impose the median zero and median independence assumptions for each coordinate of the vector $U$ as a sequence of moment conditions. In particular, for $k=1\ldots,K$, we will impose the moment conditions: \begin{align} \E[\left(\mathbbm{1}\{U_{k} \geq 0 \} - \mathbbm{1}\{ U_{k} \leq 0\}\right)\mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\}] \leq 0,\quad \forall z \in \mathcal{Z}, \,\, y_{-k}\in \mathcal{Y}^{K-1},\\ \E[\left(\mathbbm{1}\{ U_{k} \leq 0\} - \mathbbm{1}\{U_{k} \geq 0 \}\right)\mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\}] \leq 0,\quad \forall z \in \mathcal{Z}, \,\, y_{-k}\in \mathcal{Y}^{K-1}. \end{align} Taken together, (ref) and (ref) imply that the latent variables $U_{k}$ are both median zero and median independent of covariates $Z_{k}$ and the outcomes $Y_{-k}$.\footnote{Note that this restriction implies constraints on the joint distribution of the vector $(U_{1},\ldots,U_{K})$. Alternatively, we might instead impose only median independence of $U_{k}$ with $Z_{k}$, which restricts only the marginal distribution of $U_{k}$.} Verification of Assumption (ref), including Effros-measurability of the multifunction (ref), is provided in Appendix (ref). Turning to the counterfactual domain, there are many possible counterfactuals that may be of interest. For the sake of illustration, we will consider counterfactuals of the following form. Let $\gamma_{k}: \mathcal{Z} \times \mathcal{Y}^{K-1} \to \mathcal{Z} \times \mathcal{Y}^{K-1}$, $\gamma = (\gamma_{k})_{k=1}^{K}$, and $Y_{\gamma}^\star:=(Y_{1,\gamma}^\star,\ldots,Y_{K,\gamma}^\star)$ with typical element: \begin{align} Y_{k,\gamma}^\star&=\mathbbm{1}\{ \pi_{k} (\gamma (Z_{k},Y_{-k,\gamma}^\star);\theta ) \geq U_{k}\}. \end{align} For example, our interest may be in the properties of the counterfactual random variable $Y_{k,\gamma}^\star$, such as its mean or its conditional mean. The multifunction for the counterfactual domain is then given by: \begin{align} \bm G^\star(Z,U,\theta,\gamma):=\left\{ y^\star \in \mathcal{Y} : y_{k}^\star=\mathbbm{1}\{ \pi_{k} \left(\gamma(Z_{k},y_{-k}^\star);\theta \right) \geq U_{k}\},\,\, k=1,\ldots,K. \right\}. \end{align} Note here we take $\mathcal{Y}^\star = \mathcal{Y}$. Verification of Assumption (ref), including Effros-measurability of the multifunction in (ref), is provided in Appendix (ref).

example[Program Evaluation] Consider the problem of program evaluation. In this example, a binary variable $D \in \{0,1\}$ indicates participation in the treatment or control group for some program, and the observed real-valued outcome is given by: \begin{align} Y=U_{0}(1-D)+U_{1}D, \end{align} where $U_{0}$ and $U_{1}$ are potential outcomes that are never jointly observed. We will assume throughout that $U_{0}, U_{1} \in \mathcal{U} = [\underline{Y},\overline{Y}]$, and thus we also assume the outcome $Y$ takes values in the bounded interval $\mathcal{Y}:= [ \underline{Y},\overline{Y}]$. In the absence of a selection equation determining the values of $D$, the potential outcome model is incomplete. This case is considered in russell2019sharp, and the framework in this paper applies to this case as well. Alternatively, we will consider the more popular approach of heckman1999local and heckman2005structural, and will suppose that the treatment is determined by the equation: \begin{align} D = \mathbbm{1}\{g_{0}(Z) \geq U \}, \end{align} where $U$ is continuous, and $g_{0}(\cdot)$ is an unknown measurable function of the observable covariates $Z \in \mathcal{Z}\subset \mathbb{R}^{d_{z}}$, where $d_{z}$ is the dimension of the vector $Z$. We will assume that $\mathcal{Z}$ is finite, and will allow for the case when the vector $Z$ can be decomposed as $Z=(X,Z_{0})$ with (i) $U\independent Z_{0}|X$ (conditional independence) and (ii) $\E[U_{d}|Z] = \E[U_{d}|X]$ for $d \in\{0,1\}$ (mean independence). We will thus decompose $\mathcal{Z}$ as $\mathcal{Z} = \mathcal{Z}_{0} \times \mathcal{X}$, where $\mathcal{Z}_{0}$ is the support of $Z_{0}$ and $\mathcal{X}$ is the support of $X$. Under these assumptions, it is without loss of generality that $U$ be taken to be uniformly distributed on $[0,1]$ conditional on $Z$. As shown in vytlacil2002independence, these assumptions, combined with the additive separability of the selection equation in (ref), are equivalent to the assumptions required to estimate the local average treatment effect (LATE) of imbens1994. This connects this model with a large body of empirical work that focuses on obtaining estimates of the LATE. Set the parameter space as $\Theta= \mathcal{G}\times \mathcal{T}$. Here $\mathcal{G}$ can be taken equal to the space of all positive measurable functions on $\mathcal{Z}$ which is a metric space with the sup norm (for example); under finiteness of $\mathcal{Z}$, this space is Polish. Furthermore, we will take the component $\mathcal{T}$ of the parameter space to be the space of all possible measurable functions on $\mathcal{Z}$. This component of the parameter space will be used in the moment functions below. Finally, we shall denote a generic pair $(g,t) \in \Theta$ as $\theta$. We will denote the support of $(U_{0},U_{1},U)$ as $\mathcal{U}:= [\underline{Y},\overline{Y}]^{2} \times [0,1]$. We also assume that the random variables in the vector $(Y,D,Z,U_{0},U_{1},U)$ are all supported on the same probability space $(\Omega,\mathfrak{A},P)$. Under these conditions, Assumption (ref) is verified in Appendix (ref). For the factual domain we have the multifunction: \begin{align} \bm G^{-} \left( Y,D,Z,\theta \right):= cl\left\{ (U_{0},U_{1},U) \in \mathcal{U} : \begin{array}{l} Y = U_{0}(1-D) + U_{1}D, \\ D = \mathbbm{1}\{g(Z) \geq U\}. \end{array} \right\}. \end{align} Note the closure is taken to ensure that $\bm G^{-}(\cdot,\theta)$ is a closed set for each $\theta$. However, this introduces no additional structure and serves merely as a technical simplification, since $\bm G^{-}(\cdot,\theta)$ as defined above will be almost surely equal to the right hand side of (ref) without taking the closure, which follows from the assumption that $U$ is continuously distributed. \todo{How will you enforce this?} Close inspection of this multifunction provides some simplification: \begin{align} \bm G^{-} \left( Y,D,Z,\theta \right)=\begin{cases} \{ Y \} \times [Y,\overline{Y}] \times [g(Z),1], & if D=0,\\ [Y,\overline{Y}] \times \{ Y \} \times [0,g(Z)], &\text{ if } D=1. \end{cases} \end{align} To complete the description of the factual domain, we will impose the independence condition $U \independent Z_{0}|X$ and the mean independence condition $\E[U_{d}|Z] = \E[U_{d}|X]$, for $d \in\{0,1\}$, as a sequence of moment conditions. In particular, since $\mathcal{Z}$ is assumed to be finite, let us partition $\mathcal{Z}$ into the product $\mathcal{Z}=\mathcal{Z}_{0}\times \mathcal{X}$, where $\mathcal{Z}_{0}:=\left\{z_{01}, \ldots, z_{0K} \right\}$ and $\mathcal{X}:= \left\{x_{1},\ldots,x_{L} \right\}$. Now consider the following sequence of moment inequalities: \begin{align} \E[\left(D - g(z_{0},x)\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] \leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X},\\ \E[\left(g(z_{0},x) - D\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] \leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X}, \end{align} and: \begin{align} \E[\left(\mathbbm{1}\{U \leq g(z_{0},x)\} - g(z_{0},x)\right)\mathbbm{1}\{X=x\}] \leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X},\, \\ \E[\left(g(z_{0},x) - \mathbbm{1}\{U \leq g(z_{0},x)\}\right)\mathbbm{1}\{X=x\}] \leq 0, \qquad\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X}. \end{align} Together (ref) and (ref) imply $P(D=1|Z=z) = g(z)$ for all $z \in \mathcal{Z}$, and (ref) and (ref) imply $P(U\leq g(z)|Z=z) = P(U \leq g(z)|X=x) = g(z)$ for all $z_{0} \in \mathcal{Z}_{0}$ and $x \in \mathcal{X}$. Under finiteness of the support $\mathcal{Z}$, these moment inequalities represent the only observable implications of the independence condition $U \independent Z_{0}|X$. In addition, we will impose the following moment conditions: \begin{align} \E\left[t(z_{0},x)-\mathbbm{1}\{Z=z_{0},X=x\}\right] &\leq 0,\qquad\forall z_{0} \in \mathcal{Z}_{0},\, \forall x \in \mathcal{X}, \\ \E\left[\mathbbm{1}\{Z=z_{0},X=x\}-t(z_{0},x)\right] &\leq 0,\qquad\forall z_{0} \in \mathcal{Z}_{0},\, \forall x \in \mathcal{X}, \end{align} and: \begin{align} \E\left[U_{d}\left(\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{X=x\}t(z_{0},x) \right)\right] &\leq 0, \,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\},\\ \E\left[U_{d}\left(\mathbbm{1}\{X=x\}t(z_{0},x)-\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x) \right) \right] &\leq 0, \,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\}. \end{align} Together (ref) - (ref) imply the mean independence condition: $\E[U_{d}|Z] = \E[U_{d}|X]$ for $d \in\{0,1\}$. In particular, (ref) and (ref) ensure $t(z_{0},x) = P(Z_{0}=z_{0},X=x)$, so that the moment conditions in (ref) and (ref) imply: \begin{align*} \E\left[U_{d}\left(\mathbbm{1}\{Z=z_{0},X=x\}P(X=x)- \mathbbm{1}\{X=x\}P(Z_{0}=z_{0},X=x)\right) \right] &= 0,\,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\}, \end{align*} or equivalently: \begin{align*} \E\left[U_{d}\left(\frac{\mathbbm{1}\{Z=z_{0},X=x\}}{P(Z_{0}=z_{0},X=x)}- \frac{\mathbbm{1}\{X=x\}}{P(X=x)}\right) \right] &= 0,\,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\}. \end{align*} From here, a full verification of Assumption (ref) for the factual domain, including Effros measurability of the multifunction (ref), is provided in Appendix (ref). With this setup, we might be interested in how the outcome variable changes when the factors $Z$ that determine an individual's treatment decision are modified. For example, let $\Gamma$ denote the set of all measurable functions $\gamma: \mathcal{Z} \to \mathcal{Z}$ (note that there are at most finitely many).\footnote{See carneiro2011estimating for a discussion of other possible parameters under this setting.} We can then define: \begin{align} Y_{\gamma}^\star=U_{0}(1-D_{\gamma}^\star)+U_{1}D_{\gamma}^\star, \end{align} where the random variable $D_{\gamma}^\star$ is then given by: \begin{align*} D_{\gamma}^{\star} = \mathbbm{1}\{g_{0}(\gamma(Z)) \geq U\}. \end{align*} Note that as in heckman1999local and heckman2005structural, our counterfactual $\gamma \in \Gamma$ has no direct effect on $(U_{0},U_{1})$. Our interest is in the properties of the random variable $Y_{\gamma}^\star$, such as its mean or its conditional mean. The multifunction for the counterfactual domain is given by: \begin{align} \bm G^\star(Z,U_{0},U_{1},U,\theta,\gamma):=\left\{ (Y_{\gamma}^\star,D_{\gamma}^\star) \in \mathcal{Y} \times \{0,1\} : \begin{array}{l} Y_{\gamma}^\star = U_{0}(1-D_{\gamma}^\star) + U_{1}D_{\gamma}^\star, \\ D_{\gamma}^\star = \mathbbm{1}\{g(\gamma(Z)) \geq U\}. \end{array} \right\}. \end{align} Note here we take $\mathcal{Y}^\star = \mathcal{Y}$. Again, close inspection of this multifunction provides some simplification: \begin{align} \bm G^\star(Z,U_{0},U_{1},U,\theta,\gamma)=\begin{cases} (U_{1},1), &\text{ if } U \leq g(\gamma(Z)),\\ (U_{0},0), &\text{ if } g(\gamma(Z)) < U. \end{cases} \end{align} A full verification of Assumption (ref) for the counterfactual domain, including Effros measurability of the multifunction (ref), is provided in Appendix (ref).

definition[Policy Transform and Conditional Policy Transform] Let $\varphi: \Omega \times \Gamma \to \mathbb{R}$ be a bounded and measurable function. The policy transform of $\varphi$ is a function $I[\varphi](\gamma): \Gamma \to \mathbb{R}$ given by: \begin{align} I[\varphi](\gamma):= \int \varphi(\omega,\gamma) \,dP. \end{align} Furthermore, if $\mathfrak{A}' \subset \mathfrak{A}$ is a $\sigma-$algebra, then a conditional policy transform of $\varphi$ given $\mathfrak{A}'$ is a function $\tilde{I}[\varphi]: \Omega \times \Gamma \to \mathbb{R}$ such that (i) $\tilde{I}[\varphi]:\Omega\times\Gamma \to \mathbb{R}$ is $\mathfrak{A}'\otimes\Gamma-$measurable, and (ii) $I[\tilde{I}[\varphi](\cdot,\gamma)\mathbbm{1}_{A}](\gamma) = I[\varphi \mathbbm{1}_{A}](\gamma)$ for every $A \in \mathfrak{A}'$.

definition[The Decision Problem] The policymaker's decision problem is characterized by: \begin{enumerate}[label=(\roman*)] • The population, represented by the probability space $(\Omega,\mathfrak{A},P)$. • The action (or policy) space, given by $(\Gamma,\mathfrak{B}(\Gamma))$. • The sample space, given by $(\Psi_{n},\Sigma_{\Psi_{n}}, P_{Y,Z}^{\otimes n})$, where $\Psi_{n} := (\mathcal{Y}\times \mathcal{Z})^{n}$, with typical element $\psi=\{(y_{i},z_{i})\}_{i=1}^{n}$, equipped with the product Borel $\sigma-$algebra $\Sigma_{\Psi_{n}} := (\mathfrak{B}(\mathcal{Y})\otimes\mathfrak{B}(\mathcal{Z}))^{\otimes n}$ and the product measure $P_{Y,Z}^{\otimes n}$. • The state space, given by $\mathcal{S}\times \mathcal{P}_{Y,Z}$, where $\mathcal{P}_{Y,Z}$ is the set of all Borel probability measures on $\mathcal{Y}\times \mathcal{Z}$, and $\mathcal{S}$ is the set of all triples $s=(\theta,P_{U|Y,Z},P_{Y_{\gamma}^\star|Y,Z,U})$ such that the pair $(s,P_{Y,Z})$ satisfies: \begin{enumerate} • $\theta \in \Theta$, • $P_{U|Y,Z}(U \in \bm G^{-}\left(Y,Z,\theta\right) | Y=y,Z=z)=1$, $(y,z)$-a.s., • $P_{Y_{\gamma}^\star|Y,Z,U}(Y_{\gamma}^\star \in \bm G^{\star}(Y,Z,U,\theta,\gamma) | Y=y,Z=z,U=u )=1$, $(y,z,u)$-a.s., and • the elements $\theta \in \Theta$ and $P_{U|Y,Z}$ satisfy: \begin{align} \max_{j=1,\ldots,J} \E_{P_{U|Y,Z} \times P_{Y,Z} } [ m_{j}(Y,Z,U,\theta) ] \leq 0. \end{align} \end{enumerate} • The feasible statistical decision rules $\mathcal{D}$, with typical element $d$, given by the set of all measurable functions $d: \Psi_{n} \to \Gamma$.\note{Note that the composition of universally measurable functions is universally measurable. Thus, it seems Borel measurability of $d$ is sufficient for the map $\psi\mapsto \inf_{s \in \mathcal{S}} I[\varphi](d(\psi),s)$ to be measurable if $\gamma \mapsto \inf_{s\in \mathcal{S}} I[\varphi](\gamma,s)$ is universally measurable.} • The objective function, given by a function $I[\varphi]:\Gamma \times \mathcal{S}\times \mathcal{P}_{Y,Z}\to \mathbb{R}$, called the state-dependent policy transform, which has the expression: \begin{align} I[\varphi](\gamma,s):= \int \varphi(v) \, d(P_{Y_{\gamma}^\star|Y,Z,U} \times P_{U|Y,Z}\times P_{Y,Z}) \end{align} where $\varphi: \mathcal{V} \to \mathbb{R}$ is a measurable function (where $P_{Y,Z}$ is left implicit when writing $I[\varphi](\gamma,s)$).\todo{Varphi should depend on $\theta$, no?} \end{enumerate}

definition[PAC Maximin Preference Relation] Fix a sample size $n$. For any $\kappa \in (0,1)$ and any $d \in \mathcal{D}$, let $c_{n}(\cdot,\kappa): \mathcal{D} \to \mathbb{R}_{++}$ be the smallest value satisfying: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left(\inf_{s \in \mathcal{S}} I[\varphi](d(\psi),s) + c_{n}(d,\kappa) \geq \sup_{\gamma \in \Gamma} \inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) \right) \geq \kappa. \end{align} Then decision rule $d:\Psi_{n} \to \Gamma$ is weakly preferred to (or weakly dominates) decision rule $d':\Psi_{n} \to \Gamma$ at level $\kappa$ and sample size $n$, denoted by $d' \preccurlyeq_{\kappa} d$, if and only if $c_{n}(d,\kappa) \leq c_{n}(d',\kappa)$. The decision rule $d:\Psi_{n} \to \Gamma$ is strictly preferred to (or strictly dominates) decision rule $d':\Psi_{n} \to \Gamma$, denoted by $d' \prec_{\kappa} d$, if and only if $c_{n}(d,\kappa) < c_{n}(d',\kappa)$. A decision rule $d \in \mathcal{D}$ will be called admissible with respect to $\preccurlyeq_{\kappa}$ if there is no decision rule $d' \in \mathcal{D}$ that is strictly preferred to (or strictly dominates) $d$.\note{It seems this will actually be equivalent to Wald's decision problem when $\mathcal{S}=\{s_{0}\}$.}

propositionSuppose that Assumptions (ref), (ref) and (ref) hold, and that $\varphi: \mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}] \subseteq \mathbb{R}$ is a bounded and measurable function. Also, suppose that $\gamma \mapsto \inf_{s \in \mathcal{S}} I[\varphi](\gamma,s)$ is (universally) measurable. Let $d,d'\in \mathcal{D}$ be two decision rules, and suppose that for every $P_{Y,Z} \in \mathcal{P}_{Y,Z}$ we have $I[\varphi](d'(\psi),s) \leq I[\varphi](d(\psi),s)$ a.s. for every $s \in \mathcal{S}$. Then for any $\kappa \in (0,1)$ we have $d' \preccurlyeq_{\kappa} d$, where $\preccurlyeq_{\kappa}$ is the preference relation from Definition (ref); that is, the preference relation $\preccurlyeq_{\kappa}$ respects weak dominance.

proofSee Appendix (ref).

remarkUniversal measurability is a weaker requirement then Borel measurability, and is defined in Appendix (ref). Also, in Appendix (ref) we show that the map $\gamma \mapsto \inf_{s \in \mathcal{S}} I[\varphi](\gamma,s)$ is universally measurable, although the result and proof relies on Assumption (ref) introduced in the next section. Since Assumption (ref) has not yet been introduced at this point, we impose (universal) measurability of $\gamma \mapsto \inf_{s \in \mathcal{S}} I[\varphi](\gamma,s)$ as a separate assumption in this proposition.

example[Simultaneous Discrete Choice (Cont'd)] For the simultaneous discrete choice example, recall that our interest is in the properties of the counterfactual random variable $Y_{k,\gamma}^\star$, such as its mean or its conditional mean. For the sake of illustration, we will focus on the quantity: \begin{align} I[\varphi](\gamma) = \int_{\Omega} \mathbbm{1}\{Y_{k,\gamma}^\star(\omega) = 1 \} \,dP, \end{align} which is a counterfactual choice probability. Note this quantity is the policy transform of the function $\varphi(\omega,\gamma) = \mathbbm{1}\{Y_{k,\gamma}^\star(\omega) = 1 \}$. Without much additional complication, we might instead be interested in the conditional choice probability $\E[\mathbbm{1}\{Y_{k,\gamma}^\star(\omega) = 1 \}|Z]$; it is easily verified that $\tilde{I}[\varphi](\omega,\gamma)= \E[\varphi(\omega,\gamma)|Z](\omega)$, with $\varphi(\omega,\gamma) = \mathbbm{1}\{Y_{k,\gamma}^\star(\omega) = 1 \}$, is a conditional policy transform.\footnote{Indeed, by definition this quantity is measurable with respect to $\sigma(Z)$, and satisfies: \begin{align} I[\tilde{I}[\varphi](\cdot,\gamma)\mathbbm{1}_{A}](\gamma) = \int \E[\varphi(\omega,\gamma)|Z](\omega) \mathbbm{1}_{A}(\omega) \, dP = \int \mathbbm{1}\{Y_{k,\gamma}^\star(\omega) = 1 \} \mathbbm{1}_{A}(\omega) \, dP =I[\varphi \mathbbm{1}_{A}](\gamma), \end{align} for every $A \in \sigma(Z)$. } Throughout we will suppose the policymaker is interested in selecting the policy $\gamma \in \Gamma$ that maximizes the quantity (ref). We can now formally define the policymaker's decision problem. The population is given by the probability space $(\Omega,\mathfrak{A},P)$ and the action space is given by $(\Gamma,\mathfrak{B}(\Gamma))$, where $\Gamma$ is the set of all functions $\gamma=(\gamma_{k})_{k=1}^{K}$ with $\gamma_{k}: \mathcal{Z}\times \mathcal{Y}^{K-1} \to \mathcal{Z}\times \mathcal{Y}^{K-1}$ and $\mathfrak{B}(\Gamma)$ can be taken as the power set of $\Gamma$.\footnote{Since $\mathcal{Z}$ and $\mathcal{Y}$ are finite, both $\Gamma$ and $\mathfrak{B}(\Gamma)$ contain at most finitely many elements.} The sample space in this example is given by $\Psi_{n}$, which is all possible realizations of the $n$ vectors $\{(y_{i},z_{i})\}_{i=1}^{n}$. Each state of the world is indexed by a pair $(\theta,P_{U|Y,Z})$ satisfying the support restriction given by (ref) and the moment conditions (ref) and (ref). The state dependent policy transform is given by: \begin{align*} I[\varphi](\gamma,s):= \int \mathbbm{1}\{U_{k} \leq \pi_{k}(\gamma(Z_{k},Y_{-k});\theta) \} \,dP_{U|Y,Z}dP_{Y,Z}. \end{align*} A feasible statistical decision rule is then any measurable function $d:\Psi_{n} \to \Gamma$ that selects a policy indexed by $\gamma$ given access to an $n-$sample from $\Psi_{n}$.

example[Program Evaluation (Cont'd)] For the program evaluation example, recall that our interest is in the properties of the random variable $Y_{\gamma}^\star$, such as its mean or its conditional mean. For the sake of illustration, we will focus on the average outcome under some counterfactual policy $\gamma \in \Gamma$, given by $\E[Y_{\gamma}^\star]$. Note that taking $\varphi(\omega,\gamma) = Y_{\gamma}^\star(\omega) \left(:= Y^\star(\omega,\gamma)\right)$, it is then clear that $\E[Y_{\gamma}^\star] = I[\varphi](\gamma)$, so that the average effect of a counterfactual policy is the policy transform of the random variable $Y_{\gamma}^\star(\omega)$. Without much additional complication, we might instead be interested in the conditional average effect $\E[Y_{\gamma}^\star|X]$. It is easily verified that $\tilde{I}[\varphi](\omega,\gamma)= \E[\varphi(\omega,\gamma)|X](\omega)$, with $\varphi(\omega,\gamma) = Y_{\gamma}^\star(\omega)$, is a conditional policy transform.\footnote{Indeed, by definition this quantity is measurable with respect to $\sigma(X)$, and satisfies: \begin{align} I[\tilde{I}[\varphi](\cdot,\gamma)\mathbbm{1}_{A}](\gamma) = \int \E[\varphi(\omega,\gamma)|X](\omega) \mathbbm{1}_{A}(\omega) \, dP = \int Y_{\gamma}^\star(\omega) \mathbbm{1}_{A}(\omega) \, dP =I[\varphi \mathbbm{1}_{A}](\gamma), \end{align} for every $A \in \sigma(X)$. } We will assume throughout that the policymaker is interested in maximizing the value of $\E[Y_{\gamma}^\star]$. We can now formally define the policymaker's decision problem. The population is given by the probability space $(\Omega,\mathfrak{A},P)$ and the action space is given by $(\Gamma,\mathfrak{B}(\Gamma))$, where $\Gamma$ is the set of all functions $\gamma: \mathcal{Z} \to \mathcal{Z}$ and $\mathfrak{B}(\Gamma)$ is the power set of $\Gamma$.\footnote{Since $\mathcal{Z}$ is finite, both $\Gamma$ and $\mathfrak{B}(\Gamma)$ contain at most finitely many elements.} The sample space is given by $\Psi_{n} = (\mathcal{Y} \times \{0,1\} \times \mathcal{Z})^{n}$ with a typical element $\psi= ((y_{i},d_{i},z_{i}))_{i=1}^{n}$. The state space $\mathcal{S}$ is given by $s=(\theta,P_{U_{0},U_{1},U|Y,Z}, P_{Y_{\gamma}^{\star}|U_{0},U_{1},U,Y,Z})$, where $P_{U_{0},U_{1},U|Y,Z}$ and $P_{Y_{\gamma}^{\star}|U_{0},U_{1},U,Y,Z}$ are any random variables that satisfy the support restriction (ref) and moment conditions (ref) - (ref). Finally, a feasible statistical decision rule is any measurable function $d:\Psi_{n} \to \Gamma$ that selects a policy indexed by $\gamma$ given access to an $n-$sample from $\Psi_{n}$.

definition[PAMPAC Learnability] Under Assumptions (ref), (ref), and (ref), a policy space $\Gamma$ is policy agnostic maximin PAC-learnable (PAMPAC) with respect to the policy transform of $\varphi:\mathcal{V} \to \mathbb{R}$ if there exists a function $\zeta_{\Gamma}: \mathbb{R}_{++}\times (0,1) \to \mathbb{N}$ such that, for any $(c,\kappa) \in \mathbb{R}_{++}\times (0,1)$ and any distribution $P_{Y,Z}$ over $\mathcal{Y}\times \mathcal{Z}$, if $n \geq \zeta_{\Gamma}(c,\kappa)$ then there is some decision procedure $d: \Psi_{n} \to \Gamma$ satisfying: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s) + c \geq \sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) \right) \geq \kappa. \end{align}

example[Simultaneous Discrete Choice (cont'd)] Consider the general setup of Example (ref). Suppose for simplicity that $K=1$, and consider the following modifications. Let $\mathcal{Z}=[-1,1]$ and $\Theta=[-1,1]$ and let $\pi_{k}(Z_{k},Y_{-k};\theta)=\pi_{k}(Z_{k};\theta) = \sin(Z_{k}/\theta)$. Then $Y_{k}$ is determined by the equation: \begin{align*} Y_{k} = \mathbbm{1}\{\sin(Z_{k}/\theta) \geq U_{k} \}. \end{align*} Now consider a policy space $\Gamma$ that consists of all functions $\gamma: \mathcal{Z} \to \mathcal{Z}$, and suppose we are interested in the policy transform: \begin{align*} I[\varphi](\gamma):= \int_{\Omega} \varphi(\omega,\gamma)\,dP= \int_{\Omega} \mathbbm{1}\{ Y_{k,\gamma}^\star(\omega)=1\} \,dP, \end{align*} where $\varphi(\omega,\gamma)= \mathbbm{1}\{ Y_{k,\gamma}^\star(\omega)=1\}$ and: \begin{align*} Y_{k,\gamma}^\star = \mathbbm{1}\{\sin(\gamma(Z_{k})/\theta) \geq U_{k} \}. \end{align*} In this case, we claim the policy space $\Gamma$ may not be PAMPAC learnable with respect to the policy transform of $\varphi$. \todo{Prove rigorously}

example[Program Evaluation (cont'd)] Consider the general setup of Example (ref), with the following modifications. Let $\mathcal{Z}=[-1,1]$ and let $\Theta$ denote the space of continuous functions with values in $[-1,1]$. Otherwise, keep all other aspects of the factual domain the same. Now consider a policy space $\Gamma$ that consists of all continuous functions $\gamma: \mathcal{Z} \to \mathcal{Z}$. Suppose we are still interested in the policy transform of $\varphi(\omega,\gamma) = Y_{\gamma}^\star(\omega)$, where: \begin{align} Y_{\gamma}^\star=U_{0}(1-D_{\gamma}^\star)+U_{1}D_{\gamma}^\star, \end{align} and where the random variable $D_{\gamma}^\star$ is given by: \begin{align*} D_{\gamma}^{\star} = \mathbbm{1}\{\theta_{0}(\gamma(Z)) \geq U\}. \end{align*} In this case, we claim the policy space $\Gamma$ may not be PAMPAC learnable with respect to the policy transform of $\varphi$. \todo{Prove rigorously}

definition[Distributions of Selections] The collection $\mathcal{P}_{U|Y,Z}(\theta)$ contains all regular conditional probability measures $P_{U|Y,Z}$ such that each $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$ is the distribution of some selection $U \in Sel(\bm G^{-}(\cdot,\theta))$; that is:\footnote{Clearly the collection $\mathcal{P}_{U|Y,Z}(\theta)$ also depends on $P_{Y,Z}$, although we suppress this dependence for notational simplicity throughout.} \begin{align} \mathcal{P}_{U|Y,Z}(\theta) := \left\{P_{U|Y,Z} : U \sim P_{U|Y,Z} for some U \in Sel(\bm G^{-}(\cdot,\theta))\right\}. \end{align} Furthermore, the collection $\mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)$ contains all regular conditional probability measures $P_{Y_{\gamma}^\star|Y,Z,U}$ such that each $P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)$ is the distribution of some selection $Y_{\gamma}^\star \in Sel(\bm G^{\star}(\cdot,\theta,\gamma))$; that is:\footnote{Clearly the collection $\mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)$ also depends on $P_{Y,Z,U}$, although we suppress this dependence for notational simplicity throughout.} \begin{align} \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma) := \left\{P_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma) : Y_{\gamma}^\star \sim P_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma) for some Y_{\gamma}^\star \in Sel(\bm G^{\star}(\cdot,\theta,\gamma))\right\}. \end{align}

definition[Identified Set of Structural Parameters] Under Assumptions (ref) and (ref), the identified set $\Theta^{*}$ of structural parameters (with respect to the distribution $P_{Y,Z}$) is given by: \begin{align} \Theta^{*}:= \left\{\theta \in \Theta :\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} \,\,\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right] \leq 0 \right\}. \end{align}

definition[Identified Set for Policy Transforms] Under Assumptions (ref), (ref), and (ref), for any $\gamma \in \Gamma$ the identified set for $I[\varphi](\gamma)$ (with respect to the distribution $P_{Y,Z}$) is given by: \begin{align} \mathcal{I}^{*}[\varphi](\gamma) := \bigcup_{\theta \in \Theta^{*}} \mathcal{I}[\varphi](\theta,\gamma), \end{align} where: \begin{align} \mathcal{I}[\varphi](\theta,\gamma):= \bigg\{ I \in \mathbb{R} : \exists P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta) and &P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma) \nonumber\\ & satisfying I^{*}[\varphi]\left( \theta,\gamma,I, P_{Y_{\gamma}^\star|Y,Z,U}, P_{U|Y,Z}\right)\leq 0\bigg\}. \end{align}

assumption[Error Bounds] (i) (Linear Minorant) There exists values $\delta>0$ and $C_{1}>0$ such that for every $\theta \in \Theta$: \begin{align} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+} \geq C_{1} \min\{\delta,d(\theta,\Theta^{*})\}. \end{align} (ii) (Local Counterfactual Robustness) There exists a value $C_{2} \geq 0$ such that for any $\theta \in \Theta_{\delta}^{*} := \{ \theta : d(\theta, \Theta^{*}) \leq \delta\} $: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}\nonumber\\ &\qquad\qquad\qquad\geq \inf_{\theta^{*} \in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta^{*})}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta^{*},\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} - C_{2} d(\theta,\Theta^{*}), \end{align} and: \begin{align} &\sup_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\sup_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}\nonumber\\ &\qquad\qquad\qquad\leq \sup_{\theta^{*} \in \Theta^{*}}\sup_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta^{*})}\sup_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta^{*},\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} + C_{2} d(\theta,\Theta^{*}). \end{align}

assumption[Error Bounds (2)(ii)] For some $\delta > 0$, there exists values $\ell_{1},\ell_{2} \geq 0$ (possibly depending on $\delta$) such that: \begin{align} d(u,\bm G^{-}(y,z,\theta)) &\leq \ell_{1} \cdot d(\theta,\Theta^{-}(y,z,u)\cap \Theta_{\delta}^{*}), \,\,&&(y,z)-a.s. \,\, for all u \in \mathcal{U} and \theta \in \Theta_{\delta}^{*}, \\ d(y^\star,\bm G^{\star}(y,z,u,\theta,\gamma)) &\leq \ell_{2} \cdot d(\theta,\Theta^{\star}(v,\gamma)\cap \Theta_{\delta}^{*}), \,\,&&(y,z,u)-a.s. \,\, for all y^\star \in \mathcal{Y}^\star and \theta \in \Theta_{\delta}^{*}. \end{align} where $\Theta^{-}(y,z,u)$ and $\Theta^{\star}(v,\gamma)$ are defined by: \begin{align*} \Theta^{-}(y,z,u)&:= \left\{ \theta : u \in \bm G^{-}(y,z,\theta)\right\}, &&\Theta^{\star}(v,\gamma):= \left\{ \theta : y^\star \in \bm G^{\star}(y,z,u,\theta,\gamma)\right\}. \end{align*} Furthermore, the function $\varphi: \mathcal{V} \to \mathbb{R}$ is bounded, measurable, and Lipschitz continuous in $(u,y^\star)$ with Lipschitz constant $L_{\varphi}$.

lemmaSuppose that Assumptions (ref), (ref) and (ref) are satisfied. Finally, suppose that $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Then Assumption (ref) implies Assumption (ref)(ii) with $C_{2} = L_{\varphi} \max\{\ell_{1},\ell_{2}\}$.

proofSee Appendix (ref).

theorem[Bounds on the Policy Transform] Suppose that Assumptions (ref), (ref), (ref) and (ref) all hold. Also, suppose that $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}]\subset \mathbb{R}$ is a bounded, measurable function, and that for each $\gamma \in \Gamma$, the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Then $\overline{\text{co}}\, \mathcal{I}^{*}[\varphi](\gamma)=[I_{\ell b}[\varphi](\gamma),I_{u b}[\varphi](\gamma)]$, with: \begin{align} I_{\ell b}[\varphi](\gamma) &= \inf_{\theta \in \Theta} \max_{\lambda_{j} \in \{0,1\}}\int\inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg) \, dP_{Y,Z},\\ I_{ub}[\varphi](\gamma) &= \sup_{\theta \in \Theta} \min_{\lambda_{j} \in \{0,1\}} \int \sup_{u \in \bm G^{-}(y,z,\theta)} \sup_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) - \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg) \, dP_{Y,Z}, \end{align} where $\mu^{*} \in \mathbb{R}_{+}$ is any value satisfying: \begin{align} \mu^{*} \geq \max\left\{\frac{C_{2}}{C_{1}}, \frac{(\varphi_{ub} - \varphi_{\ell b})}{C_{1}\delta} \right\}, \end{align} and where $C_{1}$, $C_{2}$ and $\delta$ are from Assumption (ref).

proofSee Appendix (ref).

example[Simultaneous Discrete Choice (cont'd)] Consider again Example (ref) on simultaneous discrete choice, and recall that we have imposed a median zero and median independence restriction using the moment conditions in (ref) and (ref). This example presents challenges for the verification of Assumption (ref) because of the discontinuity of the function $\varphi(v)=\mathbbm{1}\{\pi_{k}(\gamma(z,y_{-k});\theta)\geq u \}$. Indeed, under our current assumptions, Assumption (ref) is not satisfied. To appreciate the intuition, focus on Assumption (ref)(ii). The issue for this assumption arises only when for some $k \in \{1,\ldots,K\}$ and some $z \in \mathcal{Z}$ and $y_{-k} \in \mathcal{Y}^{K-1}$ we have (i) the counterfactual cutoff value $\pi_{k}(\gamma(z,y_{-k});\theta^{*})=0$ at some $\theta^{*} \in \partial\Theta^{*}$, and if (ii) $P(Y_{k}=1|Z_{k}=z',Y_{-k}=y_{-k}')\neq 0.5$, where $(z',y_{-k}')=\gamma(z,y_{-k})$. In this knife-edge case, a very small change from $\theta^{*}\in \partial\Theta^{*}$ to some $\theta \notin \Theta^{*}$ can cause a discontinuous change in $P(Y_{\gamma,k}^\star=1)$. A full description of this failure, including illustrations of various cases, is presented in Appendix (ref). However, by slightly strengthening our moment conditions we can satisfy Assumption (ref) in this example. The key is to introduce additional assumptions on the degree of smoothness of the distribution of $U_{k}$ around zero. In particular, we will replace the moment conditions in (ref) and (ref) with the following conditions: \begin{align} \E \left[\left(\mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\} - \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\} - 0.5\right)\mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\} \right]\leq 0,\\ \E\left[\left(0.5 - \mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\}-\max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\right) \mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\} \right] \leq 0, \end{align} for $k=1,\ldots,K$, for all $z,z' \in \mathcal{Z}$ and all $y_{-k},y_{-k}' \in \mathcal{Y}^{K-1}$. In addition to implying the median zero/median independence assumption, these new moment conditions also limit the amount of probability mass on $\mathcal{U}$ that is arbitrarily close to zero, which turns out to be key to satisfying Assumption (ref). Also note that, despite the fact that these moment conditions will implicitly impose constraints on the obtainable counterfactual choice probabilities, it is easily verified that they do not impose any additional constraints on the set of structural parameters $\theta \in \Theta$ that can rationalize the observed distribution (in the sense of Definition (ref)), and thus do not violate the no-backtracking principle introduced in Remark (ref). With these new moment conditions, Assumption (ref) can be shown to be satisfied. Recall that when first introducing Example (ref) we assumed $\pi_{k}$ is a known measurable function of $(Z_{k},Y_{-k})$ that is linear in parameters $\theta$, and has a gradient (with respect to $\theta$) bounded away from zero for each $(z,y_{-k})$. We conclude that $\pi_{k}$ is Lipschitz in $\theta$, and also satisfies a “reverse Lipschitz” condition; that is, for each $(z,y_{-k})$ we have: \begin{align*} L_{k}'||\theta-\theta^{*}||\leq |\pi_{k}(z,y_{-k};\theta)-\pi_{k}(z,y_{-k};\theta^{*})| \leq L_{k}||\theta-\theta^{*}||, \end{align*} for some $L_{k}',L_{k}>0$. Now define: \begin{align} \tau:= \min_{k}\min_{(z,y_{-k})} |0.5 - P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})| && s.t. &&|0.5 - P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})|>0. \end{align} Then the analysis in Appendix (ref) shows that Assumption (ref) is verified for $C_{1} = L_{0} L'$, $C_{2}=L_{0}L$ and $\delta = \tau/(L_{0} L')$, where $L=\min_{k} L_{k}$ and $L' = \min_{k} L_{k}'$. In Theorem (ref) we can thus take the penalty $\mu^{*}$ to be any value satisfying: \begin{align*} \mu^{*}\geq \max\left\{\frac{L}{L'}, \frac{1}{\tau} \right\}. \end{align*} Theorem (ref) says that the lower and upper envelopes on $I[\varphi](\gamma)= P(Y_{\gamma}^\star=1)$, as a function of $\gamma$, are given by (ref) and (ref), respectively.

remark[Counterfactual Coherency] Recall that Theorem (ref) applies only if the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. In the simultaneous discrete choice example, the counterfactual map $\bm G^{\star}(\cdot,\theta,\gamma)$ can fail to be almost-surely non-empty, which is related to the well known problem of coherency in these models. In particular, for a given instantiation of a vector of unobservables $(u_{1},\ldots,u_{K})$, there may not exist any vector of counterfactual endogenous outcome variables $(y_{1,\gamma}^\star,\ldots,y_{K,\gamma}^\star)$ that solves the system of equations represented by (ref). However, we note that this issue is unrelated to our particular approach, and might be resolved by (i) conditioning the analysis on the subset of $\mathcal{U}$ that ensures a solution to the system of equations in (ref), or (ii) imposing certain constraints on the parameter space that ensures the existence of a solution to the system of equations in (ref). We refer the reader to chesher2020structural for a thorough discussion of this issue. However, whether this “counterfactual coherency” problem can be resolved without violating the no-backtracking principle from Remark (ref) appears to be an open question.\todolt{Does this violate the no back-tracking argument?}

example[Program Evaluation (cont'd)] Consider again Example (ref) on program evaluation. Verification of Assumption (ref) is presented in Appendix (ref), and uses Lemma (ref) to verify Assumption (ref)(ii). Remarkably, we show that Assumption (ref) is satisfied for any value of $\delta>0$ with $C_{1}=C_{2}=1$. Thus we can take the penalty $\mu^{*}=1$. Then Theorem (ref) says that the lower and upper envelopes on $I[\varphi](\gamma)= \E[Y_{\gamma}^\star]$, as a function of $\gamma$, are given by (ref) and (ref), respectively.

propositionSuppose Assumptions (ref), (ref), (ref), and (ref) hold. Also, suppose that $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}]\subset \mathbb{R}$ is a bounded, measurable function, and that for each $\gamma \in \Gamma$, the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Then a policy space $\Gamma$ is PAMPAC learnable with respect to the policy transform of $\varphi$ if and only if: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \sup_{\gamma \in \Gamma} I_{\ell b}[\varphi](\gamma) - I_{\ell b}[\varphi](d(\psi)) \leq c \right) \geq \kappa, \end{align} where $I_{\ell b}[\varphi]: \Gamma \to \mathbb{R}$ is the lower envelope function from Theorem (ref).

remarkBy Proposition (ref) in Appendix (ref), the map $\psi \mapsto I_{\ell b}[\varphi](d(\psi))$ is universally measurable; that is, measurable with respect to the completion of any $P_{Y,Z} \in \mathcal{P}_{Y,Z}$. Thus, the event in (ref) can always be assigned a unique probability using outer measures, if necessary.

definition[Covering Number, Metric Entropy] Let $(\mathcal{T},\rho)$ be a semi-metric space. A cover of $\mathcal{T}$ is any collection of sets whose union contains $\mathcal{T}$ as a subset. For any $\varepsilon>0$, the covering number for $\mathcal{T}$, denoted by $N(\varepsilon,\mathcal{T},\rho)$, is the smallest number of $\rho-$balls needed to form a $\varepsilon-$cover. The metric entropy is the logarithm of the covering number.

definition[Entropy Growth Condition] Let $\mathcal{F}$ be a measurable class of real-valued functions on a measurable space $(\mathcal{X},\mathfrak{A}_{\mathcal{X}})$ with envelope $F$. The class $\mathcal{F}$ satisfies the entropy growth condition if: \begin{align} \sup_{Q \in \mathcal{Q}_{n}} \log N(\varepsilon,\mathcal{F},||\cdot||_{Q,2})=o(n), \end{align} for every $\varepsilon>0$, with the supremum taken over all discrete probability measures $\mathcal{Q}_{n}$ on $\mathcal{X}$ with atoms that have probabilities that are integer multiples of $1/n$.

comment\begin{definition}[VC Type Classes, chernozhukov2013gaussian Definition 2.1] Let $\mathcal{F}$ be a measurable class of functions with envelope $F$ on a measurable space $(\mathcal{X},\mathfrak{A}_{\mathcal{X}})$. The class $\mathcal{F}$ is VC type with envelope $F$ if there are constants $A,\nu >0$ such that: \begin{align*} \sup_{Q} N(\varepsilon ||F||_{Q,2},\mathcal{F},||\cdot||_{Q,2}) \leq \left( \frac{A}{\varepsilon} \right)^\nu, \end{align*} for every $\varepsilon>0$, with the supremum taken over all finitely discrete probability measures on $(\mathcal{X},\mathfrak{A}_{\mathcal{X}})$. \end{definition} Pollard Manageabaility?

theoremSuppose that Assumptions (ref), (ref), (ref) and (ref) hold. Also, suppose that $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}]\subset \mathbb{R}$ is a bounded, measurable function, and that for each $\gamma \in \Gamma$, the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Fix any $\varepsilon>0$. (i) If the class of functions $\mathcal{H}_{\ell b}$ satisfies the entropy growth condition, then every policy space is PAMPAC learnable with respect to the policy transform of $\varphi$. Furthermore, for any $c>0$ we have: \begin{align} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) - \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s) \geq c \right) =O(r_{1}(n)), \end{align} where: \begin{align} r_{1}(n):=\max\left\{n^{-1/2}, n^{-1/2} \sup_{Q \in \mathcal{Q}_{n}}\sqrt{ \log N(\varepsilon,\mathcal{H}_{\ell b},||\cdot||_{Q,2})}\right\}. \end{align} (ii) If the class of functions: \begin{align} \Phi &:= \{ \varphi(\cdot,u,y^\star):\mathcal{Y}\times\mathcal{Z} \to \mathbb{R} : (u,y^\star) \in \mathcal{U} \times \mathcal{Y}^\star \},\\ \mathcal{M}_{j}&:= \left\{ m_{j}(\cdot,u,\theta):\mathcal{Y}\times\mathcal{Z} \to \mathbb{R} : (u,\theta) \in \mathcal{U}\times\Theta \right\},\quad j=1,\ldots,J, \end{align} are uniformly bounded, and satisfy the entropy growth condition, then so does $\mathcal{H}_{\ell b}$. Furthermore, for any $c>0$ we have:\todolt{Eventually might be nice to have a lower bound on the rate. Also, note that often $y^\star=f(y,z,u,\theta,\gamma)$...how to verify entropy growth condition for $\varphi$ in this case? Finally, what are necessary conditions for learning?} \begin{align} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) - \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s) \geq c \right) =O(r_{2}(n)), \end{align} where: \begin{align} r_{2}(n):=\max\left\{n^{-1/2}, n^{-1/2} \sup_{Q \in \mathcal{Q}_{n}} \sqrt{ \log N (\varepsilon/4,\Phi,||\cdot||_{Q,2}) + \sum_{j=1}^{J} \log N (\varepsilon/2,\mathcal{M}_{j},||\cdot||_{Q,2})}\right\}. \end{align}

proofSee Appendix (ref).

example[Simultaneous Discrete Choice (cont'd)] Consider again Example (ref) on simultaneous discrete choice. In this case we have: \begin{align} \Phi &:= \{ \mathbbm{1}\{\pi_{k}(\gamma(\,\cdot\,);\theta) \geq u \} : (u,\theta) \in \mathcal{U}\times\Theta \}, \end{align} with the moment conditions: \begin{align} \E \left[\left(\mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\} - \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\} - 0.5\right)\mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\} \right]\leq 0,\\ \E\left[\left(0.5 - \mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\}-\max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\right) \mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\} \right] \leq 0, \end{align} for $k=1,\ldots,K$, for all $z,z' \in \mathcal{Z}$ and all $y_{-k},y_{-k}' \in \mathcal{Y}^{K-1}$. Details on the verification of the entropy growth condition for both $\Phi$ and the class of moment functions associated with the moment conditions above are presented in Appendix (ref). Furthermore, under our assumptions for this example, the rate of convergence derived from Theorem (ref) is found to be $O(n^{-1/2})$.

example[Program Evaluation (cont'd)] Consider again Example (ref) on program evaluation. In this case we have: \begin{align} \Phi &:= \{ \mathbbm{1}\{g(\gamma(z)) \geq u \}(u_{1}-u_{0}) + u_{0} : (u_{0},u_{1},u,g) \in \mathcal{U}\times\mathcal{G} \}, \end{align} with the moment conditions: \begin{align} \E[\left(D - g(Z_{0},X)\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] &\leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X},\\ \E[\left(g(Z_{0},X) - D\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] &\leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X},\\ \E[\left(\mathbbm{1}\{U \leq g(z_{0},x)\} - g(z_{0},x)\right)\mathbbm{1}\{X=x\}] &\leq 0,\qquad \forall z_{0} \in \mathcal{Z}_{0}, \, x \in \mathcal{X}, \\ \E[\left(g(z_{0},x) - \mathbbm{1}\{U \leq g(z_{0},x)\}\right)\mathbbm{1}\{X=x\}] &\leq 0, \qquad\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\\ \E\left[t(z_{0},x)-\mathbbm{1}\{Z=z_{0},X=x\}\right] &\leq 0,\qquad\forall z_{0} \in \mathcal{Z}_{0},\, \forall x \in \mathcal{X}, \\ \E\left[\mathbbm{1}\{Z=z_{0},X=x\}-t(z_{0},x)\right] &\leq 0,\qquad\forall z_{0} \in \mathcal{Z}_{0},\, \forall x \in \mathcal{X}, \end{align} and: \begin{align} \E\left[U_{d}\left(\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{X=x\}t(z_{0},x) \right)\right] &\leq 0, \,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\},\\ \E\left[U_{d}\left(\mathbbm{1}\{X=x\}t(z_{0},x)-\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x) \right) \right] &\leq 0, \,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\}. \end{align} Details on the verification of the entropy growth condition for both $\Phi$ and the class of functions associated with the moment functions above are presented in Appendix (ref). Furthermore, under our assumptions for this example, the rate of convergence derived from Theorem (ref) is found to be $O(n^{-1/2})$.

definition[$\varepsilon-$maximin empirical welfare] Fix any $\varepsilon\geq 0$ and let $\widehat{I}_{\ell b}[\varphi](\gamma)$ denote the lower envelope from Theorem (ref) evaluated at the empirical measure for $(Y,Z)$. Then $d: \Psi_{n} \to \Gamma$ is a $\varepsilon-$maximin empirical (eME) rule if: \begin{align} \widehat{I}_{\ell b}[\varphi](d(\psi)) + \varepsilon \geq \sup_{\gamma \in \Gamma} \widehat{I}_{\ell b}[\varphi](\gamma). \end{align}

remarkNote that in general the “$\varepsilon$” is necessary (although it can be made arbitrarily small), owing to the fact that the supremum of $\widehat{I}_{\ell b}[\varphi](\cdot)$ may not be obtained.

definition[Empirical Rademacher Complexity] Let $\mathcal{F}$ be a class of measurable functions $f:\mathcal{Y}\times\mathcal{Z} \to \mathbb{R}$. The empirical Rademacher complexity of $\mathcal{F}$ is given as: \begin{align} ||\mathfrak{R}_{n}||(\mathcal{F}) := \sup_{f \in \mathcal{F}} \left|\frac{1}{n} \sum_{i=1}^{n} \xi_{i} \cdot f(y_{i},z_{i})\right|, \end{align} where $\xi_{i}$ are realizations of Rademacher random variables; that is, $\xi \in \{-1,1\}$ and $P(\xi_{i}=-1)=P(\xi=1)=1/2$.

remarkA technical point worth emphasizing is that, when seen as a function of the underlying product probability space, the empirical Rademacher complexity may not be a measurable function. We suppress these difficulties in the statement of our results, although we show in Appendix (ref) that the Rademacher complexity $||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})$ is universally measurable (with respect to the product Borel $\sigma-$algebra on $(\mathcal{Y}\times\mathcal{Z})^{n}$), which is sufficient for the purposes in this paper.

theoremSuppose that Assumptions (ref), (ref), (ref), and (ref) hold. Let $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}]\subset \mathbb{R}$ be a bounded, measurable function, and suppose that for each $\gamma \in \Gamma$, the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Let $\{(y_{i},z_{i})\}_{i=1}^{n}$ be i.i.d. from some distribution $P_{Y,Z}$ satisfying our assumptions and let $d:\Psi_{n} \to \Gamma$ be an eME decision rule for some $\varepsilon>0$. Furthermore, let $\overline{H}<\infty$ satisfy $|h| \leq \overline{H}$ for every $h \in \mathcal{H}_{\ell b}$, and let: \begin{align} c_{n}(\kappa)&= 4 ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) + \sqrt{\frac{72 \ln(2/(2-\kappa)) \overline{H}^{2}}{n}} + 5\varepsilon. \end{align} Then for any sample size $n$, and any $\kappa \in (0,1)$ we have: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) - \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s) \leq c_{n}(\kappa) \right) \geq \kappa. \end{align}

proofSee Appendix (ref).

theoremSuppose that Assumptions (ref), (ref), (ref), and (ref) hold. Also suppose that $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}]\subset \mathbb{R}$ is a bounded, measurable function, and that for each $\gamma \in \Gamma$, the random sets $\bm G^{-}(\cdot,\theta)$ and $\bm G^{\star}(\cdot,\theta,\gamma)$ are almost-surely non-empty for each $\theta \in \Theta^{*}$. Let $\overline{H}<\infty$ satisfy $|h| \leq \overline{H}$ for every $h \in \mathcal{H}_{\ell b}$, and suppose that $\{(y_{i},z_{i})\}_{i=1}^{n}$ is i.i.d. from some distribution $P_{Y,Z}$ satisfying our assumptions. Define: \begin{align*} \mathcal{H}_{n,\ell b}'(\delta)&:=\{ h_{\ell b}(\cdot,\cdot,\theta,\gamma,\lambda) - h_{\ell b}(\cdot,\cdot,\theta',\gamma',\lambda') : \theta,\theta' \in \Theta,\,\, \gamma,\gamma' \in \mathscr{G}_{n}(\delta),\lambda,\lambda' \in \{0,1\}^{J} \}, \end{align*} where $\mathcal{H}_{n,\ell b}'(\delta)$ has a uniform bound $\overline{H}_{n}'(\delta)\leq 2\overline{H} <\infty$. Furthermore, let $t_{j}:=\sqrt{c_{1} \log(c_{2} j )}$ with $c_{1}=5$ and $c_{2}= (3/(2(1-\kappa)))^{2/5}$, and let $\{\delta_{j}\}_{j=0}^{\infty}$ be a sequence decreasing to zero with $\delta_{0} > 2\overline{H}$. Choose some $\mathfrak{a} \in (1,\infty)$, let $\mathfrak{b}=2-1/\mathfrak{a}$, and let: \begin{align} T_{n}(\delta)&: = \begin{cases} 2||\mathfrak{R}_{n}||(\mathcal{H}_{n,\ell b}'(\mathfrak{b}\delta_{j})) + \frac{3t_{j}\overline{H}_{n}'(\mathfrak{b}\delta_{j})}{\sqrt{n}}, & if \delta \in (\delta_{j+1},\delta_{j}] for some $j \geq 0$\\ 0, & otherwise, \end{cases} \end{align} and: \begin{align} T_{n}^\flat(\sigma)&:=\sup_{\delta \geq \sigma} \frac{T_{n}(\delta)}{\delta},\%decreasing in \sigma, discontinuous at \delta_{j}? No, discontinuity prevented by "supremum". Note that supremum always obtained a a point of discontinuity of T_{n} T_{n}^\sharp(\eta)&:=\inf\left\{ \sigma>0 : T_{n}^\flat(\sigma)\leq \eta\right\}. \end{align} Finally, set $\delta^{*}> T_{n}^\sharp(1-1/\mathfrak{a})$. Then for any $\delta \geq \mathfrak{a} \delta^{*}$ we have: \begin{align*} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left(\mathscr{G}_{n}(\delta/\mathfrak{a}) \subseteq \mathscr{G}^{*}(\delta) \subseteq \mathscr{G}_{n}(\mathfrak{b}\delta) \right) \geq \kappa. \end{align*}

proofSee Appendix (ref).

lemmaSuppose that the assumptions of Theorem (ref) all hold. Define: \begin{align*} \mathcal{H}_{\ell b}'(\delta)&:=\{ h_{\ell b}(\cdot,\cdot,\theta,\gamma,\lambda) - h_{\ell b}(\cdot,\cdot,\theta',\gamma',\lambda') : \theta,\theta' \in \Theta,\,\, \gamma,\gamma' \in \mathscr{G}^{*}(\delta),\lambda,\lambda' \in \Lambda \}, \end{align*} where $\mathcal{H}_{\ell b}'(\delta)$ has a uniform bound $\overline{H}'(\delta)\leq 2\overline{H} < \infty$. Furthermore, let $t_{j}:=\sqrt{c_{1} \log(c_{2} j )}$ with $c_{1}=5$ and $c_{2}= (3/(2(1-\kappa)))^{2/5}$, and let $\{\delta_{j}\}_{j=0}^{\infty}$ be a sequence decreasing to zero with $\delta_{0} > 2\overline{H}$. Also, let: \begin{align} T(\delta)&: = \begin{cases} 2||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}'(\delta_{j})) + \frac{3t_{j}\overline{H}'(\delta_{j})}{\sqrt{n}}, & if \delta \in (\delta_{j+1},\delta_{j}],\\ 0, & otherwise, \end{cases} \end{align} and: \begin{align} T^\flat(\sigma)&:=\sup_{\delta \geq \sigma} \frac{T(\delta)}{\delta},\\ T^\sharp(\eta)&:=\inf\left\{ \sigma>0 : T^\flat(\sigma)\leq \eta\right\}. \end{align} Finally, suppose $\delta^{**}> T^\sharp(1-1/\mathfrak{a})$ for some $\mathfrak{a}\in (1,\infty)$. Then for any $\delta \geq \mathfrak{a} \delta^{**}$ we have: \begin{align*} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left(\mathscr{G}_{n}(\delta/\mathfrak{a}) \subseteq \mathscr{G}^{*}(\delta) \subseteq \mathscr{G}_{n}((2-1/\mathfrak{a})\delta) \right) \geq \kappa. \end{align*}

proofSee Appendix (ref).

enumerate[label=(\roman*)] • The policymaker takes $n$ i.i.d. draws of a Rademacher random variable $\xi$. • At the $j^{th}$ step (beginning at step $0$) the policymaker uses $\mathscr{E}_{n}(\gamma)$ to compute the Rademacher complexity $||\mathfrak{R}_{n}||(\mathcal{H}_{n,\ell b}'(\mathfrak{b}\delta_{j}))$ with the formula (ref). • The policymaker uses $\mathscr{E}_{n}(\gamma)$ to compute a uniform upper bound $\overline{H}_{n}(\delta_{j})$ for $\mathcal{H}_{n,\ell b}'(\delta_{j})$ (or she can simply use $2\overline{H}$). • The policymaker determines if there is any value $\delta \in (\delta_{j+1},\delta_{j}]$ such that $T_{n}(\delta_{j})/\delta \geq 1-1/\mathfrak{a}$. \begin{itemize} • If so, the policymaker stops and sets $\delta^{*}= \delta+\eta$, where $\eta>0$ and $\delta\in (\delta_{j+1},\delta_{j}]$ is equal to any value satisfying $T_{n}(\delta_{j})/\delta \leq 1-1/\mathfrak{a}$. • If not, the policymaker repeats steps (i) and (ii) for iteration $j+1$. \end{itemize}

corollarySuppose the assumptions of Theorem (ref) hold, and let $\delta^{*}$ be as in Theorem (ref). For any $\varepsilon>0$ let $\hat{\gamma}\in \Gamma$ be the policy selected by the eME decision rule. If $\delta \geq \delta^{*}\geq \varepsilon>0$, then: \begin{align*} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \mathscr{E}^{*}(\hat{\gamma})\leq \delta \right) \geq \kappa. \end{align*} That is $\hat{\gamma} \in \mathscr{G}^{*}(\delta)$ with high probability when $\delta \geq \delta^{*}\geq \varepsilon>0$.\note{It feels like there is some way to turn this into necessary conditions for learnability. In particular, it seems that the eME rule is always eventually in the $\delta-$minimal set. If $\delta^{*}$ tends to zero, does it suffice to focus on the eME rule?}

appendix\section{Preliminaries} \subsection{Preliminaries on Random Set Theory} This Appendix introduces some key elements of random set theory. Since measurability issues play a significant role in random set theory, we begin by providing the definition of an Effros-measurable multifunction, and show its connection with the definition of a random set. \begin{definition}[Effros-Measurability, Random Set] Let $(\Omega,\mathfrak{A},P)$ be a probability space, let $\mathcal{V}$ be a Polish space, and let $\mathcal{O}_{\mathcal{V}}$ denote the collection of all open sets on $\mathcal{V}$. A multifunction $\bm V: \Omega \to \mathfrak{F}_{\mathcal{V}}$ is called Effros-measurable if for every $A \in \mathcal{O}_{\mathcal{V}}$ we have $\bm V^{-}(A):=\{ \omega \in \Omega : \bm V(\omega) \cap A \neq \emptyset \} \in \mathfrak{A}$. An Effros-measurable closed-valued multifunction on a probability space $(\Omega,\mathfrak{A},P)$ is called a random closed set. \end{definition} From this definition, we see that a random closed set is an Effros-measurable closed multifunction which takes elements from the underlying probability space to the collection of closed sets on some Polish space $\mathcal{V}$. An Effros-measurable closed multifunction is also sometimes called weakly measurable.\footnote{See aliprantis2006infinite Ch. 18} When the underlying probability space $(\Omega,\mathfrak{A},P)$ is complete Effros-measurability is equivalent to both (i) $\bm V^{-}(B) \in \mathfrak{A}$ for all $B \in \mathfrak{B}(\mathcal{V})$ (Borel measurability) and (ii) $\bm V^{-}(F) \in \mathfrak{A}$ for all $F \in \mathfrak{F}_{\mathcal{V}}$ (strong measurability).\footnote{See molchanov2017theory Theorem 1.3.3, p.59. } Our main interest in the paper is in the case when $\mathcal{V}$ is a subset of finite-dimensional euclidean space, although the framework is more general. While Effros-measurability is the proper notion of measurability for many of the results, it can be difficult to verify. There are other conditions that are sufficient for Effros measurability, but we find one condition to be particularly helpful in the examples. Let $d$ denote the metric on a Polish space $\mathcal{V}$, and let $\bm V: \Omega \to \mathfrak{F}_{\mathcal{V}}$ be a multifunction. The distance to the set $\bm V(\omega)$ on $\mathcal{V}$ is given by: \begin{align*} d(v, \bm V(\omega)) := \inf\{ d(v,v') : v' \in \bm V(\omega) \}. \end{align*} By a result of himmelberg1975measurable, Effros measurability of the multifunction $\bm V$ is equivalent to measurability of $d(v, \bm V(\omega))$ (as a random variable from $\Omega$ to $[0,\infty]$) for each $v \in \mathcal{V}$. Throughout the paper it is also important to understand what it means for two random sets to be identically distributed, which is provided in the next definition. \begin{definition}[Identically Distributed Random Sets] Let $(\Omega,\mathfrak{A},P)$ be a probability space, let $\mathcal{V}$ be a Polish space. We say that two random sets $\bm V$ and $\bm V^{*}$ are identically distributed, denoted by $\bm V \sim \bm V^{*}$, if for every $A \in \mathcal{O}_{\mathcal{V}}$ we have $P(\omega : \bm V(\omega) \cap A \neq \emptyset) = P(\omega : \bm V^{*}(\omega) \cap A \neq \emptyset)$. \end{definition} Finally, an important concept in random set theory is that of a selection from a random set. Intuitively, a random set $\bm V$ can be understood as a collection of random variables $V$ satisfying $V(\omega) \in \bm V(\omega)$ $P-$a.s. Such random variables are called selections from the random set $\bm V$, which is made precise in the following definition. \begin{definition}[Selections, Conditional Selections] A random element $V: \Omega \to \mathcal{V}$ is called a (measurable) selection of $\bm V$ if $V(\omega) \in \bm V(\omega)$ for $P-$almost all $\omega \in \Omega$. The family of all measurable selections of a random set $\bm V$ will be denoted by $Sel(\bm V)$. \end{definition} Although it is suppressed in the notation, the family of selections $Sel(\cdot)$ depends both on the distribution of the random set $\bm V$, and on the underlying probability space. Indeed, two identically distributed random sets on the same probability space may have different families of selections.\footnote{See Example 1.4.2 in molchanov2017theory, p. 79.} However, the weak closed convex hulls of the family of selections from two random closed sets on the same probability space coincide. In addition, when the underlying probability space is non-atomic, it is not necessary to take convex hulls. See the discussion following Definition (ref) in the main text. \subsection{PAC Learnability} As described in the introduction, our definition of learnability is related to the definition of learnability prescribed in valiant1984theory. It will thus be useful to understand the concept of learnability from computational learning theory. We will omit technical details in the pursuit of clarity. In a supervised learning problem, the researcher is presumed to have an i.i.d. sample $\psi= ((y_{i},z_{i}))_{i=1}^{n}$ from the true measure $P_{Y,Z}$. The researcher is also assumed to have a class of functions $\mathcal{F}$ in mind, called the hypothesis space. The researcher's objective is to select a function $f: \mathcal{Z} \to \mathcal{Y}$, called a hypothesis (or a classifier or a predictor), from the hypothesis space $\mathcal{F}$ that can accurately predict values in $\mathcal{Y}$ given values in $\mathcal{Z}$. The performance of a given function $f \in \mathcal{F}$ is measured according to a loss function. That is, it is assumed the researcher has some function $L: \mathcal{Y} \times \mathcal{Y} \to \mathbb{R}$ such that $L(y,f(z))$ measures the loss incurred when a prediction $f(z)$ is made and the true value of the outcome is $y$. The problem of selecting a good hypothesis $f$ is then translated into the problem of choosing $f \in \mathcal{F}$ to minimize expected loss, or risk. A decision rule in this context is a measurable map $d: \Psi_{n} \to \mathcal{F}$ that selects a hypothesis from the hypothesis space; in learning theory, this decision rule is called an algorithm. So far the reader should note a resemblance to decision problems seen in statistics and econometrics. However, important differences between the fields arise when evaluating a given statistical decision rule. In particular, computer scientists are interested in rules that achieve close to the minimum possible risk with high probability in finite samples. To define this rigorously, let $\hat{f} \in \mathcal{F}$ be the hypothesis selected by some decision rule (or algorithm) $d: \Psi_{n} \to \mathcal{F}$. Since $\hat{f} \in \mathcal{F}$ depends on the observed sample, ex-ante it will be a random variable. Now fix any values $(c,\kappa) \in \mathbb{R}_{++}\times (0,1)$. Then $\hat{f}$ closely approximates the performance of the optimal decision rule in finite samples if: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \left| \inf_{f \in \mathcal{F}} \E [L(y,f(z))] - \E [L(y,\hat{f}(z))] \right| \leq c \right) \geq \kappa, \end{align} for a small value of $c \in \mathbb{R}_{+}$ and a large value of $\kappa \in (0,1)$ at sample size $n$. Here $\mathcal{P}_{Y,Z}$ is the collection of all Borel probability measures on $\mathcal{Y}\times\mathcal{Z}$, and thus the performance of a decision rule is uniform over all possible distributions $P_{Y,Z} \in \mathcal{P}_{Y,Z}$.\footnote{Note that taking the outer probability is necessary because the sampling uncertainty from the choice of $\hat{f}$ is not resolved by the inner expectation.} We can now introduce the notion of (agnostic) PAC learnability initially proposed by haussler1992decision. \begin{definition}[Agnostic PAC Learnability] A hypothesis class $\mathcal{F}$ is (agnostic) probably approximately correct (PAC) learnable with respect to the loss function $L$ if there exists a function $\zeta_{\mathcal{F}}: \mathbb{R}_{+}\times (0,1) \to \mathbb{N}$ such that, for any $(c,\kappa) \in \mathbb{R}_{++} \times (0,1) \to \mathbb{N}$, if $n \geq \zeta_{\mathcal{F}}(c,\kappa)$ then there is some decision procedure $d: \Psi_{n} \to \mathcal{F}$ such that $\hat{f}:=d(\psi)$ satisfies (ref). \end{definition} \begin{remark} This definition omits an important component of the original definition of PAC learnability found in the paper of valiant1984theory, which also requires that the algorithm (decision rule) can be processed in polynomial time (relative to the length of its input). For some this may be a serious omission, as the requirement that an algorithm can be efficiently processed is seen as a core component of learnability in computational learning theory.\footnote{This perspective is apparent in valiant2013probably.} \end{remark} In other words, a hypothesis space is (agnostic) PAC learnable if we can guarantee that (ref) holds for any choice of the pair $(c,\kappa) \in \mathbb{R}_{++} \times (0,1)$ for large enough $n$. Here $c$ is called the error tolerance parameter, and $\kappa$ is called the confidence parameter. The “agnostic” component of the definition refers to the fact that the hypothesis class $\mathcal{F}$ may or may not include the true labelling function $f^{*}:\mathcal{Z} \to \mathcal{Y}$; indeed, such a “true” labelling function may not even exist. One major advantage of the PAC framework---relative to other frequentist methods of evaluating decision rules---is its analytical tractability and amenability to analysis via concentration inequalities, and techniques from empirical process theory. Indeed, in the case when the decision rule $d:\Psi_{n}\to \mathcal{F}$ corresponds to the empirical risk minimization rule, it is well known that PAC learnability is implied by uniform convergence (over both $\mathcal{P}_{Y,Z}$ and $\mathcal{F}$) of the empirical risk to the population risk.\footnote{See, for example, shalev2014understanding Lemma 4.2.} In specific learning problems this uniform convergence is equivalent to learnability (see the discussion in alon1997scale and shalev2010learnability). This means well-developed tools in empirical process theory can be used to establish the learnability of a particular class of functions. Intuitively, whether or not a particular class of functions $\mathcal{F}$ is learnable depends on the “complexity” of the function class. There are various ways to measure the complexity of $\mathcal{F}$, some of which are encountered in the current paper. In general, classes that exhibit less complexity are easier to learn than classes that exhibit more complexity, and if a class of functions is too complex, it may not be learnable.\todolt{Finish the section: “On the Relation Between PAC Learnability and Decision Theory” Discuss stochastic dominance.} \begin{comment} \subsection{Comparison of PAC Learnability and Minimax Regret} In this subsection we consider a brief comparison between the PAC criterion and the minimax regret criterion in the framework of wald1950statistical. Let us consider a more general loss function than the above, given by $L: \mathcal{X} \times \mathcal{F} \to \mathbb{R}$, where $\mathcal{X} = \mathcal{Y} \times \mathcal{Z}$. The expected loss when using the function $f \in \mathcal{F}$ when the true probability measure on $\mathcal{X}$ is $P$ is given by: \begin{align*} \int L(X,f)\,dP. \end{align*} Now suppose that a decision $d(\psi)=\hat{f}$ is made on the basis of i.i.d. data $\psi =\{(y_{i},z_{i})\}_{i=1}^{n}$. Then the regret associated with decision $d(\psi)$ when the true probability measure on $\mathcal{X}$ is $P$ is given by: \begin{align*} R_{n}(d,P):= \E_{P^{\otimes n}} \left[ \int L(X,d(\psi))\,dP - \inf_{h \in \mathcal{H}} \int L(X,f)\,dP \right]. \end{align*} The minimax regret criterion is: \begin{align*} \inf_{d \in \mathcal{D}} \sup_{P \in \mathscr{P}} R_{n}(d,P), \end{align*} where $\mathcal{D}$ is the collection of possible decision rules and $\mathscr{P}$ is the set of all probability measures on $\mathcal{X}$. The following proposition then follows from two simple applications of Markov's inequality: \begin{proposition} Suppose that the environment is as above, and that $L$ is uniformly bounded by some value $\overline{L}$. Then $\mathcal{F}$ is (agnostic) PAC-learnable if and only if there exists a decision rule $d \in \mathcal{D}$ such that: \begin{align*} \sup_{P \in \mathscr{P}} R_{n}(d,P)\to 0, \end{align*} as $n\to \infty$. \end{proposition} This proposition shows that any decision rule obtaining zero asymptotic regret uniformly over $\mathscr{P}$ also satisfies the definition of (agnostic) PAC learnability. Furthermore, any decision rule that satisfies the definition of (agnostic) PAC learnability obtains an asymptotic regret of zero. While this comparison is basic, it demonstrates that there are likely even deeper connections between the PAC learning model of valiant1984theory and the framework for making statistical decisions proposed by wald1950statistical. We believe this to be an important avenue of future research. \end{comment} \section{Proofs} \begin{remark}[Common Notation] To avoid repetition we introduce some common notation for use in the proofs of Theorem (ref), Theorem (ref), Lemma (ref) and Lemma (ref). In particular, for any $\theta \in \Theta$ and $\gamma \in \Gamma$ let $\lambda^{*}(\theta,\gamma)$, and $\hat{\lambda}(\theta,\gamma)$ satisfy: \begin{align} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda^{*}(\theta,\gamma)) = \max_{\lambda \in \Lambda} Ph_{\ell b}(\cdot,\theta,\gamma,\lambda),\\ \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\hat{\lambda}(\theta,\gamma)) = \max_{\lambda \in \Lambda} \P_{n}h_{\ell b}(\cdot,\theta,\gamma,\lambda). \end{align} Now for any $\gamma \in \Gamma$, let $\theta^{*}$ and $\hat{\theta}$ satisfy: \begin{align} Ph_{\ell b}(\cdot, \theta^{*}(\gamma),\gamma,\lambda^{*}(\theta^{*}(\gamma),\gamma)) \leq \inf_{\theta \in \Theta} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda^{*}(\theta,\gamma))+\varepsilon,\\ \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma),\gamma,\hat{\lambda}(\hat{\theta}(\gamma),\gamma)) \leq \inf_{\theta \in \Theta} \P_{n}h_{\ell b}(\cdot,\theta,\gamma,\hat{\lambda}(\theta,\gamma)) +\varepsilon, \end{align} Finally, let $\gamma^{*}$ and $\hat{\gamma}$ satisfy: \begin{align} Ph_{\ell b}(\cdot, \theta^{*}(\gamma^{*}),\gamma^{*},\lambda^{*}(\theta^{*}(\gamma^{*}),\gamma^{*})) \geq \sup_{\gamma \in \Gamma} Ph_{\ell b}(\cdot, \theta^{*}(\gamma),\gamma,\lambda^{*}(\theta^{*}(\gamma),\gamma)) -\varepsilon,\\ \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\hat{\theta}(\hat{\gamma}),\hat{\gamma})) \geq \sup_{\gamma \in \Gamma} \P_{n}h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\hat{\lambda}(\hat{\theta}(\gamma),\gamma)) -\varepsilon. \end{align} \todoltinline{Whether there exists a Borel measurable $d:\Psi_{n} \mapsto \Gamma$ such that $d(\psi) = \hat{\gamma}$ appears to be unresolved. The issue is that $\hat{\theta}:\Gamma\to \Theta$ may only be analytically measurable (e.g. see the stinchcombe and white paper). Even if $h_{\ell b}$ is a Borel measurable function of $\theta$ (which it is generally not) the composition of $h_{\ell b}$ and $\hat{\theta}(\cdot)$ may not be analytically measurable (since it is the composition of a Borel function and analytic function). This might be resolved by assuming (i) the supremum over $\Theta$ is obtained, (ii) $\Gamma$ has at most countably many points, and (iii) allowing $d$ to be analytically measurable: then the function $\P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma),\gamma,\hat{\lambda}(\hat{\theta}(\gamma),\gamma))$ will be lower semi-analytic, and the countable maximum of lower semi-analytic functions is lower semi-analytic.} With these definitions, it is straightforward to show: \begin{align} \sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda) &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) +3\varepsilon,\\ \sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) +3\varepsilon. \end{align} Furthermore, we can always choose $\gamma^{*}$ and $\hat{\gamma}$ to satisfy:\todo{Verify} \begin{align} \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) \leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda),\\ \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) \leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda). \end{align} \end{remark} \begin{remark}[Measurability] We will not comment on measurability issues in every proof, and instead we refer readers to the discussion Appendix (ref) (namely, Proposition (ref) and Corollary (ref)). There it is shown that certain quantities in this paper that are not typically (Borel) measurable are still universally measurable. This allows us to use outer measures to resolve measurability issues, although this is left implicit in many of the proofs. However, we also note that all measurability issues can also be resolved by restricting $\Theta$ and $\Gamma$ to have at most countably many points. \end{remark} \subsection{Proofs of the Main Results} \begin{proof}[Proof of Proposition (ref)] Recall by assumption we have $\gamma\mapsto \inf_{s \in \mathcal{S}} I[\varphi](\gamma,s)$ is universally measurable. By (Borel) measurability of each decision rule $d:\Psi_{n} \to \Gamma$ (and thus universal measurability), and the fact that universally measurable functions are closed under composition, this implies that the map $\psi\mapsto \inf_{s \in \mathcal{S}} I[\varphi](d(\psi),s)$ is universally measurable. The result then follows from Lemma (ref) after noting that $\sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s)$ is a constant for each $P_{Y,Z} \in \mathcal{P}_{Y,Z}$ (and thus plays the role of “$c(P)$” from Lemma (ref)). \end{proof} \begin{proof}[Proof of Lemma (ref)] Fix a value of $\delta>0$ satisfying Assumption (ref). We will focus on proving (ref) holds, as the proof of (ref) is similar. By iterated application of Lemma (ref), (ref) can be rewritten as: \begin{align*} \inf_{\theta^{*} \in \Theta^{*}} \int \inf_{u \in \bm G^{-}(y,z,\theta^{*})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \, dP_{Y,Z} - \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \, dP_{Y,Z} \leq C_{2} d(\theta,\Theta^{*}). \end{align*} Note that this inequality is trivially satisfied for any $C_{2} \geq 0$ when $\theta \in \Theta^{*}$. Thus, it suffices to focus on the case when $\theta \in \Theta_{\delta}^{*}\setminus \Theta^{*}$. Furthermore, for this latter case it suffices to find a value of $C_{2}\geq 0$ satisfying: \begin{align*} \int \left( \inf_{u \in \bm G^{-}(y,z,\theta_{1})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{1},\gamma)} \varphi(v) - \inf_{u \in \bm G^{-}(y,z,\theta_{2})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{2},\gamma)} \varphi(v) \right) \, dP_{Y,Z} \leq C_{2} d(\theta_{1},\theta_{2}), \end{align*} for any $\theta_{1},\theta_{2} \in \Theta_{\delta}^{*}$. However, to find $C_{2}$ in the previous display, it suffices to find $C_{2}$ such that: \begin{align} \inf_{u \in \bm G^{-}(y,z,\theta_{1})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{1},\gamma)} \varphi(v) - \inf_{u \in \bm G^{-}(y,z,\theta_{2})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{2},\gamma)} \varphi(v) \leq C_{2} d(\theta_{1},\theta_{2}), \end{align} $(y,z)-a.s.$ Fix any $\varepsilon>0$ and let $(y,z) \in \mathcal{Y}\times\mathcal{Z}$ be any pair (outside the null sets in (ref) and (ref)). For any $\theta_{1},\theta_{2} \in \Theta_{\delta}^{*}$ let $u_{1}^{*}$, $u_{2}^{*}$, $y_{1}^{*}$ and $y_{2}^{*}$ satisfy: \begin{align*} u_{1}^{*}\in \bm G^{-}(y,z,\theta_{1}),&&y_{1}^{*}\in \bm G^{\star}(y,z,u_{1}^{*},\theta_{1},\gamma),\\ u_{2}^{*}\in \bm G^{-}(y,z,\theta_{2}),&&y_{2}^{*}\in \bm G^{\star}(y,z,u_{2}^{*},\theta_{2},\gamma), \end{align*} and: \begin{align*} \varphi(y,z,u_{1}^{*},y_{1}^{*}) \leq \inf_{u \in \bm G^{-}(y,z,\theta_{1})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{1},\gamma)} \varphi(v)+\varepsilon,\\ \varphi(y,z,u_{2}^{*},y_{2}^{*}) \leq \inf_{u \in \bm G^{-}(y,z,\theta_{2})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{2},\gamma)} \varphi(v)+\varepsilon. \end{align*} For simplicity we will denote $v_{1}^{*}:=(y,z,u_{1}^{*},y_{1}^{*})$ and $v_{2}^{*}:=(y,z,u_{2}^{*},y_{2}^{*})$. Now, by Proposition 3C.1 in dontchev2009implicit, condition (ref) implies: \begin{align*} d_{H}(\bm G^{\star}(y,z,u,\theta_{1},\gamma),\bm G^{\star}(y,z,u,\theta_{2},\gamma)) \leq \ell_{2} d(\theta_{1},\theta_{2}), &&\forall \theta_{1},\theta_{2} \in \Theta_{\delta}^{*} \end{align*} $(y,z,u)-$a.s.\footnote{Recall the Hausdorff distance between two non-empty subsets $A$ and $B$ of a metric space $(\mathcal{X},d)$ is given by: \begin{align*} d_{H}(A,B):= \max\left\{\sup_{a\in A}\inf_{b \in B} d(a,b), \sup_{b\in B}\inf_{a \in A} d(a,b)\right\}. \end{align*} } Thus, since $y_{2}^{*}\in \bm G^{\star}(y,z,u,\theta_{2},\gamma)$ by assumption, there exists $y_{1} \in \bm G^{\star}(y,z,u,\theta_{1},\gamma)$ such that $d(y_{1},y_{2}^{*}) \leq \ell_{2} d(\theta_{1},\theta_{2})$. Furthermore, by Proposition 3C.1 in dontchev2009implicit, condition (ref) implies: \begin{align*} d_{H}(\bm G^{-}(y,z,\theta_{1}),\bm G^{-}(y,z,\theta_{2})) \leq \ell_{1} d(\theta_{1},\theta_{2}), &&\forall \theta_{1},\theta_{2} \in \Theta_{\delta}^*. \end{align*} Thus, since $u_{2}^{*}\in \bm G^{-}(y,z,\theta_{2})$ by assumption, there exists $u_{1} \in \bm G^{-}(y,z,\theta_{1})$ such that $d(u_{1},u_{2}^{*}) \leq \ell_{1} d(\theta_{1},\theta_{2})$. Now let us define $v_{1}:=(y,z,u_{1},y_{1})$. Then we have: \begin{align*} &\inf_{u \in \bm G^{-}(y,z,\theta_{1})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{1},\gamma)} \varphi(v) - \inf_{u \in \bm G^{-}(y,z,\theta_{2})} \inf_{y^\star \in \bm G^{\star}(y,z,u,\theta_{2},\gamma)}\varphi(v)\\ &\qquad\qquad\leq \varphi(v_{1}^{*}) - \varphi(v_{2}^{*}) +\varepsilon \\ &\qquad\qquad\leq \varphi(v_{1}) - \varphi(v_{2}^{*}) +2\varepsilon\\ &\qquad\qquad\leq L_{\varphi} d((y_{1},u_{1}),(u_{2}^{*},y_{2}^{*})) +2\varepsilon\\ &\qquad\qquad\leq L_{\varphi} \max\{d(y_{1},y_{2}^{*}),d(u_{1},u_{2}^{*})\} +2\varepsilon\\ &\qquad\qquad\leq L_{\varphi} \max\{\ell_{1},\ell_{2}\}d(\theta_{1},\theta_{2}) +2\varepsilon, \end{align*} which holds for all $\theta_{1}, \theta_{2} \in \Theta_{\delta}^{*}$.\footnote{Here we take the product metric as the sup metric; that is, if $(\mathcal{X},d)$ and $(\mathcal{X}',d')$ are two metric spaces, then the product metric $d_{\infty}$ on $\mathcal{X} \times \mathcal{X}'$ is defined as $d_{\infty}((x_{1},x_{1}'),(x_{2},x_{2}')) = \max\left\{d(x_{1},x_{2}),d'(x_{1}',x_{2}') \right\}$.} Since $\varepsilon>0$ is arbitrary, we conclude that $C_{2}$ in (ref) can be taken equal to $L_{\varphi} \max\{\ell_{1},\ell_{2}\}$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem (ref)] We will show the lower bound, as the proof for the upper bound is symmetric. We will prove the following sequence of equalities and inequalities: \begin{align} I[\varphi](\gamma)&:=\int \varphi(v) \, dP_{V_{\gamma}} \nonumber\\ &\geq \inf_{\theta \in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}\\ &=\inf_{\theta \in \Theta} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \Bigg(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg)\\ &=\inf_{\theta \in \Theta} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \Bigg(\int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)}\varphi(v) \, dP_{Y,Z,U}\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg)\\ &=\inf_{\theta \in \Theta} \int \Bigg(\inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)}\varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg) \, dP_{Y,Z}\\ &=\inf_{\theta \in \Theta} \max_{\lambda_{j} \in \{0,1\}} \int \Bigg(\inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)}\varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg) \, dP_{Y,Z}. \end{align} Inequality (ref) is obvious. Equality (ref) follows from Lemma (ref). Equalities (ref) and (ref) follow from Lemma (ref). Finally, (ref) follows from Lemma (ref). \end{proof} \begin{proof}[Proof of Theorem (ref)] Let $\mathcal{F}$ be a class of real-valued functions, and let $\psi = ((y_{i},z_{i}))_{i=1}^{n}$ denote a particular sample vector taking values in the sample space $\Psi_{n}$. For any $f ,f' \in \mathcal{F}$ define the norm: \begin{align*} ||f-f'||_{\psi,2} := \left( \sum_{i=1}^{n} \left( f(y_{i},z_{i}) - f'(y_{i},z_{i}) \right)^{2} \right)^{1/2} . \end{align*} Recall that: \begin{align*} h_{\ell b}(y,z,\theta,\gamma,\lambda):= \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg). \end{align*} For notational simplicity we will define: \begin{align*} \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \frac{1}{n}\sum_{i=1}^{n} \inf_{u_{i} \in \bm G^{-}(y_{i},z_{i},\theta)} \inf_{y_{i}^{\star} \in \bm G^{\star}(y_{i},z_{i},u_{i},\theta,\gamma)} \Bigg( \varphi(v_{i}) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y_{i},z_{i},u_{i},\theta)\Bigg),\\ P h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\,dP_{Y,Z}. \end{align*} For any decision rule $d:\Psi_{n} \to \Gamma$ and any $P_{Y,Z} \in \mathcal{P}_{Y,Z}$, we have by Markov's inequality and Theorem (ref):\footnote{To be mindful of measurability issues, we can use the outer-measures version of Markov's inequality given in Lemma 6.10 in kosorok2008introduction.} \begin{align} P_{Y,Z}^{\otimes n} \left( \sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) - \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s) \geq c \right) &\leq \frac{1}{c} \E \left(\sup_{\gamma \in \Gamma}\inf_{s \in \mathcal{S}} I[\varphi](\gamma,s) - \inf_{s\in \mathcal{S}} I[\varphi](d(\psi),s)\right)\nonumber\\ &=\frac{1}{c} \E \left(\sup_{\gamma \in \Gamma} I_{\ell b}[\varphi](\gamma) - I_{\ell b}[\varphi](d(\psi))\right). \end{align} Now note by symmetrization (e.g. van1996weak Lemma 2.3.1) we have: \begin{align} &\sup_{\gamma \in \Gamma}\sup_{\theta \in \Theta} \max_{\lambda \in \Lambda}\bigg| \E \left(\P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) - Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) \right)\bigg|\nonumber\\ &\qquad\qquad\qquad\qquad\leq \E \sup_{\gamma \in \Gamma}\sup_{\theta \in \Theta} \max_{\lambda \in \Lambda}\bigg| \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) - Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) \bigg| \leq 2\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}), \end{align} where the final outer expectation is a joint expectation that is also taken over the Rademacher random variables. Now let $\lambda^{*}(\theta,\gamma)$, $\hat{\lambda}(\theta,\gamma)$, $\theta^{*}(\gamma)$, $\hat{\theta}(\gamma)$, $\gamma^{*}$ and $\hat{\gamma}$ be as in Remark (ref), and set $d(\psi)=\hat{\gamma}$.\todolt{It seems this may not always be possible, since there may not exist a Borel measurable $d$ satisfying this requirement.} Then we have: \begin{align*} &\E I_{\ell b}[\varphi](d(\psi))\\ &=\E \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} Ph_{\ell b}(\cdot, \theta,d(\psi),\lambda), &&(by Theorem (ref)),\\ &=\E \inf_{\theta \in \Theta} Ph_{\ell b}(\cdot, \theta,d(\psi),\lambda^{*}(\theta,d(\psi))), &&(since $\lambda^{*}$ is optimal at $P$ for any $(\theta,\gamma)$),\\ &=\E Ph_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\lambda^{*}(\theta^{*},d(\psi)))-\varepsilon, &&(since $\theta^{*}$ is $\varepsilon-$optimal at $(P,\lambda^{*})$ for any $\gamma$),\\ &\geq \E Ph_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\hat{\lambda}(\theta^{*}(d(\psi)),d(\psi)))-\varepsilon, &&(since $\lambda^{*}$ was optimal at $P$ for any $(\theta,\gamma)$),\\ &\geq \E \P_{n}h_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\hat{\lambda}(\theta^{*}(d(\psi)),d(\psi))) -2\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})-\varepsilon, &&\text{(by (ref)),}\\ &\geq \E \P_{n}h_{\ell b}(\cdot, \hat{\theta}(d(\psi)),d(\psi),\hat{\lambda}(\hat{\theta}(d(\psi)),d(\psi))) -2\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})-2\varepsilon, &&\text{(since $\hat{\theta}$ is $\varepsilon$-optimal at $(\P_{n},\hat{\lambda})$ for any $\gamma$),}\\ &\geq \E \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\hat{\lambda}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -2\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})-3\varepsilon, &&\text{(since $d(\psi)$ was $\varepsilon$-optimal at $(\P_{n},\hat{\lambda},\hat{\theta})$),}\\ &\geq \E \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -2\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})-3\varepsilon, &&\text{(since $\hat{\lambda}$ was optimal at $\P_{n}$ for any $(\theta,\gamma)$),}\\ &\geq \E Ph_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 3\varepsilon, &&\text{(by (ref)),}\\ &\geq \E Ph_{\ell b}(\cdot, \theta^{*}(\gamma^{*}),\gamma^{*},\lambda^{*}(\theta^{*}(\gamma^{*}),\gamma^{*})) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 4\varepsilon, &&\text{(since $\theta^{*}$ is $\varepsilon$-optimal at $(P,\lambda^{*})$ for any $\gamma$),}\\ &\geq \E \sup_{\gamma \in \Gamma} Ph_{\ell b}(\cdot, \theta^{*}(\gamma),\gamma,\lambda^{*}(\theta^{*}(\gamma),\gamma)) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 5\varepsilon, &&\text{(since $\gamma^{*}$ was $\varepsilon$-optimal at $(P,\lambda^{*},\theta^{*})$),}\\ &\geq \E \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda^{*}(\theta,\gamma)) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 5\varepsilon, &&\text{(since $\theta^{*}$ was $\varepsilon$-optimal at $(P,\lambda^{*})$ for any $\gamma$),}\\ &\geq \E \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 5\varepsilon, &&\text{(since $\lambda^{*}$ was optimal at $P$ for any $(\theta,\gamma)$),}\\ &\geq \E \sup_{\gamma \in \Gamma} I_{\ell b}[\varphi](\gamma) -4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) - 5\varepsilon, &&\text{(by Theorem (ref)).} \end{align*} Since $\varepsilon>0$ can be taken arbitrarily small, we conclude that: \begin{align} \E \left(\sup_{\gamma \in \Gamma}I[\varphi](\gamma) -I[\varphi](d(\psi))\right) \leq 4\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}). \end{align} It thus suffices to bound the Rademacher complexity, given by: \begin{align*} &\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) \\ &= \E \left( \sup_{\gamma \in \Gamma} \sup_{\theta \in \Theta}\max_{\lambda \in \Lambda} \left| \frac{1}{n}\sum_{i=1}^{n} \xi_{i} \left(\inf_{u_{i} \in \bm G^{-}(y_{i},z_{i},\theta)} \Bigg( \inf_{y_{i}^{\star} \in \bm G^{\star}(y_{i},z_{i},u_{i},\theta,\gamma)} \varphi(v_{i}) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y_{i},z_{i},u_{i},\theta)\Bigg)\right)\right|\right). \end{align*} If $\mathcal{H}_{\ell b}$ is not closed under symmetry, then redefine it as $\mathcal{H}_{\ell b}\cup (- \mathcal{H}_{\ell b})$; for our purposes this is without loss of generality, since this operation can only increase the value of $\E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b})$. We then have from Lemma (ref) that for any $\varepsilon>0$: \begin{align} \E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) \leq \frac{2\varepsilon}{\sqrt{n}} + 2\text{Diam}_{\psi,2}(\mathcal{H}_{\ell b}) \sqrt{ \frac{\log N(\varepsilon,\mathcal{H}_{\ell b},||\cdot||_{\psi,2})}{n}}. \end{align} Since the class of functions $\mathcal{H}_{\ell b}$ is uniformly bounded, we have $\text{Diam}_{\psi,2}(\mathcal{H}_{\ell b})<\infty$. It remains to bound the metric entropy. To do so, we will define: \begin{align} &\mathcal{H}_{I}:=\Bigg\{ h(\cdot,u,\theta,\gamma,\lambda): \mathcal{Y}\times \mathcal{Z} \to \mathbb{R} : \nonumber\\ &\qquad\qquad\qquad\qquad h(y,z,u,\theta,\gamma)= \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta),\nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(u,\theta,\gamma,\lambda) \in \mathcal{U} \times \Theta \times \Gamma \times \Lambda\Bigg\},\\ \nonumber\\ &\mathcal{H}_{II}:=\left\{ h(\cdot,u,\theta,\gamma): \mathcal{Y}\times \mathcal{Z} \to \mathbb{R} : h(y,z,u,\theta,\gamma)=\inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v), \quad (u,\theta,\gamma) \in \mathcal{U} \times \Theta \times \Gamma \right\},\\ \nonumber\\ &\mathcal{H}_{III}:=\left\{ h(\cdot,u,y^{\star}): \mathcal{Y}\times \mathcal{Z} \to \mathbb{R} : h(y,z,u,y^\star)=\varphi(y,z,u,y^\star), \quad (u,y^\star) \in \mathcal{U} \times \mathcal{Y}^{\star} \right\},\\ \nonumber\\ &\mathcal{H}_{IV}:=\left\{ h(\cdot,u,\theta,\lambda): \mathcal{Y}\times \mathcal{Z} \to \mathbb{R} : h(y,z,u,\theta)=\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta), \,\, (u,\theta,\lambda) \in \mathcal{U} \times \Theta \times \Lambda \right\}. \end{align} By Lemma (ref), we have: \begin{align*} N (\varepsilon,\mathcal{H}_{\ell b},||\cdot||_{\psi,2})\leq N (\varepsilon/2,\mathcal{H}_{I},||\cdot||_{\psi,2}). \end{align*} By Lemma (ref) we also have: \begin{align*} N (\varepsilon/2,\mathcal{H}_{I},||\cdot||_{\psi,2}) \leq N (\varepsilon/2,\mathcal{H}_{II},||\cdot||_{\psi,2}) N (\varepsilon/2,\mathcal{H}_{IV},||\cdot||_{\psi,2}). \end{align*} Applying Lemma (ref) again we have: \begin{align*} N (\varepsilon/2,\mathcal{H}_{II},||\cdot||_{\psi,2}) \leq N (\varepsilon/4,\mathcal{H}_{III},||\cdot||_{\psi,2}). \end{align*} Finally, from iterated application of Lemma (ref): \begin{align*} N (\varepsilon/2,\mathcal{H}_{IV},||\cdot||_{\psi,2}) \leq \prod_{j=1}^{J} N (\varepsilon/(2J),\mathcal{M}_{j},||\cdot||_{\psi,2}), \end{align*} We conclude that: \begin{align*} \log N (\varepsilon,\mathcal{H}_{\ell b},||\cdot||_{\psi,2})&\leq \log N (\varepsilon/4,\mathcal{H}_{III},||\cdot||_{\psi,2}) + \sum_{j=1}^{J} \log N (\varepsilon/(2J),\mathcal{M}_{j},||\cdot||_{\psi,2})\\ &\leq \sup_{Q \in \mathcal{Q}_{n}} \log N (\varepsilon/4,\mathcal{H}_{III},||\cdot||_{Q,2}) + \sum_{j=1}^{J} \sup_{Q \in \mathcal{Q}_{n}} \log N (\varepsilon/(2J),\mathcal{M}_{j},||\cdot||_{Q,2}), \end{align*} with the supremum taken over all discrete probability measures $\mathcal{Q}_{n}$ on $\mathcal{X}$ with atoms that have probabilities that are integer multiples of $1/n$. Since by assumption $\mathcal{H}_{III}$ and $\mathcal{M}_{j}$ satisfy the entropy growth condition, the right side of the previous display is of order $o(n)$. Combining this with (ref), we see that for any $(c,\kappa)$ pair, there exists some $n$ such that $4 \E ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) \leq c(1-\kappa)$. Combining this with (ref) and (ref), the proof is complete. \end{proof} \begin{proof}[Proof of Theorem (ref)] Recall that: \begin{align*} h_{\ell b}(y,z,\theta,\gamma,\lambda):= \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg). \end{align*} For notational simplicity we will define: \begin{align*} \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \frac{1}{n}\sum_{i=1}^{n} \inf_{u_{i} \in \bm G^{-}(y_{i},z_{i},\theta)} \inf_{y_{i}^{\star} \in \bm G^{\star}(y_{i},z_{i},u_{i},\theta,\gamma)} \Bigg( \varphi(v_{i}) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y_{i},z_{i},u_{i},\theta)\Bigg),\\ P h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\,dP_{Y,Z}. \end{align*} We claim that it suffices to set $c_{n}(\kappa)=2\tilde{c}_{n}(\psi,\kappa) + 5\varepsilon$, where $\tilde{c}_{n}(\psi,\kappa)$ satisfies: \begin{align} \sup_{\gamma \in \Gamma}\sup_{\theta \in \Theta} \max_{\lambda \in \Lambda} \bigg| \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) - Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) \bigg| \leq \tilde{c}_{n}(\psi,\kappa), \end{align} with probability at least $\kappa/2$. Let $\lambda^{*}(\theta,\gamma)$, $\hat{\lambda}(\theta,\gamma)$, $\theta^{*}(\gamma)$, $\hat{\theta}(\gamma)$, $\gamma^{*}$ and $\hat{\gamma}$ be as in Remark (ref) and set $d(\psi)=\hat{\gamma}$.\todolt{It seems this may not always be possible, since there may not exist a Borel measurable $d$ satisfying this requirement.} Then we have: \begin{align*} &I_{\ell b}[\varphi](d(\psi))\\ &=\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} Ph_{\ell b}(\cdot, \theta,d(\psi),\lambda), &&\text{(by Theorem (ref)),}\\ &=\inf_{\theta \in \Theta} Ph_{\ell b}(\cdot, \theta,d(\psi),\lambda^{*}(\theta,d(\psi))), &&\text{(since $\lambda^{*}$ is optimal at $P$ for any $(\theta,\gamma)$),}\\ &=Ph_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\lambda^{*}(\theta^{*},d(\psi)))-\varepsilon, &&\text{(since $\theta^{*}$ is $\varepsilon-$optimal at $(P,\lambda^{*})$ for any $\gamma$),}\\ &\geq Ph_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\hat{\lambda}(\theta^{*}(d(\psi)),d(\psi)))-\varepsilon, &&\text{(since $\lambda^{*}$ was optimal at $P$ for any $(\theta,\gamma)$),}\\ &\geq_{(\kappa/2)} \P_{n}h_{\ell b}(\cdot, \theta^{*}(d(\psi)),d(\psi),\hat{\lambda}(\theta^{*}(d(\psi)),d(\psi))) -\tilde{c}_{n}(\psi,\kappa)-\varepsilon, &&\text{(by (ref)),}\\ &\geq \P_{n}h_{\ell b}(\cdot, \hat{\theta}(d(\psi)),d(\psi),\hat{\lambda}(\hat{\theta}(d(\psi)),d(\psi))) -\tilde{c}_{n}(\psi,\kappa)-2\varepsilon, &&\text{(since $\hat{\theta}$ is $\varepsilon$-optimal at $(\P_{n},\hat{\lambda})$ for any $\gamma$),}\\ &\geq \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\hat{\lambda}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -\tilde{c}_{n}(\psi,\kappa)-3\varepsilon, &&\text{(since $d(\psi)$ was $\varepsilon$-optimal at $(\P_{n},\hat{\lambda},\hat{\theta})$),}\\ &\geq \P_{n}h_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -\tilde{c}_{n}(\psi,\kappa)-3\varepsilon, &&\text{(since $\hat{\lambda}$ was optimal at $\P_{n}$ for any $(\theta,\gamma)$),}\\ &\geq_{(\kappa/2)} Ph_{\ell b}(\cdot, \hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) -2\tilde{c}_{n}(\psi,\kappa) - 3\varepsilon, &&\text{(by (ref)),}\\ &\geq Ph_{\ell b}(\cdot, \theta^{*}(\gamma^{*}),\gamma^{*},\lambda^{*}(\theta^{*}(\gamma^{*}),\gamma^{*})) -2\tilde{c}_{n}(\psi,\kappa) - 4\varepsilon, &&\text{(since $\theta^{*}$ is $\varepsilon$-optimal at $(P,\lambda^{*})$ for any $\gamma$),}\\ &\geq\sup_{\gamma \in \Gamma} Ph_{\ell b}(\cdot, \theta^{*}(\gamma),\gamma,\lambda^{*}(\theta^{*}(\gamma),\gamma)) -2\tilde{c}_{n}(\psi,\kappa) - 5\varepsilon, &&\text{(since $\gamma^{*}$ was $\varepsilon$-optimal at $(P,\lambda^{*},\theta^{*})$),}\\ &\geq \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda^{*}(\theta,\gamma)) -2\tilde{c}_{n}(\psi,\kappa) - 5\varepsilon, &&\text{(since $\theta^{*}$ was $\varepsilon$-optimal at $(P,\lambda^{*})$ for any $\gamma$),}\\ &\geq \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) -2\tilde{c}_{n}(\psi,\kappa) - 5\varepsilon, &&\text{(since $\lambda^{*}$ was optimal at $P$ for any $(\theta,\gamma)$),}\\ &\geq \sup_{\gamma \in \Gamma} I_{\ell b}[\varphi](\gamma) -2\tilde{c}_{n}(\psi,\kappa) - 5\varepsilon, &&\text{(by Theorem (ref)).} \end{align*} where each inequality “$\geq_{(\kappa/2)}$” holds with probability at least $\kappa/2$. Note that this shows: \begin{align*} \sup_{\gamma \in \Gamma} I_{\ell b}[\varphi](\gamma)- I_{\ell b}[\varphi](\hat{\gamma}) \leq 2\tilde{c}_{n}(\psi,\kappa) +5\varepsilon, \end{align*} with probability at least $\kappa$. To satisfy (ref) it clearly suffices to choose $\tilde{c}_{n}(\psi,\kappa)$ to satisfy: \begin{align} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n}\bigg(\sup_{\gamma \in \Gamma}\sup_{\theta \in \Theta}\max_{\lambda \in \Lambda} \bigg| \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) - Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) \bigg|\geq \tilde{c}_{n}(\psi,\kappa)\bigg) \leq 1-\kappa/2. \end{align} From koltchinskii2011oracle Theorem 4.6 we have for any $t>0$: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n}\left( \sup_{\gamma \in \Gamma}\sup_{\theta \in \Theta}\max_{\lambda \in \Lambda} \bigg| \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) - Ph_{\ell b}(\cdot, \theta,\gamma,\lambda) \bigg| \geq 2 ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) + \frac{3t \overline{H} }{\sqrt{n}} \right) &\leq \exp \left( -\frac{t^{2}}{2} \right). \end{align*} Now set: \begin{align*} \tilde{c}_{n}(\psi,\kappa)&= 2||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) + \sqrt{\frac{18 \ln(2/(2-\kappa)) \overline{H}^{2}}{n}}. \end{align*} Then we have: \begin{align*} c_{n}(\kappa)&= 4 ||\mathfrak{R}_{n}||(\mathcal{H}_{\ell b}) + \sqrt{\frac{72 \ln(2/(2-\kappa)) \overline{H}^{2}}{n}} + 5\varepsilon. \end{align*} Then we conclude (ref). \end{proof} \begin{proof}[Proof of Theorem (ref)] Let $T, T^\flat$, and $T^\sharp$ be as defined in Lemma (ref).\note{Note that the objective of the proof is to show $\delta^{**} \leq \delta^{*}$ on $E_{n}$, the same event for which Lemma (ref) applies. Knowing the feasible-to-compute $\delta^{*}$ then allows us to apply Lemma (ref).} In this proof, it is useful to note the following facts: \begin{enumerate}[label=(\roman*)] • The functions $\delta \mapsto T_{n}(\delta), T(\delta)$ are non-decreasing left-continuous step functions that are greater than or equal to zero on the interval $[0,\delta_{0}]$, and zero otherwise. • The functions $\sigma \mapsto T_{n}^\flat(\sigma), T^\flat(\sigma)$, are non-increasing and left-continuous with their only possible points of discontinuity at the points $\{\delta_{j}\}_{j=0}^{\infty}$.\noteinline{It seems they are continuous on any interval where $T_{n}(\delta)/\delta$ and $T(\delta)/\delta$ are increasing, and left continuous otherwise. The left-continuity seems to come from the fact that, if $T(\delta)/\delta$ is decreasing in $\delta$, then it is a decreasing and left-continuous function, so that also $\sigma\mapsto\sup_{\delta \geq \sigma} T(\delta)/\delta$ is also left continuous.} • The functions $\eta \mapsto T_{n}^\sharp(\eta), T^\sharp(\eta)$ are non-increasing and continuous.\noteinline{In fact $T^\sharp$ will be constant at the points of discontinuity of $T^\flat$. The continuity comes from the fact that $T^\flat$ is a non-increasing left-continuous function. At any point $\sigma^{*}$ of discontinuity of $T^\flat$ we will have $T^\flat(\sigma^{*}) > \lim_{\sigma \downarrow \sigma^{*}} T^\flat(\sigma)$, and can have a situation where $T^\flat(\sigma^{*}) > \eta \geq T^\flat(\sigma)$ for $\sigma>\sigma^{*}$. However, in such a case we will have $T^\sharp(\eta) = \sigma^{*}$, since the infimum in the definition of $T^\sharp$ behaves in a continuous way. At such points of discontinuity of $T^\flat$, it seems $T^\sharp$ will be constant.} \end{enumerate} Now for any $\eta>0$, let: \begin{align*} \delta^{*}&= T_{n}^{\sharp}(1-1/\mathfrak{a})+\eta',\\ \delta^{**}&= T^{\sharp}(1-1/\mathfrak{a})+\eta, \end{align*} where $\eta'=\eta+\varepsilon$ for some $\varepsilon>0$. Note that choosing $\delta^{**}$ slightly larger than $T^\sharp(1-1/\mathfrak{a})$ ensures that $T^\flat(\delta^{**})\leq 1-1/\mathfrak{a}$. A similar note applies to $\delta^{*}$ and $T_{n}^{\sharp}(1-1/\mathfrak{a})$. From the proof of Lemma (ref) we know there exists an event $E_{n}$ with $P_{Y,Z}^{\otimes n} \left(E_{n} \right) \geq \kappa$ such that on $E_{n}$ we have $\mathscr{G}^{*}(\delta) \subseteq \mathscr{G}_{n}(\mathfrak{b}\delta)$ for every $\delta \geq \delta^{**}$. Thus, for every $\delta \geq \delta^{**}$ we have on $E_{n}$ that $T(\delta)\leq T_{n}(\delta)$, which implies: \begin{align*} \frac{T(\delta)}{\delta} \leq \frac{T_{n}(\delta)}{\delta}, \end{align*} for all $\delta \geq \delta^{**}$. Thus, on $E_{n}$ we have $T^\flat(\sigma) \leq T_{n}^\flat(\sigma)$ for any $\sigma \geq \delta^{**}$, and in particular we have: \begin{align} T^\flat(\delta^{**}):= \sup_{\delta \geq \delta^{**}}\frac{T(\delta)}{\delta} \leq \sup_{\delta \geq \delta^{**}}\frac{T_{n}(\delta)}{\delta}=:T_{n}^\flat(\delta^{**}), \end{align} Recall our choice of $\delta^{**}$ ensures that $T^\flat(\delta^{**})\leq 1-1/\mathfrak{a}$. We can now distinguish two cases on the event $E_{n}$: \begin{enumerate} • We have: \begin{align*} \sup_{\delta \geq \delta^{**}}\frac{T(\delta)}{\delta} \leq 1-\frac{1}{\mathfrak{a}} \leq \sup_{\delta \geq \delta^{**}}\frac{T_{n}(\delta)}{\delta}. \end{align*} In this case, we have $T_{n}^\flat(\delta^{**}) \geq 1-1/\mathfrak{a}$, and thus $T_{n}^\sharp(1-1/\mathfrak{a}) \geq \delta^{**}$, so that $\delta^{*}>\delta^{**}$ (see the definitions of $\delta^{*}$ and $\delta^{**}$ above). • We have: \begin{align*} \sup_{\delta \geq \delta^{**}}\frac{T(\delta)}{\delta} \leq \sup_{\delta \geq \delta^{**}}\frac{T_{n}(\delta)}{\delta} < 1-\frac{1}{\mathfrak{a}}. \end{align*} This implies either (i) $T^\sharp(1-1/\mathfrak{a}) \leq T_{n}^\sharp(1-1/\mathfrak{a}) < \delta^{**}$, or (ii) $T_{n}^\sharp(1-1/\mathfrak{a}) < T^\sharp(1-1/\mathfrak{a}) < \delta^{**}$. In case (i) we clearly have $\delta^{*} \geq \delta^{**}$. In case (ii), let: \begin{align*} c := T^\sharp(1-1/\mathfrak{a}) - T_{n}^\sharp(1-1/\mathfrak{a})>0. \end{align*} Then: \begin{align*} \delta^{**} - \delta^{*} &= T^\sharp(1-1/\mathfrak{a}) +\eta - T_{n}^\sharp(1-1/\mathfrak{a}) - \eta'\\ &=c -\varepsilon, \end{align*} where the last line follows from the definition of $\eta'$. Now suppose that $c>\varepsilon$ for our $\varepsilon>0$ chosen at the beginning of the proof. We will show that this produces a contradiction. To understand the approach, note the value of $c$ does not depend on the value of $\eta>0$, so the assumption that $c>\varepsilon$ must trivially hold for every $\eta>0$. If we can show that $c<\varepsilon$ for some $\eta>0$, we will have arrived at our desired contradiction. Recall that on $E_{n}$ we have $T^\flat(\sigma) \leq T_{n}^\flat(\sigma)$ for any $\sigma \geq \delta^{**}$. This implies that, for any $r>0$, if $T^\sharp(r) \geq \delta^{**}$ then $T_{n}^\sharp(r) \geq T^\sharp(r)$. Now choose a value $r_{\eta}\in \mathbb{R}$ closest to $1-1/\mathfrak{a}$ such that $r_{\eta} \leq 1-1/\mathfrak{a}$ and: \begin{align*} T^\sharp(r_{\eta}) =T^{\sharp}(1-1/\mathfrak{a})+\eta =\delta^{**}. \end{align*} Such a choice is always possible by continuity of $T^{\sharp}$, and by the fact that $T^\sharp$ is non-increasing. By taking $\eta$ (and thus also $\delta^{**}$) small enough we conclude by continuity of $T^\sharp$ that the point $r_{\eta}$ can also always be chosen arbitrarily close to $1-1/\mathfrak{a}$.\todo{You better check this, I am not certain it is correct.} Recall by continuity of $T_{n}^\sharp$ that there exists $\varepsilon'>0$ such that $T_{n}^\sharp(x) - T_{n}^\sharp(1-1/\mathfrak{a}) < \varepsilon$ whenever $(1-1/\mathfrak{a}) < x + \varepsilon'$. Now by choosing $r_{\eta} \leq 1-1/\mathfrak{a}$ such that $1-1/\mathfrak{a}<r_{\eta}+\varepsilon'$, we have: \begin{align*} c &= T^\sharp(1-1/\mathfrak{a}) - T_{n}^\sharp(1-1/\mathfrak{a})\\ &< T^\sharp(r_{\eta}) - T_{n}^\sharp(1-1/\mathfrak{a})\\ &\leq T_{n}^\sharp(r_{\eta}) - T_{n}^\sharp(1-1/\mathfrak{a})\\ &<\varepsilon. \end{align*} This of course contradicts the fact that $c>\varepsilon$ for every choice of $\eta>0$. We conclude that $c\leq\varepsilon$, and since $\delta^{**}-\delta^{*}=c-\varepsilon$, we have $\delta^{**}\leq \delta^{*}$.\note{This proof is extremely tricky, but essentially seems related to the fact that $\delta^{**}$ is a function of $\eta$, and the fact that if $T^\sharp(r) \geq \delta^{**}(\eta)$ then $T_{n}^\sharp(r) \geq T^\sharp(r)$. } \end{enumerate} We conclude in all cases that $\delta^{**}\leq \delta^{*}$ on $E_{n}$. The result then follows directly from Lemma (ref). \end{proof} \begin{proof}[Proof of Lemma (ref)] Recall that: \begin{align*} h_{\ell b}(y,z,\theta,\gamma,\lambda):= \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg). \end{align*} For notational simplicity we will define: \begin{align*} \P_{n}h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \frac{1}{n}\sum_{i=1}^{n} \inf_{u_{i} \in \bm G^{-}(y_{i},z_{i},\theta)} \inf_{y_{i}^{\star} \in \bm G^{\star}(y_{i},z_{i},u_{i},\theta,\gamma)} \Bigg( \varphi(v_{i}) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y_{i},z_{i},u_{i},\theta)\Bigg),\\ P h_{\ell b}(\cdot, \theta,\gamma,\lambda) &:= \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\,dP_{Y,Z}. \end{align*} Define the events: \begin{align*} E_{n,j} := \left\{\sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\delta_{j})} \sup_{\lambda,\lambda' \in \Lambda}\left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right| \leq T(\delta_{j})\right\}, \end{align*} and: \begin{align} E_{n}:= \bigcap_{\{ j : \delta_{j}\geq \delta^{**}\}} E_{n,j}. \end{align} Note the value $2 \overline{H}$ is an upper bound for any function in $\mathcal{H}_{\ell b}'(\delta)$ for any $\delta> 0$. By our choice of $\delta_{0}> 2\overline{H}$ we have: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( E_{n,0}^{c}\right)=0. \end{align*} Furthermore, from the uniform version of Hoeffding's inequality (e.g. koltchinskii2011oracle Theorem 4.6, p.71) we have: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n}\left( E_{n,j}^{c}\right) \leq \exp \left( -\frac{t_{j}^{2}}{2} \right), \end{align*} for each $j\in \mathbb{N}$. We conclude by the union bound that: \begin{align*} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n}\left( E_{n}\right) \geq 1- \sum_{\{j : \delta_{j} \geq \delta^{**}\}} \exp \left( -\frac{t_{j}^{2}}{2} \right). \end{align*} Now note that with $c_{1}=5$, $c_{2}= (3/(2\kappa))^{2/5}$ and $t_{j} = \sqrt{c_{1} \log(c_{2} \cdot j)}$, we have: \begin{align*} \sum_{\{j : \delta_{j} \geq \delta^{**}\}} \exp \left( -\frac{t_{j}^{2}}{2} \right) &\leq \sum_{j=1}^{\infty} \exp \left( -\frac{t_{j}^{2}}{2} \right)\\ &= \sum_{j=1}^{\infty} \exp \left( -\frac{c_{1} \log(c_{2} \cdot j)}{2} \right) \\ &= \sum_{j=1}^{\infty} (c_{2}\cdot j)^{-\frac{c_{1}}{2}} \\ &=\frac{2(1-\kappa)}{3} \sum_{j=1}^{\infty} \left( \frac{1}{j} \right)^{5/2}\\ &\leq \frac{2(1-\kappa) }{3} \left( \frac{3}{2}\right)\\ &=1-\kappa. \end{align*} Thus we conclude: \begin{align} \inf_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( E_{n} \right) \geq \kappa. \end{align} The remainder of the proof proceeds in two parts: \begin{enumerate} • We will show that on the event $E_{n}$ we have for any $\gamma \in \Gamma$, $\mathscr{E}_{n}(\gamma) \leq (2-1/\mathfrak{a})\left(\mathscr{E}^{*}(\gamma)\vee \delta^{**}\right)$. We will then use this fact to argue that, on $E_{n}$, for any $\delta \geq \delta^{**}$ we have $\mathscr{G}^{*}(\delta) \subseteq \mathscr{G}_{n}((2-1/\mathfrak{a})\delta)$. • We will show that on the event $E_{n}$ we have for any $\gamma \in \Gamma$, $\mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\left(\mathscr{E}_{n}(\gamma)\vee \delta^{**}\right)$. We will then use this fact to argue that, on $E_{n}$, for any $\delta \geq \mathfrak{a}\delta^{**}$ we have $\mathscr{G}_{n}(\delta/\mathfrak{a}) \subseteq \mathscr{G}^{*}(\delta)$. \end{enumerate} Throughout this proof, let $\lambda^{*}(\theta,\gamma)$, $\hat{\lambda}(\theta,\gamma)$, $\theta^{*}(\gamma)$, $\hat{\theta}(\gamma)$, $\gamma^{*}$ and $\hat{\gamma}$ be as in Remark (ref).\\ \\ \textbf{Part 1:} We will prove that on the event $E_{n}$ we have $\mathscr{E}_{n}(\gamma) \leq (2-1/\mathfrak{a})\left(\mathscr{E}^{*}(\gamma)\vee \delta^{**}\right)$ for any $\gamma \in \Gamma$. First, consider any $\gamma$ with $\sigma:= \mathscr{E}^{*}(\gamma) \geq \delta^{**}$. Pick any $\varepsilon>0$ such that $\delta^{**}\geq\varepsilon$, which is possible since $\delta^{**}>T^\sharp(1-1/\mathfrak{a}) \geq 0$. Then on the event $E_{n}$ we have: \begin{align*} \mathscr{E}_{n}(\gamma)&:=\sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)+3\varepsilon\\ &= \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\qquad\qquad+ \left( \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) \right)\\ &\qquad\qquad\qquad\qquad-\left(\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right)+3\varepsilon\\ &\leq\sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\qquad\qquad+ \left( \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) \right)\\ &\qquad\qquad\qquad\qquad-\left(\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right)+3\varepsilon\\ &=\mathscr{E}^{*}(\gamma)+ \left( \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) \right)\\ &\qquad\qquad\qquad\qquad-\left(\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right)+3\varepsilon. \end{align*} Now note: \begin{align*} &\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda)+\varepsilon\\ &\leq \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda)- \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda)+2\varepsilon\\ &\leq \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda)- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))+2\varepsilon\\ &= P h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma))- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))+2\varepsilon. \end{align*} Similarly: \begin{align*} &\inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\leq \inf_{\theta \in \Theta}\max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda) + \varepsilon\\ &\leq \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda) - \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda) + 2\varepsilon\\ &\leq \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda) - \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma)) + 2\varepsilon\\ &= \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma})) - \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma)) + 2\varepsilon. \end{align*} Thus we conclude: \begin{align*} \mathscr{E}_{n}(\gamma)&\leq \mathscr{E}^{*}(\gamma)+ 7\varepsilon + P h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma))- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))\\ &\qquad\qquad\qquad\qquad\qquad\qquad- \left(\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma)) - \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))\right). \end{align*} However, $\gamma \in \mathscr{G}^{*}(\sigma)$ by assumption, and by Lemma (ref) we have $\hat{\gamma} \in \mathscr{G}^{*}(\sigma)$ on the event $E_{n}$. Thus, the right side of the previous display can be bounded above: \begin{align*} &P h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma))- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))- \left(\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma),\gamma,\lambda^{*}(\hat{\theta}(\gamma),\gamma)) - \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma})\right)\\ &\leq \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)} \sup_{\lambda,\lambda' \in \Lambda} \left| \left(\P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right) \right|. \end{align*} Furthermore, for any $\sigma\geq \delta^{**}$, on the event $E_{n}$ this final quantity is bounded above by $T(\sigma)$; this follows from the definition of $T(\sigma)$ and the monotonicity of the map: \begin{align*} x \mapsto \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(x)} \max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|. \end{align*} Thus on $E_{n}$: \begin{align*} \mathscr{E}_{n}(\gamma)&\leq \mathscr{E}^{*}(\gamma)+ T(\sigma)+7\varepsilon\\ &= \mathscr{E}^{*}(\gamma)+ \frac{T(\sigma)}{\sigma} \sigma+7\varepsilon\\ &\leq \mathscr{E}^{*}(\gamma)+ \sup_{\delta \geq \sigma}\left( \frac{T(\delta)}{\delta}\right)\sigma+7\varepsilon\\ &= \mathscr{E}^{*}(\gamma) + T^\flat(\sigma)\sigma+7\varepsilon\\ &= \mathscr{E}^{*}(\gamma) + T^\flat(\sigma)\mathscr{E}^{*}(\gamma)+7\varepsilon. \end{align*} Now, since $\sigma \geq \delta^{**} > T^\sharp(1-1/\mathfrak{a})$ we have $T^\flat(\sigma)\leq T^\flat(\delta^{**}) \leq 1- 1/\mathfrak{a}$. Thus, on the event $E_{n}$, if $\gamma$ is such that $\mathscr{E}^{*}(\gamma) \geq \delta^{**}$, we have: \begin{align*} \mathscr{E}_{n}(\gamma) \leq \left(2-\frac{1}{\mathfrak{a}}\right)\mathscr{E}^{*}(\gamma)+7\varepsilon. \end{align*} Since $\varepsilon>0$ is any value such that $\delta^{**}\geq \varepsilon$, and thus can be made arbitrarily small, we conclude that on the event $E_{n}$ we have for any $\gamma$ with $\mathscr{E}^{*}(\gamma) \geq \delta^{**}$: \begin{align*} \mathscr{E}_{n}(\gamma) \leq \left(2-\frac{1}{\mathfrak{a}}\right)\mathscr{E}^{*}(\gamma). \end{align*} Now consider the case when $\sigma:=\mathscr{E}^{*}(\gamma) \leq \delta^{**}$. By the same derivation as above we obtain: \begin{align*} &\mathscr{E}_{n}(\gamma)\\ &\leq \mathscr{E}^{*}(\gamma) + \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)}\max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right| +7\varepsilon. \end{align*} By monotonicity, we have: \begin{align*} &\sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)}\max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|\\ &\leq \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\delta^{**})}\max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|. \end{align*} Furthermore, on the event $E_{n}$ we have: \begin{align*} \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\delta^{**})}\max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right| \leq T(\delta^{**}). \end{align*} Thus, on the event $E_{n}$: \begin{align*} \mathscr{E}_{n}(\gamma)&\leq \mathscr{E}^{*}(\gamma) + T(\delta^{**})+7\varepsilon\\ &\leq \mathscr{E}^{*}(\gamma) + \sup_{\delta \geq \delta^{**}}\left( \frac{T(\delta)}{\delta} \right) \delta^{**} +7\varepsilon\\ &= \mathscr{E}^{*}(\gamma) + T^\flat(\delta^{**})\delta^{**} +7\varepsilon\\ &\leq \mathscr{E}^{*}(\gamma) + \left( 1-\frac{1}{\mathfrak{a}} \right) \delta^{**} +7\varepsilon\\ &\leq \delta^{**} + \left( 1-\frac{1}{\mathfrak{a}} \right) \delta^{**} +7\varepsilon\\ &=\left( 2-\frac{1}{\mathfrak{a}} \right) \delta^{**} +7\varepsilon. \end{align*} Since $\varepsilon>0$ is any value such that $\delta^{**}\geq \varepsilon$, and thus can be made arbitrarily small, we conclude that on the event $E_{n}$ we have for any $\gamma$: \begin{align*} \mathscr{E}_{n}(\gamma) \leq \left(2-\frac{1}{\mathfrak{a}}\right)\left( \mathscr{E}^{*}(\gamma)\vee \delta^{**}\right). \end{align*} We will use this result to argue that, on the event $E_{n}$, if $\delta\geq \delta^{**}$ then $\mathscr{E}^{*}(\gamma) \leq \delta \implies \mathscr{E}_{n}(\gamma) \leq (2-1/\mathfrak{a})\delta$. There are two cases: \begin{enumerate}[label=(\roman*)] • $\mathscr{E}^{*}(\gamma) \leq \delta^{**} \leq \delta$, which implies on the event $E_{n}$: \begin{align*} \mathscr{E}_{n}(\gamma) \leq \left(2-\frac{1}{\mathfrak{a}}\right)\left( \mathscr{E}^{*}(\gamma)\vee \delta^{**}\right) =\left(2-\frac{1}{\mathfrak{a}}\right)\delta^{**} \leq \left(2-\frac{1}{\mathfrak{a}}\right)\delta. \end{align*} • $\delta^{**} \leq \mathscr{E}^{*}(\gamma) \leq \delta$, which implies on the event $E_{n}$: \begin{align*} \mathscr{E}_{n}(\gamma) \leq \left(2-\frac{1}{\mathfrak{a}}\right)\left( \mathscr{E}^{*}(\gamma)\vee \delta^{**}\right) = \left(2-\frac{1}{\mathfrak{a}}\right) \mathscr{E}^{*}(\gamma)\leq \left(2-\frac{1}{\mathfrak{a}}\right) \delta. \end{align*} \end{enumerate} Thus we conclude that for any $\delta \geq \delta^{**}$, on $E_{n}$ we have that $\mathscr{E}^{*}(\gamma) \leq \delta \implies \mathscr{E}_{n}(\gamma) \leq (2-1/\mathfrak{a})\delta$. Now recall that we have $\mathscr{E}^{*}(\gamma) \leq \delta \iff \gamma \in \mathscr{G}^{*}(\delta)$ and $\mathscr{E}_{n}(\gamma) \leq (2-1/\mathfrak{a})\delta \iff \gamma \in \mathscr{G}_{n}((2-1/\mathfrak{a})\delta)$. Thus, we conclude that for any $\delta \geq \delta^{**}$, on the event $E_{n}$: \begin{align*} \mathscr{G}^{*}(\delta) \subseteq \mathscr{G}_{n}((2-1/\mathfrak{a})\delta), \end{align*} as desired.\\ \textbf{Part 2:} We will prove that on the event $E_{n}$ we have $\mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\left(\mathscr{E}_{n}(\gamma)\vee \delta^{**}\right)$ for any $\gamma \in \Gamma$. If $\gamma$ is such that $\mathscr{E}^{*}(\gamma) \leq \delta^{**}$ then this is trivially true (since $\mathfrak{a}>1$). Now consider any $\gamma$ with $\sigma:= \mathscr{E}^{*}(\gamma) \geq \delta^{**}$. Pick any $\varepsilon>0$ such that $\delta^{**}\geq\varepsilon$, which is possible since $\delta^{**}>T^\sharp(1-1/\mathfrak{a}) \geq 0$. Then on the event $E_{n}$ we have: \begin{align*} \mathscr{E}^{*}(\gamma)&:=\sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)+3\varepsilon\\ &= \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\qquad\qquad+ \left( \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)- \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) \right)\\ &\qquad\qquad\qquad\qquad-\left(\sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)\right)+3\varepsilon\\ &=\mathscr{E}^{*}(\gamma)+ \left( \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)\right)\\ &\qquad\qquad\qquad\qquad-\left(\sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)\right)+3\varepsilon. \end{align*} Now note: \begin{align*} &\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda) \\ &\leq\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) - \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\lambda)+\varepsilon \\ &\leq\max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda) - \max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\lambda)+2\varepsilon \\ &\leq\max_{\lambda \in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda) - P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))+2\varepsilon \\ &\leq P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))+2\varepsilon. \end{align*} Similarly: \begin{align*} &\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \sup_{\gamma \in \Gamma}\inf_{\theta \in \Theta} \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda)\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda) + 3\varepsilon\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda) + 4\varepsilon\\ &\leq \max_{\lambda \in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\lambda) - \max_{\lambda \in \Lambda}\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda) + 5\varepsilon\\ &\leq \P_{n} h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma)) - \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) + 5\varepsilon. \end{align*} Thus we conclude: \begin{align*} \mathscr{E}^{*}(\gamma)&\leq \mathscr{E}_{n}(\gamma)+ 10\varepsilon + P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))\\ &\qquad\qquad\qquad\qquad\qquad\qquad- \left(\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - \P_{n} h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))\right). \end{align*} However, $\gamma \in \mathscr{G}^{*}(\sigma)$ by assumption, and $\mathscr{E}^{*}(\gamma^{*}) \leq \varepsilon \leq \mathscr{E}^{*}(\gamma) = \sigma$ implies $\gamma^{*} \in \mathscr{G}^{*}(\sigma)$. Thus, the right side of the previous display can be bounded above: \begin{align*} &P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - P h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))\\ &\qquad\qquad\qquad\qquad\qquad\qquad- \left(\P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - \P_{n} h_{\ell b}(\cdot,\theta^{*}(\gamma),\gamma,\hat{\lambda}(\theta^{*}(\gamma),\gamma))\right)\\ &\leq \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)}\max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|. \end{align*} Furthermore, for any $\sigma\geq \delta^{**}$, on the event $E_{n}$ this final quantity is bounded above by $T(\sigma)$; this follows from the definition of $T(\sigma)$ and the monotonicity of the map: \begin{align*} x \mapsto \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(x)} \max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|. \end{align*} Thus on $E_{n}$: \begin{align*} \mathscr{E}^{*}(\gamma)&\leq \mathscr{E}_{n}(\gamma)+ T(\sigma)+10\varepsilon\\ &= \mathscr{E}_{n}(\gamma)+ \frac{T(\sigma)}{\sigma} \sigma+10\varepsilon\\ &\leq \mathscr{E}_{n}(\gamma)+ \sup_{\delta \geq \sigma}\left( \frac{T(\delta)}{\delta}\right) \sigma+10\varepsilon\\ &= \mathscr{E}_{n}(\gamma) + T^\flat(\sigma)\sigma+10\varepsilon\\ &= \mathscr{E}_{n}(\gamma) + T^\flat(\sigma)\mathscr{E}^{*}(\gamma)+10\varepsilon. \end{align*} Now, since $\sigma \geq \delta^{**} > T^\sharp(1-1/\mathfrak{a})$ we have $T^\flat(\sigma)\leq T^\flat(\delta^{**}) \leq 1- 1/\mathfrak{a}$.\note{This seems to be the main reason why you pick $\delta^{**} > T^\sharp(1-1/\mathfrak{a}$.} Thus, on the event $E_{n}$, if $\gamma$ is such that $\sigma=\mathscr{E}^{*}(\gamma) \geq \delta^{**}$, we have: \begin{align*} &\mathscr{E}^{*}(\gamma) \leq \mathscr{E}_{n}(\gamma) + \left(1-\frac{1}{\mathfrak{a}}\right)\mathscr{E}^{*}(\gamma)+10\varepsilon \\ &\qquad\qquad\qquad\qquad\implies \mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\mathscr{E}_{n}(\gamma) +10\mathfrak{a}\varepsilon. \end{align*} Since $\varepsilon>0$ is any value such that $\delta^{**}\geq \varepsilon$, and thus can be made arbitrarily small, we conclude that on the event $E_{n}$ we have for any $\gamma$: \begin{align*} \mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\left(\mathscr{E}_{n}(\gamma)\vee \delta^{**}\right). \end{align*} We will use this result to argue that, on the event $E_{n}$, if $\delta/\mathfrak{a}\geq \delta^{**}$ then $\mathscr{E}^{*}(\gamma) \leq \delta$. There are two cases: \begin{enumerate}[label=(\roman*)] • $\mathscr{E}_{n}(\gamma) \leq \delta^{**} \leq \delta/\mathfrak{a}$, which implies on the event $E_{n}$: \begin{align*} \mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\left( \mathscr{E}_{n}(\gamma)\vee \delta^{**}\right) =\mathfrak{a}\delta^{**} \leq \delta. \end{align*} • $\delta^{**} \leq \mathscr{E}_{n}(\gamma) \leq\delta/\mathfrak{a}$, which implies on the event $E_{n}$: \begin{align*} \mathscr{E}^{*}(\gamma) \leq \mathfrak{a}\left( \mathscr{E}_{n}(\gamma)\vee \delta^{**}\right) = \mathfrak{a}\mathscr{E}^{*}(\gamma)\leq \delta. \end{align*} \end{enumerate} Thus we conclude that for any $\delta/\mathfrak{a} \geq \delta^{**}$, on $E_{n}$ we have that $\mathscr{E}_{n}(\gamma) \leq \delta/\mathfrak{a} \implies \mathscr{E}^{*}(\gamma) \leq \delta$. Now recall that we have $\mathscr{E}_{n}(\gamma) \leq \delta/\mathfrak{a} \iff \gamma \in \mathscr{G}_{n}(\delta/\mathfrak{a})$ and $\mathscr{E}^{*}(\gamma) \leq \delta \iff \gamma \in \mathscr{G}^{*}(\delta)$. Thus, we conclude that for any $\delta \geq \mathfrak{a}\delta^{**}$, on the event $E_{n}$: \begin{align*} \mathscr{G}_{n}(\delta/\mathfrak{a}) \subseteq \mathscr{G}^{*}(\delta) , \end{align*} as desired. This completes the proof. \end{proof} \subsection{Auxiliary Results and Proofs} \subsubsection{On Issues of Measurability} The following discussion mirrors the discussion in dudley2010real Section 3.3 and dudley2014uniform Section 5.3. Let $\mathcal{X}$ be a Polish space, and let $\mathfrak{B}(\mathcal{X})$ be the Borel $\sigma-$algebra on $\mathcal{X}$. Then $(\mathcal{X},\mathfrak{B}(\mathcal{X}))$ is a measurable space. If $P$ is a probability law on $\mathfrak{B}(\mathcal{X})$, then $(\mathcal{X},\mathfrak{B}(\mathcal{X}),P)$ is a probability space. Now for any $B \subset \mathcal{X}$, we can define the outer measure $P^{*}$ on $B$ as: \begin{align*} P^{*}(B) := \inf\{P(C) : B \subset C \text{ and } C \in \mathfrak{B}(\mathcal{X})\}. \end{align*} By Theorem 3.3.1 in dudley2010real, there always exists $C \in \mathfrak{B}(\mathcal{X})$ such that $P^{*}(B) = P(C)$, and such a set $C$ is called a measurable cover of $B$. Now define the collection of null sets for $P$ as: \begin{align*} Null(P):= \{A \subset \mathcal{X} : P^{*}(A) =0 \}. \end{align*} Furthermore, let $\mathfrak{B}_{P}^{*}(\mathcal{X})$ denote the smallest $\sigma-$algebra generated by $\mathfrak{B}(\mathcal{X})\cup Null(P)$. By Proposition 3.3.2 in dudley2010real, we have: \begin{align*} \mathfrak{B}_{P}^{*}(\mathcal{X}):= \{B \subset \mathcal{X} : B \Delta C \in Null(P) \text{ for some } C \in \mathfrak{B}(\mathcal{X}) \}, \end{align*} where $B \Delta C = (B\setminus C)\cup(C\setminus B)$. We can now extend the measure $P$ from $\mathfrak{B}(\mathcal{X})$ to a measure $\overline{P}$ on $\mathfrak{B}_{P}^{*}(\mathcal{X})$ as follows: if $B\Delta C \in Null(P)$ and $C \in \mathfrak{B}(\mathcal{X})$, then set $\overline{P}(B) = P(C)$. Proposition 3.3.3 in dudley2010real verifies this is a valid extension; that is, $\overline{P}$ is a measure on $\mathfrak{B}_{P}^{*}(\mathcal{X})$ and $\overline{P}$ agrees with $P$ for all sets in $\mathfrak{B}(\mathcal{X})$. However, note that the collection $\mathfrak{B}_{P}^{*}(\mathcal{X})$ clearly depends on the probability measure $P$. Indeed, if $Q$ is another measure on $\mathfrak{B}(\mathcal{X})$, and $\mathfrak{B}_{Q}^{*}(\mathcal{X})$ is defined in an analogous manner to $\mathfrak{B}_{P}^{*}(\mathcal{X})$, then it is possible for the two collections $\mathfrak{B}_{P}^{*}(\mathcal{X})$ and $\mathfrak{B}_{Q}^{*}(\mathcal{X})$ to differ because the null sets of $P$ and $Q$ differ. On the other hand, clearly both $\mathfrak{B}_{P}^{*}(\mathcal{X})$ and $\mathfrak{B}_{Q}^{*}(\mathcal{X})$ must have many elements in common; for example, both collections must contain the Borel sets $\mathfrak{B}(\mathcal{X})$. A set $B \in \mathfrak{B}_{P}^{*}(\mathcal{X})$ is called \textit{measurable for the completion of $P$}. If for every probability measure $P$ the set $B$ is measurable for the completion of $P$, then we call $B$ \textit{universally measurable}. We will denote the universally measurable sets as $\mathfrak{B}^{*}(\mathcal{X})$; it is easily verified that $\mathfrak{B}^{*}(\mathcal{X})$ is also a $\sigma-$algebra.\footnote{This follows from the fact that an arbitrary intersection of $\sigma-$algebras is a $\sigma-$algebra.} By definition, for any two probability measures $P$ and $Q$, both $\mathfrak{B}_{P}^{*}(\mathcal{X})$ and $\mathfrak{B}_{Q}^{*}(\mathcal{X})$ contain the universally measurable sets. Also note that, in our example, clearly the Borel sets $\mathfrak{B}(\mathcal{X})$ are universally measurable. A subset $A\subset \mathcal{X}$ of a Polish space $\mathcal{X}$ (with the Borel $\sigma-$field) is called $\mathfrak{B}(\mathcal{X})$-\textit{analytic} if there exists a compact metric space $\mathcal{Y}$ such that $A$ is the projection onto $\mathcal{X}$ of some $B \in \mathfrak{B}(\mathcal{X})\otimes \mathfrak{B}(\mathcal{Y})$.\footnote{We note that this is one of many equivalent definitions of an analytic set. See Chapter 8 of cohn2013measure. Our definition is from stinchcombe1992some.} A function $f:A \to [-\infty,\infty]$ is called lower (or upper) semi-analytic if $A$ is an analytic set and $\{x \in A : f(x) < c\}$ (or $\{x \in A : f(x) \geq c\}$) is an analytic set for every $c \in \mathbb{R}$; that is, if the epigraph (or hypograph) of $f$ is an analytic set. In a Polish space, every analytic set is universally measurable. We refer the reader to Chapter 8 of cohn2013measure for additional details. \begin{lemma}[Infimum over Random Sets is Lower Semi-Analytic] Suppose that Assumptions (ref), (ref) and (ref) hold. Then for any lower semi-analytic function $f:\mathcal{V}\times\Gamma\times\Theta\times\{0,1\}^{J} \to \mathbb{R}$, the function $f_{lb,1}(y,z,u,\theta,\gamma,\lambda)$ given by: \begin{align} f_{lb,1}(y,z,u,\theta,\gamma,\lambda)&:=\inf_{y^{\star} \in \bm G^\star(y,z,u,\theta,\gamma)} f(v,\theta,\gamma,\lambda), \end{align} is lower semi-analytic; that is, $\{(y,z,u,\gamma,\theta,\lambda): f_{lb,1}(y,z,u,\theta,\gamma,\lambda) <r\}$ is an analytic set for every $r \in \mathbb{R}$, and thus is universally measurable. In addition, the function $f_{lb,2}(y,z,\theta,\gamma,\lambda)$ given by: \begin{align} f_{lb,2}(y,z,\theta,\gamma,\lambda)&:=\inf_{u \in \bm G^{-}(y,z,\theta)}f_{lb,1}(y,z,u,\theta,\gamma,\lambda), \end{align} is also lower semi-analytic; that is, $\{(y,z,\theta,\gamma,\lambda): f_{lb,2}(y,z,\theta,\gamma,\lambda)<r\}$ is an analytic set for every $r \in \mathbb{R}$, and thus is universally measurable. \end{lemma} \begin{remark} Defining $f_{ub,1}(y,z,u,\theta,\gamma,\lambda)$ and $f_{ub,2}(y,z,u,\theta,\gamma,\lambda)$ as the analogous functions with the infimum replaced with the supremum, it is possible to show that $f_{ub,1}(y,z,u,\theta,\gamma,\lambda)$ and $f_{ub,2}(y,z,u,\theta,\gamma,\lambda)$ are upper semi-analytic. \end{remark} \begin{proof}[Proof of Lemma (ref)] Recall that under Assumption (ref), the multifunction $\bm G^{\star}(y,z,u,\theta,\gamma)$ is Effros measurable with respect to the product Borel $\sigma-$algebra $\mathfrak{B}(\mathcal{Y})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\mathcal{U})\otimes\mathfrak{B}(\Gamma)$. By molchanov2017theory Theorem 1.3.3 this implies that: \begin{align*} \text{Graph}(\bm G^{\star}) \in \mathfrak{B}(\mathcal{Y})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\mathcal{U})\otimes\mathfrak{B}(\Theta)\otimes\mathfrak{B}(\Gamma). \end{align*} Thus $\text{Graph}(\bm G^{\star})$ is a Borel (and thus also an analytic) set. Now note that $\bm G^\star(y,z,u,\theta,\gamma)$ can be rewritten as: \begin{align*} \bm G^\star(y,z,u,\theta,\gamma):= \left\{y^\star \in \mathcal{Y}^\star : (y,z,u,y^\star,\theta,\gamma) \in \text{Graph}(\bm G^{\star}) \right\}. \end{align*} The fact that $f_{lb,1}: \mathcal{V}\times\Gamma\times\Theta\times\{0,1\}^{J} \to \mathbb{R}$ is lower semi-analytic then follows directly from the selection Theorem of shreve1978alternative, p. 968.\footnote{See also bertsekas1978stochastic Proposition 7.47, p. 179.} Taking $f_{lb,1}(y,z,u,\theta,\gamma,\lambda)$ as lower semi-analytic, a nearly identical proof shows that $f_{lb,2}(y,z,\theta,\gamma,\lambda)$ is also lower semi-analytic. \end{proof} \begin{proposition} Suppose the assumptions of Theorem (ref) hold. Then the maps $\gamma \mapsto I_{\ell b}[\varphi](\gamma), I_{u b}[\varphi](\gamma)$ are universally measurable. \end{proposition} \begin{proof} We will focus on the map $\gamma \mapsto I_{\ell b}[\varphi](\gamma)$, as the proof for the upper envelope function is symmetric. By Theorem (ref) we have: \begin{align*} I_{\ell b}[\varphi](\gamma) = \inf_{\theta \in \Theta} \max_{\lambda \in \{0,1\}^{J}} \int h_{\ell b}(y,z,\theta,\gamma,\lambda) \, dP_{Y,Z}, \end{align*} where: \begin{align*} h_{\ell b}(y,z,\theta,\gamma,\lambda):= \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg). \end{align*} Suppose that $h_{\ell b}(y,z,\theta,\gamma,\lambda)$ is lower semi-analytic (we will return to this in a moment). Then by proposition 7.46 in bertsekas1978stochastic, the map: \begin{align} (\theta,\gamma,\lambda) \mapsto \int h_{\ell b}(y,z,\theta,\gamma,\lambda) \, dP_{Y,Z}, \end{align} is lower semi-analytic. Furthermore, suppose that $g_{1}: \mathbb{R} \to \mathbb{R}$ and $g_{2}:\mathbb{R} \to \mathbb{R}$ are lower semi-analytic. The function $g(x) = g_{1}(x)\vee g_{2}(x)$ satisfies: \begin{align*} g^{-1}((-\infty,r)) = g_{1}^{-1}((-\infty,r))\cup g_{2}^{-1}((-\infty,r)). \end{align*} Since analytic sets are closed under (countable) unions and intersections (parthasarathy2005probability Theorem 3.1), we have that $g$ is lower semi-analytic whenever $g_{1}$ and $g_{2}$ are lower semi-analytic. From this we conclude that the function: \begin{align*} (\theta,\gamma) \mapsto \max_{\lambda \in \{0,1\}^{J}} \int h_{\ell b}(y,z,\theta,\gamma,\lambda) \, dP_{Y,Z}, \end{align*} is lower semi-analytic, being the pointwise maximum of at most $2^{J}$ lower semi-analytic functions of the form (ref). Finally, by the selection Theorem of shreve1978alternative, p. 968 (see also bertsekas1978stochastic Proposition 7.47) we have the map: \begin{align*} \gamma \mapsto \sup_{\theta \in \Theta} \max_{\lambda \in \{0,1\}^{J}} \int h_{\ell b}(y,z,\theta,\gamma) \, dP_{Y,Z}, \end{align*} is lower semi-analytic, and thus universally measurable. It thus remains only to show that $h_{\ell b}(y,z,\theta,\gamma,\lambda)$ is lower semi-analytic. By Lemma (ref), $h_{\ell b}(y,z,\theta,\gamma)$ will be lower semi-analytic if we can show the function: \begin{align} (v,\theta,\gamma,\lambda)\mapsto \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta), \end{align} is lower semi-analytic. Since both $\varphi(v)$ and $\{m_{j}(y,z,u,\theta)\}_{j=1}^{J}$ are Borel measurable by assumption, since the composition of Borel measurable functions are Borel measurable we conclude that (ref) is Borel measurable. The conclusion then follows from the fact that every Borel measurable function is lower semi-analytic. \end{proof} A nearly identical argument shows that, for every fixed sequence $(\xi_{1},\ldots,\xi_{n})\in \{-1,1\}^{n}$, the Rademacher complexity: \begin{align*} ((y_{1},z_{1}),\ldots,(y_{n},z_{n})) \mapsto ||\mathfrak{R}||(\mathcal{H}_{\ell b}), \end{align*} is universally measurable. This is stated as a corollary of the previous result for easy reference. \begin{corollary} Suppose the assumptions of Theorem (ref) hold, and suppose that the sequence $(Y_{1},Z_{1})$, $\ldots$, $(Y_{n},Z_{n})$ are the coordinate projections of the product probability space $((\mathcal{Y}\times\mathcal{Z})^{n},(\mathfrak{B}(\mathcal{Y})\otimes\mathfrak{B}(\mathcal{Z}))^{\otimes n},P_{Y,Z}^{\otimes n}))$. Then the map: \begin{align*} ((Y_{1},Z_{1}),\ldots,(Y_{n},Z_{n})) \mapsto ||\mathfrak{R}||(\mathcal{H}_{\ell b}), \end{align*} is universally measurable; that is, it is measurable for the completion of $P_{Y,Z}^{\otimes n}$ for any $P_{Y,Z} \in \mathcal{P}_{Y,Z}$. \end{corollary} \subsubsection{Respect for Weak Dominance of the Preference Relation in Definition (ref)} \begin{lemma} Let $(\Omega,\mathfrak{A})$ be a measurable space, and let $X_{1},X_{2}: \Omega\times \mathcal{T} \to \mathbb{R}$ be two stochastic processes such that $X_{1}(\cdot,t)$ and $X_{2}(\cdot,t)$ are measurable for each $t$, and $\omega\mapsto \inf_{t \in \mathcal{T}} X_{1}(\omega,t),\inf_{t \in \mathcal{T}} X_{2}(\omega,t)$ are universally measurable; that is, measurable with respect to the completion of any probability measure on $(\Omega,\mathfrak{A})$. Furthermore, suppose that for any probability measure on $(\Omega,\mathfrak{A})$ we have $X_{1}(\omega,t)\leq X_{2}(\omega,t)$ a.s. for every $t \in \mathcal{T}$, and let $c:\mathcal{P} \to \mathbb{R}_{++}$ be any value depending only on $P$, where $\mathcal{P}$ is the set of all probability measure on $(\Omega,\mathfrak{A})$. Finally, let $c_{1},c_{2}: (0,1)\times \mathcal{P}\to \mathbb{R}_{++}$ be the smallest values satisfying: \begin{align*} P \left( \inf_{t \in \mathcal{T}} X_{1}(\omega,t) + c_{1}(\kappa,P) \geq c(P) \right) \geq \kappa,\\ P \left( \inf_{t \in \mathcal{T}} X_{2}(\omega,t) + c_{2}(\kappa,P) \geq c(P) \right) \geq \kappa, \end{align*} for each $\kappa \in (0,1)$. Then for every $P\in \mathcal{P}$ we have $c_{2}(\kappa,P)\leq c_{1}(\kappa,P)$ for every $\kappa \in (0,1)$. \end{lemma} \begin{proof} Fix any probability measure $P\in \mathcal{P}$. Then by assumption: \begin{align*} X_{1}(\omega,t) \leq X_{2}(\omega,t)\,\, a.s.\qquad \forall t\in \mathcal{T}. \end{align*} This implies: \begin{align*} \inf_{t \in \mathcal{T}} X_{1}(\omega,t) \leq X_{2}(\omega,t)\,\, a.s.\qquad \forall t\in \mathcal{T}, \end{align*} which in turn implies: \begin{align*} \inf_{t \in \mathcal{T}} X_{1}(\omega,t) \leq \inf_{t \in \mathcal{T}} X_{2}(\omega,t)\,\, a.s., \end{align*} and thus: \begin{align*} \inf_{t \in \mathcal{T}} X_{1}(\omega,t) -c(P) \leq \inf_{t \in \mathcal{T}} X_{2}(\omega,t)-c(P)\,\, a.s. \end{align*} Let $N$ denote the null set for which this relation is not true (this set may depend on $P \in \mathcal{P}$). Then we have for every $x \in \mathbb{R}$: \begin{align*} \left\{\omega : \inf_{t \in \mathcal{T}} X_{1}(\omega,t)-c(P) > x \right\}\cap N^{c} \subseteq \left\{\omega : \inf_{t \in \mathcal{T}} X_{2}(\omega,t)-c(P) > x \right\}\cap N^{c}, \end{align*} By assumption, these events belong to the universal $\sigma-$algebra generated by $\mathfrak{A}$, and so are measurable with respect to the completion of any $P \in \mathcal{P}$. This implies that for every $x \in \mathbb{R}$: \begin{align*} P\left(\omega : \inf_{t \in \mathcal{T}} X_{1}(\omega,t)-c(P) > x \right) \leq P\left(\omega : \inf_{t \in \mathcal{T}} X_{2}(\omega,t)-c(P) > x \right). \end{align*} Now taking any $\kappa \in (0,1)$ and setting $x=-c_{1}(\kappa,P)$ we have: \begin{align*} \kappa \leq P\left(\omega : \inf_{t \in \mathcal{T}} X_{1}(\omega,t) + c_{1}(\kappa,P) > c(P) \right) \leq P\left(\omega : \inf_{t \in \mathcal{T}} X_{2}(\omega,t) +c_{1}(\kappa,P) > c(P) \right). \end{align*} By definition, this implies $c_{2}(\kappa,P)$ can be no larger than $c_{1}(\kappa,P)$; that is, $c_{2}(\kappa,P)\leq c_{1}(\kappa,P)$. Since $\kappa \in (0,1)$ was arbitrary, we conclude that $c_{2}(\kappa,P)\leq c_{1}(\kappa,P)$ for every $\kappa \in (0,1)$. Since $P \in \mathcal{P}$ was also arbitrary we conclude that for every $P \in \mathcal{P}$ we have $c_{2}(\kappa,P)\leq c_{1}(\kappa,P)$ for every $\kappa \in (0,1)$. This completes the proof. \end{proof} \subsubsection{Results on Interchanging Integrals and Supremum/Infimum} \begin{lemma}[Interchange of Integral and Supremum/Infimum] Let $(\mathcal{V},\mathfrak{B}(\mathcal{V}))$ and $(\mathcal{V}',\mathfrak{B}(\mathcal{V}'))$ be measurable spaces with $\mathcal{V}$ and $\mathcal{V}'$ as Polish spaces. Let $V \in \mathcal{V}$ be any random variable defined on the probability space $(\Omega,\mathfrak{A},P)$ with (marginal) distribution $P_{V} = P \circ V^{-1}$. Furthermore, let $\bm G: \mathcal{V} \to \mathcal{V}'$ be an almost surely non-empty Effros-measurable multifunction. Then for any bounded and measurable function $\varphi: \mathcal{V} \times \mathcal{V}'\to \mathbb{R}$, we have: \begin{align} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \sup_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \,dP_{V}, \\ \int \inf_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \inf_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \,dP_{V}, \end{align} In particular, if: \begin{align*} \mathcal{P}_{V'|V}:= \{P_{V'|V} : V' \sim P_{V'|V}, \, \,V': \mathcal{V} \to \mathcal{V}' \text{ is measurable and } P_{V'|V}(V' \in \bm G(V)|V=v)=1\,\,a.s. \}, \end{align*} then: \begin{align} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \sup_{P_{V'|V} \in \mathcal{P}_{V'|V}} \int \varphi(v,v') \,d(P_{V'|V} \times P_{V}), \\ \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \inf_{P_{V'|V} \in \mathcal{P}_{V'|V}} \int \varphi(v,v') \,d(P_{V'|V} \times P_{V}). \end{align} \end{lemma} \begin{proof}[Proof of Lemma (ref)] Since $\bm G$ is Effros measurable, by Theorem 1.3.3 in molchanov2017theory we have that $\text{gr}(\bm G) \in \mathfrak{B}(\mathcal{V})\otimes \mathfrak{B}(\mathcal{V}')$, and thus $\text{gr}(\bm G)$ is trivially an analytic set. Now define: \begin{align*} \text{gr}_{v}(\bm G):= \{ v' \in \mathcal{V}' : (v,v') \in \text{gr}(\bm G) \}. \end{align*} Now let: \begin{align*} \varphi^{*}(v):=\sup_{v' \in \bm G(v)} \varphi(v,v') = \sup_{v' \in \text{gr}_{v}(\bm G)} \varphi(v,v'). \end{align*} Furthermore, define the set: \begin{align*} M:= \left\{v \in \Pi_{\mathcal{V}}(\text{gr}(\bm G)) : \exists v' \in \text{gr}_{v}(\bm G) \text{ s.t. } \varphi(v,v') = \varphi^{*}(v) \right\}. \end{align*} where $\Pi_{\mathcal{V}}: \mathcal{V} \times \mathcal{V}' \to \mathcal{V}$ is the projection operator. Fix any $\varepsilon>0$. By the Exact Selection Theorem (shreve1979universally, p.16) there exists a universally measurable function $\tilde{v}':\mathcal{V} \to \mathcal{V}'$ such that $(v,\tilde{v}'(v)) \in \text{gr}(\bm G)$ for every $v \in \Pi_{\mathcal{V}}(\text{gr}(\bm G))$ and:\note{Here boundedness is essential to rule out $+\infty$} \begin{align*} \varphi(v,\tilde{v}'(v)) \begin{cases} = \varphi^{*}(v), &\text{ if } v \in M,\\ \geq \varphi^{*}(v) - \varepsilon, &\text{ if } v \notin M. \end{cases} \end{align*} This allows us to write: \begin{align*} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \, dP_{V} \leq \int \varphi(v,\widetilde{v}'(v)) \, dP_{V} + \varepsilon. \end{align*} Since $\tilde{v}'$ is a (universally) measurable selection, clearly we have: \begin{align*} \int \varphi(v,\widetilde{v}'(v)) \, dP_{V} \leq \sup_{V' \in Sel(\bm G)} \int \varphi(v,V(v)) \, dP_{V} \end{align*} It suffices to show: \begin{align*} \sup_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \, dP_{V} \leq \int \sup_{v' \in \bm G(v)} \varphi(v,v') \, dP_{V} . \end{align*} For any $\varepsilon>0$, let $V_{\varepsilon}' \in Sel(\bm G)$ satisfy: \begin{align*} \sup_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \, dP_{V} \leq \int \varphi(v,V_{\varepsilon}'(v)) \, dP_{V} + \varepsilon. \end{align*} Furthermore, let $N:= \{v : V_{\varepsilon}(v) \notin \bm G(v)\}$. Then by definition of $Sel(\bm G)$ we have $P(N)=0$. Thus: \begin{align*} \int \varphi(v,V_{\varepsilon}(v)) \, dP_{V} = \int_{N^c} \varphi(v,V_{\varepsilon}'(v)) \, dP_{V} \leq \int_{N^{c}} \sup_{v' \in \bm G(v)} \varphi(v,v')\,dP_{V} \leq \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V}. \end{align*} Combining everything we have: \begin{align*} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \, dP_{V} \leq \sup_{V' \in Sel(\bm G)} \int \varphi(v,V(v)) \, dP_{V} + \varepsilon \leq \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} + 2 \varepsilon \end{align*} Since $\varepsilon>0$ was arbitrary, we conclude: \begin{align*} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \sup_{V' \in Sel_{u.m.}(\bm G)} \int \varphi(v,V'(v)) \,dP_{V}. \end{align*} Since each $V' \in Sel_{u.m.}(\bm G)$ is universally measurable, each $V'$ can be associated with a $\mathfrak{B}(\mathcal{V})-$measurable random variable $\tilde{V}'$ such that $V' = \tilde{V}'$ a.s. Thus we can conclude: \begin{align*} \int \sup_{v' \in \bm G(v)} \varphi(v,v') \,dP_{V} = \sup_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \,dP_{V}. \end{align*} To show the final claim, note that for any $V':\mathcal{V} \to \mathcal{V}'$ we have: \begin{align*} P_{V'|V}(V'=v'|V=v) = \mathbbm{1}\{V'(v)=v'\}. \end{align*} i.e. the conditional distribution of $V'$ is degenerate. Thus for any $V' \in Sel(\bm G)$: \begin{align*} \int \int \varphi(v,v') \,d(P_{V'|V}\times P_{V}) &= \int \varphi(v,v') \mathbbm{1}\{V'(v)=v'\} \,dP_{V}\\ &= \int \varphi(v,V'(v)) \,dP_{V}. \end{align*} By definition of $\mathcal{P}_{V'|V}$, we conclude that:\todo{Check with Joon, as this still feels a bit shaky.} \begin{align*} \sup_{V' \in Sel(\bm G)} \int \varphi(v,V'(v)) \,dP_{V} = \sup_{P_{V'|V} \in \mathcal{P}_{V'|V}} \int \varphi(v,v') \,d(P_{V'|V}\times P_{V}). \end{align*} \noteinline{Recall your previous approach. You tried to show that $\int \sup_{v' \in \bm G(v)} \varphi(v,v') $ could be written as an integral with respect to a capacity functional. You could then use the interchange results of Graf or Miranda to show the integral with respect to the capacity functional could be written as $\sup_{V \in Sel(\bm G)} \int \varphi(v,V(v)) \P_{V}$. However, this argument only works well for problems like $\int \sup_{v' \in \bm G(v)} \varphi(v')$, in which case it seems all the measurability problems with projections do not arise; for example, with $\varphi\geq 0$ this can be written as $\int_{0}^{\infty} P( \bm G(v) \cap \{ v' : \varphi(v') \geq t\}\neq \emptyset )$, whereas in your problem you get $\int_{0}^{\infty} P( \text{gr}(\bm G(v)) \cap \{(v,v') : \varphi(v,v') \geq t\}\neq \emptyset )$, which is not necessarily the capacity functional of a random closed set. } \end{proof} \subsubsection{Results on Error Bounds} In the next Lemma we focus on the lower envelope function, although clearly an analogous result is true for the upper envelope function. For notational simplicity, denote: \begin{align} \varphi^{*}:= \inf_{\theta \in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}. \end{align} We now have the following result: \begin{lemma}[Equality Between Primal and Penalized Problems] Suppose the Assumptions of Theorem (ref) hold. Then there exists functions $\lambda_{j}^{\ell b}: \Theta \times \mathcal{P}_{Y,Z} \to \{0,1\}$, $j=1,\ldots,J$, depending only on $\theta$ and the distribution $P_{Y,Z}$, such that: \begin{align*} \varphi^{*} = \inf_{\theta \in \Theta} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\Bigg(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg). \end{align*} \end{lemma} \begin{remark} Recall that $\mathcal{P}_{Y,Z}$ is the set of all Borel probability measures on $\mathcal{Y}\times \mathcal{Z}$. \end{remark} \begin{proof}[Proof of Lemma (ref)] First, note that for any functions $\lambda_{j}^{\ell b}: \Theta \times \mathcal{P}_{Y,Z} \to \{0,1\}$, $j=1,\ldots,J$, we have: \begin{align*} \varphi^{*}&:= \inf_{\theta \in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}\\ &= \inf_{\theta\in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \sup_{\lambda \in \mathbb{R}_{+}^{J}} \Bigg(\int \varphi(v) \, dP_{V_{\gamma}} + \mu^{*} \sum_{j=1}^{J} \lambda_{j} \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg)\\ &\geq \inf_{\theta\in \Theta^{*}}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \Bigg(\int \varphi(v) \, dP_{V_{\gamma}} + \mu^{*} \sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg)\\ &\geq \inf_{\theta\in \Theta}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\Bigg(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} + \mu^{*} \sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg). \end{align*} It thus suffices to show that there exists functions $\lambda_{j}^{\ell b}: \Theta \times \mathcal{P}_{Y,Z} \to \{0,1\}$ for $j=1,\ldots,J$ satisfying the reverse inequality. This is done constructively. In particular, define: \begin{align*} \mathcal{J}^{*}(\theta,P_{Y,Z})&:= \bigg\{ j \in \{1,\ldots,J\} : \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]\\ &\qquad\qquad\qquad\qquad\qquad\qquad= \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+} \bigg\}. \end{align*} That is, the set $\mathcal{J}^{*}(\theta,P_{Y,Z})$ returns the indices of the weakly positive (i.e. weakly violated) moment functions that obtain the inner maximum in the problem:\note{Intuitively, $\lambda_{j}$ will select the most problematic moment (farthest from being satisfied for even the best case possible $P_{U|Y,Z}$)} \begin{align*} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+}. \end{align*} Now set: \begin{align} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}):= \mathbbm{1}\left\{j \in \mathcal{J}^{*}(\theta,P_{Y,Z}) \right\}. \end{align} To show why this choice works, begin by fixing any $\theta \in \Theta_{\delta}^{*}$. By Assumption (ref)(ii) we have: \begin{align} C_{2} d(\theta,\Theta^{*}) &\geq \varphi^{*}-\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}. \end{align} Furthermore, from Assumption (ref)(i), since $\theta \in \Theta_{\delta}^{*}$ by assumption, we have: \begin{align} C_{1} d(\theta,\Theta^{*})&=C_{1} \min\{\delta, d(\theta,\Theta^{*})\}\\ &\leq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+} \nonumber\\ &=\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+} \nonumber\\ &=\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right] \nonumber\\ &\leq \sum_{j=1}^{J} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right] \nonumber\\ &\leq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\sum_{j=1}^{J}\lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]. \end{align} Now by construction we have $\mu^{*}\geq C_{2}/C_{1}$. Thus: \begin{align} C_{2}d(\theta,\Theta^{*}) \leq \mu^{*}C_{1}d(\theta,\Theta^{*}). \end{align} Now use (ref) to combine (ref) and (ref) and rearrange to obtain: \begin{align*} \varphi^{*} &\leq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad+ \mu^{*}\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\sum_{j=1}^{J}\lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)] \\ &\leq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\Bigg(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}+ \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)] \Bigg), \end{align*} which holds for all $\theta \in \Theta_{\delta}^{*}$. To complete the proof, consider any $\theta \in \Theta \setminus \Theta_{\delta}^{*}$. Recall from the assumptions of Theorem (ref) that $\varphi:\mathcal{V} \to [\varphi_{\ell b}, \varphi_{ub}] \subset \mathbb{R}$. Then using Assumption (ref) we have: \begin{align*} & \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\left(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}} + \mu^{*}\sum_{j=1}^{J}\lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} [ m_{j}(y,z,u,\theta)]\right)\\ &\qquad\geq \varphi_{\ell b} + \mu^{*} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\int \left( \sum_{j=1}^{J}\lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta) \right)\, d(P_{U|Y,Z}\times P_{Y,Z})\\ &\qquad\geq \varphi_{\ell b} + \mu^{*} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} \left[ m_{j}(y,z,u,\theta) \right]|_{+}\\ &\qquad\geq \varphi_{\ell b} + \mu^{*} C_{1}\min\{\delta, d(\theta,\Theta^{*})\}\\ &\qquad= \varphi_{\ell b} + \mu^{*} C_{1} \delta\\ &\qquad\geq \varphi^{*}, \end{align*} where the last line follows since $\mu^{*} \geq (\varphi_{ub} - \varphi_{\ell b})/(C_{1}\delta) \geq (\varphi^{*} - \varphi_{\ell b})/(C_{1}\delta)$. Since we have shown the inequality holds for every $\theta \in \Theta$, we have: \begin{align*} &\inf_{\theta \in \Theta} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)}\Bigg(\inf_{P_{Y_{\gamma}^\star|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^\star|Y,Z,U}(\theta,\gamma)} \int \varphi(v) \, dP_{V_{\gamma}}+ \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{Y,Z,U}} [ m_{j}(y,z,u,\theta)]\Bigg) \geq \varphi^{*}. \end{align*} This completes the proof. \end{proof} \begin{lemma} Suppose the assumptions of Theorem (ref) hold, and define: \begin{align*} h_{\ell b}(y,z,\theta,\gamma)&:= \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg). \end{align*} where $\lambda_{j}^{\ell b}: \Theta \times \mathcal{P}_{Y,Z} \to \{0,1\}$, $j=1,\ldots,J$, are as from Lemma (ref). Then we have: \begin{align} &\int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z}\nonumber\\ &\qquad\qquad\leq \max_{\lambda_{j} \in \{0,1\}} \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J}\lambda_{j}m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}, \end{align} with equality holding in (ref) if $\theta \in \Theta^{*}$. \end{lemma} \begin{proof}[Proof of Lemma (ref)] We have: \begin{align*} &\int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z}\\ &:= \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}\\ &=\max_{\lambda_{j} \in \{0,1\} \text{ s.t } \lambda_{j} =\lambda_{j}^{\ell b}(\theta,P_{Y,Z})}\int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}\\ &\leq\max_{\lambda_{j} \in \{0,1\} } \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}. \end{align*} The first line holds by definition, the second line holds since the $\lambda_{j}(\theta,P_{Y,Z})$ depends only on $\theta$, and third line holds because the unconstrained maximum is always weakly larger than the constrained maximum. It remains only to show that the last inequality holds with equality whenever $\theta \in \Theta^{*}$. To do so it suffices to show that for any $\theta \in \Theta^{*}$: \begin{align} \int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z} \geq \max_{\lambda_{j} \in \{0,1\} } \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}. \end{align} To this end, note that by Lemma (ref) we have: \begin{align} &\int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z}\nonumber\\ &=\int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}\nonumber\\ &=\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg)\, d(P_{U|Y,Z}\times P_{Y,Z}). \end{align} Now since the infimum of the sum is larger than the sum of the infimums, we have: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) m_{j}(y,z,u,\theta)\Bigg)\, d(P_{U|Y,Z}\times P_{Y,Z})\nonumber\\ &\qquad\qquad\qquad\geq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \,d(P_{U|Y,Z}\times P_{Y,Z})\nonumber \\ &\qquad\qquad\qquad\qquad\qquad\qquad+ \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \mu^{*} \sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]. \end{align} Now recall that $\lambda_{j}^{\ell b}(\theta,P_{Y,Z})=1$ if and only if: \begin{align*} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]= \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]|_{+}. \end{align*} From here we conclude: \begin{align*} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]|_{+}\\ &=\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]\\ &\leq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)] \end{align*} Thus: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \,dP_{Y,Z,U} \nonumber\\ &\qquad\qquad\qquad\qquad+ \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \mu^{*} \sum_{j=1}^{J} \lambda_{j}^{\ell b}(\theta,P_{Y,Z}) \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]\nonumber\\ &\geq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \,dP_{Y,Z,U} \nonumber\\ &\qquad\qquad\qquad\qquad+ \mu^{*} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]|_{+}. \end{align} However, since $\theta \in \Theta^{*}$ by assumption, we have: \begin{align} \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \max_{j=1,\ldots,J} |\E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]|_{+}= 0. \end{align} Thus, combining (ref), (ref), (ref) and (ref) we can conclude: \begin{align} &\int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z}\geq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \,d(P_{U|Y,Z}\times P_{Y,Z}). \end{align} Now, applying Lemma (ref) again we have: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \int \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \varphi(v) \,d(P_{U|Y,Z}\times P_{Y,Z})\nonumber\\ &\qquad\qquad= \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \int \varphi(v)\, P_{V_{\gamma}}. \end{align} Now note for $\theta \in \Theta^{*}$: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \int \varphi(v)\, P_{V_{\gamma}}\nonumber\\ &= \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \sup_{\lambda_{j} \in \mathbb{R}_{+}} \Bigg(\int \varphi(v)\, P_{V_{\gamma}} + \mu^{*}\sum_{j=1}^{J} \lambda_{j} \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]\Bigg)\nonumber\\ &\geq \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \max_{\lambda_{j} \in \{0,1\} } \Bigg(\int \varphi(v)\, P_{V_{\gamma}} + \mu^{*}\sum_{j=1}^{J} \lambda_{j} \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]\Bigg). \end{align} Now by the minimax inequality: \begin{align} &\inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \max_{\lambda_{j} \in \{0,1\} } \Bigg(\int \varphi(v)\, P_{V_{\gamma}} + \mu^{*}\sum_{j=1}^{J} \lambda_{j} \E_{P_{U|Y,Z}\times P_{Y,Z}} [m_{j}(y,z,u,\theta)]\Bigg)\nonumber\\ &\geq \max_{\lambda_{j} \in \{0,1\} } \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \int \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\, dP_{V_{\gamma}}. \end{align} Finally, by iterated application of Lemma (ref) we have: \begin{align} &\max_{\lambda_{j} \in \{0,1\} } \inf_{P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)} \inf_{P_{Y_{\gamma}^{\star}|Y,Z,U} \in \mathcal{P}_{Y_{\gamma}^{\star}|Y,Z,U}(\theta,\gamma)} \int \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j} m_{j}(y,z,u,\theta)\Bigg)\, dP_{V_{\gamma}}\nonumber\\ &\geq \max_{\lambda_{j} \in \{0,1\} } \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}. \end{align} Combining (ref), (ref), (ref), (ref), and (ref) we have: \begin{align*} \int h_{\ell b}(y,z,\theta,\gamma)\, dP_{Y,Z}\geq\max_{\lambda_{j} \in \{0,1\} } \int \inf_{u \in \bm G^{-}(y,z,\theta)} \inf_{y^{\star} \in \bm G^{\star}(y,z,u,\theta,\gamma)} \Bigg( \varphi(v) + \mu^{*}\sum_{j=1}^{J} \lambda_{j}m_{j}(y,z,u,\theta)\Bigg)\, dP_{Y,Z}, \end{align*} whenever $\theta \in \Theta^{*}$. This concludes the proof. \end{proof} \subsubsection{Lemmas Supporting Theorem (ref) on Learnability} \begin{lemma} Suppose that $\mathcal{F}:=\{ f_{\alpha}(\cdot,\theta): \mathcal{X} \to \mathbb{R} : \theta \in \Theta, \alpha \in \mathcal{A}\}$ is a totally bounded parametric class of measurable real-valued functions on the metric space $(\mathcal{X},d)$, where $(\mathcal{A},d_{a})$ and $(\Theta,d_{\theta})$ are also metric spaces. Furthermore let $\mathcal{G}$ be a class of real-valued functions with each element $g(\cdot,\theta): \mathcal{X} \to \mathbb{R}$ defined by: \begin{align*} g(x,\theta):= \inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta), \end{align*} for some $f \in \mathcal{F}$, where $C(x,\theta)$ is a non-empty multifunction for each $(x,\theta)$ pair. Then for any probability measure $Q$ we have: \begin{align*} N(\varepsilon,\mathcal{G},||\cdot||_{Q,2}) \leq N(\varepsilon/2,\mathcal{F},||\cdot||_{Q,2}). \end{align*} \end{lemma} \begin{proof}[Proof of Lemma (ref)] As a parametric class of functions (parameterized by $(\alpha,\theta)$), the $\varepsilon/2-$cover of $\mathcal{F}$ can be characterized by a collection of points $\{(\alpha_{i},\theta_{i})\}_{i=1}^{n}$, where $n = N(\varepsilon/2,\mathcal{F},||\cdot||_{Q,2})$. Denote such a collection by $\mathcal{N}(\mathcal{F})$. We will show that for any $g \in \mathcal{G}$ there exists a pair $(\alpha',\theta') \in \mathcal{N}(\mathcal{F})$ such that: \begin{align*} |g(x,\theta) - f_{\alpha'}(x,\theta')| \leq \varepsilon. \end{align*} Since every $g \in \mathcal{G}$ can be expressed as: \begin{align*} g(x,\theta)=\inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta), \end{align*} it suffices to show there exists a pair $(\alpha',\theta') \in \mathcal{N}(\mathcal{F})$ such that: \begin{align*} \left|\inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta) - f_{\alpha'}(x,\theta')\right| \leq \varepsilon. \end{align*} Now let $\alpha^{*}$ be any value satisfying: \begin{align*} \left|\inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta) - f_{\alpha^{*}}(x,\theta)\right|\leq \varepsilon/2. \end{align*} That is, $\alpha^{*}$ is a $\varepsilon/2$ solution to the minimization problem. Now choose the pair $(\alpha',\theta') \in \mathcal{N}(\mathcal{F})$ such that $|f_{\alpha^{*}}(x,\theta) - f_{\alpha'}(x,\theta')|\leq \varepsilon/2$ (such a choice is always possible since $\mathcal{N}(\mathcal{F})$ is a $\varepsilon/2-$cover of $\mathcal{F}$). Then we have: \begin{align*} |g(x,\theta) - f_{\alpha'}(x,\theta')| &= \left|\inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta) - f_{\alpha'}(x,\theta')\right|\\ &\leq \left|\inf_{\alpha \in C(x,\theta)} f_{\alpha}(x,\theta) - f_{\alpha^{*}}(x,\theta)\right| + \left|f_{\alpha^{*}}(x,\theta) - f_{\alpha'}(x,\theta')\right|\\ &\leq\varepsilon/2 + \varepsilon/2\\ &=\varepsilon. \end{align*} This completes the proof. \end{proof} \begin{lemma} Let $\mathcal{F}$ be a symmetric class of measurable real-valued functions on a Polish space $\mathcal{X}$, and let $\psi=(x_{i})_{i=1}^{n}$ denote an arbitrary vector of $n$ points from $\mathcal{X}$. Then for any $\varepsilon>0$: \begin{align*} \E ||\mathfrak{R}_{n}||(\mathcal{F}) \leq \frac{2\varepsilon}{\sqrt{n}} + 2\text{Diam}_{\psi,2}(\mathcal{F}) \sqrt{ \frac{ \log N(\varepsilon,\mathcal{F},||\cdot||_{\psi,2})}{n}}. \end{align*} \end{lemma} \begin{proof}[Proof of Lemma (ref)] Note that: \begin{align*} n\E ||\mathfrak{R}_{n}||(\mathcal{F}) &= n \E\sup_{f \in \mathcal{F} }\left|\frac{1}{n}\sum_{i=1}^{n} \xi_{i} f(x_{i}) \right|= \E\sup_{f \in \mathcal{F} }\left|\sum_{i=1}^{n} \xi_{i} f(x_{i}) \right|. \end{align*} Now recall that the Rademacher process $\sum_{i=1}^{n} \xi_{i} f(x_{i})$ is sub-Gaussian with respect to the euclidean distance between the vectors $\left( f(x_{1}),\ldots, f(x_{n}) \right)$ and $\left( f'(x_{1}),\ldots, f'(x_{n}) \right)$ for $f,f'\in \mathcal{F}$. We denote this euclidean distance by $||f-f'||_{\psi,2}$ to emphasize that the vector $\psi = (x_{i})_{i=1}^{n}$ is fixed. Fix an minimal $\varepsilon-$net $\mathcal{F}^{*}\subset \mathcal{F}$ in the $||\cdot||_{\psi,2}$ norm. There exists at least one function $f' \in \mathcal{F}^{*}$ such that: \begin{align*} \E\sup_{f \in \mathcal{F}}\left|\sum_{i=1}^{n} \xi_{i} f(x_{i}) \right| \leq \E\sup_{f \in \mathcal{F} }\left|\sum_{i=1}^{n} \xi_{i} f(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i}) \right|+\varepsilon\sqrt{n}. \end{align*} (For example, we can always take $f'$ to be the element in $\mathcal{F}^{*}$ to be closest to $-f$ in the $||\cdot||_{\psi,2}$ norm, which is an element of $\mathcal{F}$ by symmetry.) Now for any $f \in \mathcal{F} $, let $f^{*}(f) \in \mathcal{F}^{*}$ be a function with $||f- f^{*}(f)||_{\psi,2} \leq \varepsilon$. Then: \begin{align*} &\left|\sum_{i=1}^{n} \xi_{i} f(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &= \left|\sum_{i=1}^{n} \xi_{i} f(x_{i}) - \sum_{i=1}^{n} \xi_{i} f^{*}(f)(x_{i}) + \sum_{i=1}^{n} \xi_{i} f^{*}(f)(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &\leq \left| \sum_{i=1}^{n} \xi_{i} f(x_{i}) - \sum_{i=1}^{n} \xi_{i} f^{*}(f)(x_{i})\right| + \left|\sum_{i=1}^{n} \xi_{i} f^{*}(f)(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &\leq \sup_{||f -f^{*}||_{\psi,2} \leq \varepsilon} \left| \sum_{i=1}^{n} \xi_{i} f(x_{i}) - \sum_{i=1}^{n} \xi_{i} f^{*}(x_{i})\right| + \sup_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &\leq \sup_{||f -f^{*}||_{\psi,2} \leq \varepsilon} \sum_{i=1}^{n} \left| f(x_{i}) - f^{*}(x_{i})\right| + \sup_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &\leq \sup_{||f -f^{*}||_{\psi,2} \leq \varepsilon} \sqrt{n} ||f-f^{*}||_{\psi,2} + \sup_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\\ &\leq \sqrt{n} \varepsilon + \sup_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|. \end{align*} Note we have used the inequality $||f-f'||_{\psi,1}\leq \sqrt{n} ||f-f'||_{\psi,2}$, where $||f-f'||_{\psi,1}$ denotes the $L^{1}$ distance between $f$ and $f'$ at the points $\psi=(x_{i})_{i=1}^{n}$. Now for any value $a>0$ we have: \begin{align*} &\exp \left( a \E \max_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right| \right)\\ &\leq \E \exp \left( a \max_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right| \right)\\ &= \E \max_{f^{*},f' \in \mathcal{F}^{*}} \exp \left(a \left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\right)\\ &\leq \sum_{f,f^{*}\in \mathcal{F}^{*}} \E \exp \left(a \left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right|\right)\\ &\leq \sum_{f,f^{*}\in \mathcal{F}^{*}} \exp \left(a^{2} \text{Diam}_{\psi,2}^{2}(\mathcal{F})/2\right)\\ &\leq N^{2}(\varepsilon,\mathcal{F},||\cdot||_{\psi,2}) \exp \left(a^{2} \text{Diam}_{\psi,2}^{2}(\mathcal{F})/2\right), \end{align*} where the second-last inequality follows from the fact that the Rademacher process is sub-Gaussian with parameter $\text{Diam}_{\psi,2}^{2}(\mathcal{F})$.\footnote{Recall a stochastic process $(\omega,t)\mapsto X(\omega,t)$ on a metric space $(T,d)$ is sub-Gaussian with respect to the metric $d$ if $\E \exp \left( \lambda \left( X_{t} - X_{s} \right) \right)\leq \exp(\lambda^{2}d(t,s)^{2}/2)$. For example, the Rademacher process is sub-Gaussian with respect to the euclidean metric.} Taking logs and dividing both sides by $a>0$, we have: \begin{align*} \E \max_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right| \leq \frac{2\log N(\varepsilon,\mathcal{F},||\cdot||_{\psi,2})}{a} + \frac{a\text{Diam}_{\psi,2}^{2}(\mathcal{F})}{2}. \end{align*} Minimizing the upper bound with respect to “$a$” yields:\footnote{The minimizing value is $a =2 \left( \log N(\varepsilon,\mathcal{F},||\cdot||_{\psi,2})/ \text{Diam}_{\psi,2}^{2}(\mathcal{F}) \right)^{1/2}$.} \begin{align*} \E \max_{f^{*},f' \in \mathcal{F}^{*}}\left|\sum_{i=1}^{n} \xi_{i} f^{*}(x_{i}) - \sum_{i=1}^{n} \xi_{i} f'(x_{i})\right| \leq 2\text{Diam}_{\psi,2}(\mathcal{F}) \sqrt{ \log N(\varepsilon,\mathcal{F},||\cdot||_{\psi,2})} \end{align*} We conclude that: \begin{align*} n\E ||\mathfrak{R}_{n}||(\mathcal{F}) \leq 2\sqrt{n}\varepsilon +2\text{Diam}_{\psi,2}(\mathcal{F}) \sqrt{ \log N(\varepsilon,\mathcal{F},||\cdot||_{\psi,2})}. \end{align*} \end{proof} \begin{lemma} Let $\mathcal{G}$ and $\mathcal{H}$ be two classes of functions and let $\mathcal{F} := \{ g + h : g \in \mathcal{G}, h \in \mathcal{H} \}$. Then: \begin{align*} N \left( \varepsilon, \mathcal{F}, ||\cdot|| \right) \leq N(\varepsilon/2,\mathcal{G},||\cdot||) N(\varepsilon/2,\mathcal{H},||\cdot||), \end{align*} where $||\cdot||$ is any norm. \end{lemma} \begin{remark} Note that a nearly identical proof of this result can be used to show that: $$N \left( \varepsilon, \mathcal{F}, ||\cdot|| \right) \leq N(\varepsilon\cdot a,\mathcal{G},||\cdot||) N(\varepsilon \cdot b,\mathcal{H},||\cdot||),$$ where $a, b> 0$ are any values satisfying $a+b=1$. \end{remark} \begin{proof}[Proof of Lemma (ref)] Suppose that $N \left( \varepsilon/2, \mathcal{G},||\cdot|| \right)=n$ and $N(\varepsilon/2,\mathcal{H},||\cdot|| )=m$. It suffices to show $N \left( \varepsilon, \mathcal{F}, ||\cdot|| \right)\leq nm$. Let $\mathcal{N}(\mathcal{G})$ denote the centres of the balls that obtain the $n-$cover of $\mathcal{G}$ and let $\mathcal{N}(\mathcal{H})$ denote the centres of the balls that obtain the $m-$cover of $\mathcal{H}$. Enumerate the elements of $\mathcal{N}(\mathcal{G})$ as $g_{1},\ldots, g_{n}$ and enumerate the elements of $\mathcal{N}(\mathcal{H})$ as $h_{1},\ldots,h_{m}$. Now define the following collections: \begin{align*} G_{j} := \{ g \in \mathcal{G} : ||g-g_{j}|| \leq \varepsilon/2\}, &&H_{k} := \{ h \in \mathcal{H} : ||h-h_{k}|| \leq \varepsilon/2\}, \end{align*} for $j=1,\ldots,n$ and $k=1,\ldots,m$. Then $\{G_{j}\}$ forms a $\varepsilon/2-$cover of $\mathcal{G}$ and $\{H_{k}\}$ forms a $\varepsilon/2-$cover of $\mathcal{H}$. Now for any $g_{j} \in \mathcal{N}(\mathcal{G})$ and $h_{k} \in \mathcal{N}(\mathcal{H})$ let $f_{jk}=g_{j}+h_{k}$, and define: \begin{align*} F_{jk}:= \{f : ||f-f_{jk}|| \leq \varepsilon \}. \end{align*} We will now argue that $\{F_{jk}\}$ is a $\varepsilon-$cover of $\mathcal{F}$. Note that if we can establish this, the proof will be complete, since there are only $nm$ sets $F_{jk}$. By construction each $F_{jk}$ is a $||\cdot||-$ball of radius $\varepsilon$, so it only remains to check that $\{F_{jk}\}$ covers $\mathcal{F}$. To do so, fix any $f \in \mathcal{F}$. Then by definition $f=g+h$ for some $g \in \mathcal{G}$ and $h \in \mathcal{H}$. Since $\{G_{j}\}$ forms a $\varepsilon/2-$cover of $\mathcal{G}$ and $\{H_{k}\}$ forms a $\varepsilon/2-$cover of $\mathcal{H}$, we know there is some $g_{j} \in \mathcal{N}(\mathcal{G})$ and some $h_{k} \in \mathcal{N}(\mathcal{H})$ such that $||g-g_{j}|| \leq \varepsilon/2$ and $||h-h_{k}||\leq \varepsilon/2$. But since $f_{jk}=g_{j}+h_{k}$ we have that: \begin{align*} ||f - f_{jk}|| = ||(g+h) - (g_{j}+h_{k})|| \leq ||g - g_{j}|| + ||h - h_{k}|| \leq \varepsilon/2 + \varepsilon/2 = \varepsilon, \end{align*} so that $f \in F_{jk}$, and so is an element of the cover $\{F_{jk}\}$. Since $f \in \mathcal{F}$ was arbitrary, we conclude that $\{F_{jk}\}$ covers $\mathcal{F}$. This completes the proof. \end{proof} \subsubsection{A Lemma Supporting Theorem (ref) and Lemma (ref)} \begin{lemma} Let $\delta^{**}$ be as in Lemma (ref). If $\delta \geq \delta^{**}\geq \varepsilon>0$, then: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( \mathscr{E}^{*}(\hat{\gamma})\geq \delta \right) \leq 1-\kappa. \end{align*} That is $\hat{\gamma} \in \mathscr{G}^{*}(\delta)$ with high probability when $\delta \geq \delta^{**}\geq \varepsilon>0$.\todo{Add reference} \end{lemma} \begin{proof} Throughout this proof, let $\lambda^{*}(\theta,\gamma)$, $\hat{\lambda}(\theta,\gamma)$, $\theta^{*}(\gamma)$, $\hat{\theta}(\gamma)$, $\gamma^{*}$ and $\hat{\gamma}$ be as in Remark (ref). Fix any $\delta>\delta^{**}$ (the case when $\delta=\delta^{**}$ follows from continuity). If $\sigma:=\mathscr{E}^{*}(\hat{\gamma}) \geq \delta \geq \varepsilon >0$, then: \begin{align*} \mathscr{E}^{*}(\hat{\gamma})&:= \sup_{\gamma \in \Gamma} \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma,\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) \\ &\leq \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)+3\varepsilon\\ &= \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\\ &\qquad\qquad+ \left(\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right)\\ &\qquad\qquad\qquad\qquad- \left(\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta}\max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right) +3\varepsilon\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\\ &\qquad\qquad- \left(\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta}\max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\right) +4\varepsilon \end{align*} Now note: \begin{align*} &\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)-\inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda)\\ &\leq \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)- \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda)+\varepsilon\\ &\leq \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda)- \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda)+2\varepsilon\\ &\leq \max_{\lambda\in \Lambda} P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda)- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))+2\varepsilon\\ &\leq P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*}))- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))+2\varepsilon. \end{align*} Similarly: \begin{align*} &\inf_{\theta \in \Theta}\max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \inf_{\theta \in \Theta} \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\gamma^{*},\lambda)\\ &\leq \inf_{\theta \in \Theta}\max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\theta,\hat{\gamma},\lambda) - \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda)+\varepsilon\\ &\leq \max_{\lambda\in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda) - \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda)+2\varepsilon\\ &\leq \max_{\lambda\in \Lambda}\P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\lambda) - \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*}))+2\varepsilon\\ &\leq \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma})) - \max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*}))+2\varepsilon. \end{align*} Thus we have: \begin{align*} \mathscr{E}^{*}(\hat{\gamma})&\leq P h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*}))- P h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma}))\\ &\qquad\qquad- \left(\max_{\lambda\in \Lambda} \P_{n} h_{\ell b}(\cdot,\hat{\theta}(\gamma^{*}),\gamma^{*},\lambda^{*}(\hat{\theta}(\gamma^{*}),\gamma^{*})) - \P_{n} h_{\ell b}(\cdot,\theta^{*}(\hat{\gamma}),\hat{\gamma},\hat{\lambda}(\theta^{*}(\hat{\gamma}),\hat{\gamma})) \right) + 8\varepsilon. \end{align*} Furthermore, $\sigma = \mathscr{E}^{*}(\hat{\gamma}) \geq \mathscr{E}^{*}(\gamma^{*})$ implies that $\hat{\gamma},\gamma^{*} \in \mathscr{G}(\sigma)$. Thus: \begin{align*} \mathscr{E}^{*}(\hat{\gamma}) \leq \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)} \max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|, \end{align*} which follows since $\varepsilon>0$ can be made arbitrarily small. Now define: \begin{align} E_{n,j}:= \left\{ \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)} \max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right| \leq T(\delta_{j}) \right\}, \end{align} and: \begin{align*} E_{n}:= \bigcap_{\{ j : \delta_{j}\geq \delta^{**}\}} E_{n,j}. \end{align*} Note by our choice of $\delta_{0}> 2\overline{H}$ we have: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( E_{n,0}^{c}\right)=0. \end{align*} Furthermore, from the uniform version of Hoeffding's inequality (e.g. koltchinskii2011oracle Theorem 4.6, p.71) we have: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( E_{n,j}^{c}\right) \leq \exp \left( -\frac{t_{j}^{2}}{2} \right), \end{align*} for each $j\in \mathbb{N}$. We conclude by the union bound that: \begin{align*} \sup_{P_{Y,Z} \in \mathcal{P}_{Y,Z}} P_{Y,Z}^{\otimes n} \left( E_{n}^{c}\right) \leq \sum_{\{j : \delta_{j} \geq \delta^{**}\}} \exp \left( -\frac{t_{j}^{2}}{2} \right) \leq \sum_{j=0}^{\infty} \exp \left( -\frac{t_{j}^{2}}{2} \right) \leq 1-\kappa. \end{align*} Now on the event $E_{n}$, for every $\delta \geq \delta^{**}$ we have: \begin{align*} \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\delta)} \max_{\lambda,\lambda' \in \Lambda} \left|(\P_{n}-P)\left( h_{\ell b}(\cdot,\theta,\gamma,\lambda) - h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) \right| \leq T(\delta). \end{align*} Now suppose by way of contradiction that $\{\mathscr{E}^{*}(\hat{\gamma}) \geq \delta\} \cap E_{n}\neq\emptyset$. Then on this event we have: \begin{align*} \sigma&:= \mathscr{E}^{*}(\hat{\gamma})\\ &\leq \sup_{\theta, \theta' \in \Theta} \sup_{\gamma, \gamma' \in \mathscr{G}^{*}(\sigma)} \max_{\lambda,\lambda' \in \Lambda} \left|\left( \P_{n} h_{\ell b}(\cdot,\theta,\gamma,\lambda) - \P_{n} h_{\ell b}(\cdot,\theta',\gamma',\lambda')\right) - \left( P h_{\ell b}(\cdot,\theta,\gamma,\lambda) - P h_{\ell b}(\cdot,\theta',\gamma',\lambda') \right)\right|\\ &\leq T(\sigma). \end{align*} However, note that this implies that $\sigma \leq \delta^{**}$ on the event $E_{n}$. But since $\sigma \geq \delta > \delta^{**}$ by assumption, we have a contradiction. We conclude that $\{\mathscr{E}^{*}(\hat{\gamma})\} \geq \delta\} \cap E_{n}=\emptyset$, or equivalently that $\{\mathscr{E}^{*}(\hat{\gamma}) \geq \delta\} \subseteq E_{n}^{c}$, where the event $E_{n}^{c}$ has probability at most $1-\kappa$. \end{proof} \section{Additional Details for the Examples} \subsection{Example (ref): Simultaneous Discrete Choice} \subsubsection{Verification of Assumptions (ref), (ref) and (ref)} We will now proceed to verify Assumption (ref), (ref) and (ref). First note that Assumption (ref) is trivially satisfied, since the probability space $(\Omega,\mathfrak{A},P)$ is complete, and both $\mathcal{U}$ and $\Theta$ are compact metric spaces with the euclidean norm. To verify Assumption (ref), note that the multifunction for the factual domain can be rewritten as:\todolt{aliprantis2006infinite have some other ways to check sufficient conditions} \begin{align} \bm G^{-} \left( Y,Z,\theta \right)= \left\{ u \in \mathcal{U} : \begin{array}{l} u_{k} \in [\pi_{k} ( Z_{k},Y_{-k};\theta ),1], \text{ if } Y_{k}=0,\\ u_{k} \in [-1,\pi_{k} (Z_{k},Y_{-k};\theta )], \text{ if } Y_{k}=1. \end{array}\right\}. \end{align} From here we conclude that, for any $u \in \mathcal{U}$: \begin{align} &d(u,\bm G^{-} \left( Y,Z,\theta \right))\nonumber\\ &= \max_{k} \bigg( \mathbbm{1}\{Y_{k}=0\} |\pi_{k} ( Z_{k},Y_{-k};\theta ) - u_{k}|_{+} + \mathbbm{1}\{Y_{k}=1\} |u_{k} - \pi_{k} ( Z_{k},Y_{-k};\theta ) |_{+}\bigg). \end{align} Under our assumptions, this distance is the maximum of $K$ measurable functions, and so is itself measurable. Since $u \in \mathcal{U}$ was arbitrary, by the result of himmelberg1975measurable (see also Theorem 1.3.3 in molchanov2017theory) this implies that $\bm G^{-}$ is an Effros-measurable multifunction (w.r.t. $\mathfrak{B}(\mathcal{Y})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\Theta)$), as desired. It is then easily seen that the conditional distribution of the vector $U$ given $(Y,Z)$ satisfies (ref) in Assumption (ref) using the multifunction in (ref) with $\theta=\theta_{0}$. To complete the verification of Assumption (ref), note that all the moment functions from the moment conditions in (ref) and (ref) are bounded in absolute value and Borel measurable (w.r.t. $\mathfrak{B}(\mathcal{Y})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\Theta)$). We now turn to the verification of Assumption (ref). Recall the counterfactual multifunction: \begin{align} \bm G^\star(Z,U,\theta,\gamma):=\left\{ y^\star \in \mathcal{Y} : y_{k}^\star=\mathbbm{1}\{ \pi_{k} \left(\gamma(Z_{k},y_{-k}^\star);\theta \right) \geq U_{k}\},\,\, k=1,\ldots,K. \right\}. \end{align} Close inspection reveals that: \begin{align} &d(y^\star,\bm G^{\star} \left(Z,U,\theta,\gamma\right))= \max_{k} \left|y_{k}^\star - \mathbbm{1}\{ \pi_{k} \left(\gamma(Z_{k},y_{-k}^\star);\theta \right) \geq U_{k}\}\right|. \end{align} Under our assumptions, this distance is also the maximum of $K$ measurable functions, and so is itself measurable. Since $y^\star \in \mathcal{Y}^\star$ was arbitrary, by the result of himmelberg1975measurable (see also Theorem 1.3.3 in molchanov2017theory) this implies that $\bm G^{\star}$ is an Effros-measurable multifunction (w.r.t. $\mathfrak{B}(\mathcal{Y})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\mathcal{Z})\otimes \mathfrak{B}(\Theta)\otimes \mathfrak{B}(\Gamma)$), as desired. Finally, it is easily seen that the conditional distribution of the vector $Y_{\gamma}^\star$ given $(Y,Z,U)$ satisfies (ref) in Assumption (ref) using the multifunction in (ref) with $\theta=\theta_{0}$. \subsubsection{Verification of Assumption (ref)} We will first verify Assumption (ref)(ii) for some $C_{2}\geq 0$ and $\delta>0$, and then will show that Assumption (ref)(i) is also satisfied for our choice of $\delta>0$. As was mentioned in the main text, under our current assumptions for this example Assumption (ref)(ii) is not satisfied. The issue is illustrated in Figures (ref) and (ref), and a case where Assumption (ref)(ii) is satisfied is illustrated in Figure (ref). The issue arises only when for some $k \in \{1,\ldots,K\}$ and some $z \in \mathcal{Z}$ and $y_{-k} \in \mathcal{Y}_{-k}$ we have: (i) the object of interest is $P(Y_{\gamma,k}^\star=1|Z_{k}=z',Y_{-k}=y_{-k}')$ or $P(Y_{\gamma,k}^\star=1)$, (ii) the counterfactual cutoff value $\pi_{k}(\gamma(z,y_{-k});\theta^{*})=0$ at some $\theta^{*} \in \partial \Theta^{*}$, and (iii) if $P(Y_{k}=1|Z_{k}=z',Y_{-k}=y_{-k}')\neq 0.5$, where $(z',y_{-k}')=\gamma(z,y_{-k})$. In this knife-edge case, a very small change in $\theta^{*}$ to some $\theta \notin \Theta^{*}$ can cause a discontinuous change in $P(Y_{\gamma,k}^\star=1|Z_{k}=z',Y_{-k}=y_{-k}')$ or $P(Y_{\gamma,k}^\star=1)$. \begin{figure}[!t] \caption{This figure illustrates a case violating Assumption (ref)(ii). The black dots $\bullet$ represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z,y_{-k})$. The red dots \color{red} $\bullet$ \color{black} represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z',y_{-k}')=\gamma(z,y_{-k})$. In the upper portion of the figure we have $\theta^{*} \in \Theta^{*}$, the median zero assumption is satisfied (three black dots $\bullet$ and three red dots \color{red} $\bullet$ \color{black} on either side of zero) and the maximum value of $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})$ at $\theta^{*}$ is obtained at $1/6$. However, in the bottom portion of the figure a small change in the value of $\theta^{*} \in \Theta^{*}$ to $\theta \notin \Theta^{*}$ causes a violation of the median zero assumption for the points $(z,y_{-k})$ and $(z',y_{-k}')$. At the new value $\theta \notin \Theta^{*}$ we have the maximum value of $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})$ is $1$. Note that the scale of the figure can be made arbitrarily small. } \end{figure} \begin{figure}[!t] \caption{This figure illustrates a case violating Assumption (ref)(ii). The black dots $\bullet$ represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z,y_{-k})$. The red dots \color{red} $\bullet$ \color{black} represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z',y_{-k}')=\gamma(z,y_{-k})$. In the upper portion of the figure we have $\theta^{*} \in \Theta^{*}$, the median zero assumption is satisfied (three black dots $\bullet$ and three red dots \color{red} $\bullet$ \color{black} on either side of zero) and the maximum value of $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})$ at $\theta^{*}$ is obtained at $1/2$. However, in the bottom portion of the figure a small change in the value of $\theta^{*} \in \Theta^{*}$ to $\theta \notin \Theta^{*}$ causes a violation of the median zero assumption for the point $(z',y_{-k}')$. At the new value $\theta \notin \Theta^{*}$ we have the maximum value of $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})$ is $1$. Note that the scale of the figure can be made arbitrarily small.} \end{figure} \begin{figure}[!t] \caption{This figure illustrates a case that does not violate Assumption (ref)(ii). The black dots $\bullet$ represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z,y_{-k})$. The red dots \color{red} $\bullet$ \color{black} represent equal probability masses ($1/6$) assigned by the conditional distribution of $U_{k}$ given $(z',y_{-k}')=\gamma(z,y_{-k})$. In the upper portion of the figure we have $\theta^{*} \in \Theta^{*}$, the median zero assumption is satisfied (three black dots $\bullet$ and three red dots \color{red} $\bullet$ \color{black} on either side of zero) and $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})=1/6$. In the bottom portion of the figure a small change in the value of $\theta^{*} \in \Theta^{*}$ to $\theta \notin \Theta^{*}$ causes a violation of the median zero assumption for the point $(z,y_{-k})$. However, at the new value $\theta \notin \Theta^{*}$ we still have that the maximum obtainable value of $P(Y_{\gamma}^\star=1|Z_{k}=z,Y_{-k}=y_{-k})$ is $1/6$. } \end{figure} To prevent such discontinuities in the value of the policy transform, we can introduce additional assumptions on the degree of smoothness of the distribution of $U_{k}$ around zero. In particular, instead of the moment conditions in (ref) and (ref) we propose imposing the constraints: \begin{align} P \left(U_{k}\leq \pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right) -0.5 \leq \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\},\\ 0.5 - P \left(U_{k}\leq\pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right) \leq \max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}, \end{align} for some $L_{0}> 0$, for $k=1,\ldots,K$, and for all $z,z' \in \mathcal{Z}$ and $y_{-k}, y_{-k}' \in \mathcal{Y}^{K-1}$. These constraints impose a local Lipschitzian constraint on the distribution of $U_{k}$ around zero. Note that by taking $L_{0}$ sufficiently large, these constraints will only be active when $\pi_{k}(z',y_{-k}';\theta)$ is close to zero. It is also easily verified that the new moment conditions implied by (ref) and (ref) also satisfy Assumption (ref). We claim that the constraints (ref) and (ref) imply that $U_{k}$ is median zero and median independent of $(Z,Y_{-k})$. To see this, note that $U_{k}$ has a median of zero given $(z,y_{-k})$ if and only if: \begin{enumerate}[label=(\Roman*)] • $\pi_{k}(z_{k},y_{-k};\theta) \leq 0$ and $P(U_{k}\leq \pi_{k}(z_{k},y_{-k};\theta) |Z=z_{k},Y_{-k}=y_{-k}) \leq 0.5$; or • $\pi_{k}(z_{k},y_{-k};\theta) > 0$ and $P(U_{k}> \pi_{k}(z_{k},y_{-k};\theta) |Z=z_{k},Y_{-k}=y_{-k})\leq 0.5$. \end{enumerate} The idea behind these conditions is illustrated in Figure (ref). Conversely, $U_{k}$ does not have a median of zero conditional on $(z,y_{-k})$ if and only if: \begin{enumerate}[label=(\roman*)] • $\pi_{k}(z_{k},y_{-k};\theta) > 0$ and $P(U_{k}\leq \pi_{k}(z_{k},y_{-k};\theta) |Z=z_{k},Y_{-k}=y_{-k}) < 0.5$; or • $\pi_{k}(z_{k},y_{-k};\theta) \leq 0$ and $P(U_{k}> \pi_{k}(z_{k},y_{-k};\theta) |Z=z_{k},Y_{-k}=y_{-k})< 0.5$. \end{enumerate} Note that if (i) holds then (ref) fails, and if (ii) holds then (ref) fails. This implies that if both (ref) and (ref) hold, then (i) and (ii) do not hold, and thus $U_{k}$ is median zero and median independent of $(Z,Y_{-k})$. However, note that it is possible that either (I) or (II) is satisfied but one of (ref) or (ref) fails, owing to the fact that together (ref) and (ref) are stronger than the median zero and median independence restrictions initially imposed in (ref) and (ref). \begin{figure}[!t] \caption{This figure illustrates three scenarios, each scenario involving a different allocation of probability mass for $U_{k}$, represented by the $6$ dots $\bullet$ representing equal probability mass, and a different value of the cutoff $\pi_{k}(z,y_{-k};\theta)$. In scenario (A), $\pi_{k}(z_{k},y_{-k};\theta) > 0$ and $P(Y_{k}=0|Z=z_{k},Y_{-k}=y_{-k})\leq 0.5$. In this case the median zero condition can be satisfied, for example, by the allocation of probability mass displayed in the figure. In scenario (B), $\pi_{k}(z_{k},y_{-k};\theta) > 0$ and $P(Y_{k}=0|Z=z_{k},Y_{-k}=y_{-k})> 0.5$. Here there is no way of satisfying the median zero assumption, since too much mass will always be assigned above zero. In scenario (C), $\pi_{k}(z_{k},y_{-k};\theta) < 0$ and $P(Y_{k}=0|Z=z_{k},Y_{-k}=y_{-k})> 0.5$. In this case the median zero condition can again be satisfied, for example, by the allocation of probability mass displayed in the figure. } \end{figure} We will now proceed to verify Assumption (ref). First recall from the discussion in the text that $\pi_{k}$ is a known measurable function of $(Z_{k},Y_{-k},\theta)$ that is linear in parameters $\theta$ and has a gradient (with respect to $\theta$) bounded away from zero for each $(z,y_{-k})$. Thus, $\pi_{k}$ is Lipschitz in $\theta$, and also satisfies a “reverse Lipschitz” condition; that is, for each $(z,y_{-k})$ we have: \begin{align*} L_{k}'||\theta-\theta^{*}||\leq |\pi_{k}(z,y_{-k};\theta)-\pi_{k}(z,y_{-k};\theta^{*})| \leq L_{k}||\theta-\theta^{*}||, \end{align*} for some $L_{k}',L_{k}>0$. Now, if one of the constraints (ref) or (ref) is violated, we have one of the following inequalities: \begin{align} P (\widetilde{U}_{k}\leq \pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -0.5 > \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\},\\ 0.5 -P (\widetilde{U}_{k}\leq \pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) > \max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}, \end{align} Subtracting (ref) from (ref) and taking $(z',y_{-k}')=\gamma(z,y_{-k})$, we have: \begin{align} &P \left(U_{k}\leq \pi_{k}(\gamma(z,y_{-k});\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) -P (\widetilde{U}_{k}\leq \pi_{k}(\gamma(z,y_{-k});\theta) | Z_{k}=z, Y_{-k}=y_{-k} )\nonumber\\ &\qquad=P \left(U_{k}\leq \pi_{k}(z',y_{-k}';\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) -P (\widetilde{U}_{k}\leq \pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} )\nonumber\\ &\qquad< \max\{L_{0}\pi_{k}(z',y_{-k}';\theta^{*}),0\} -\max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\nonumber\\ &\qquad\leq \max\{L_{0}\pi_{k}(z',y_{-k}';\theta^{*})-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\nonumber\\ &\qquad\leq L_{0} |\pi_{k}(z',y_{-k}';\theta^{*})-\pi_{k}(z',y_{-k}';\theta)|\nonumber\\ &\qquad\leq L_{0} L_{k} ||\theta - \theta^{*}||. \end{align} Furthermore, subtracting (ref) from (ref) and again taking $(z',y_{-k}')=\gamma(z,y_{-k})$, we have: \begin{align} &P (\widetilde{U}_{k}\leq \pi_{k}(\gamma(z,y_{-k});\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -P \left(U_{k}\leq \pi_{k}(\gamma(z,y_{-k});\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right)\nonumber\\ &\qquad=P (\widetilde{U}_{k}\leq \pi_{k}(z',y_{-k}';\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -P \left(U_{k}\leq \pi_{k}(z',y_{-k}';\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right)\nonumber\\ &\qquad< \max\{-L_{0}\pi_{k}(z',y_{-k}';\theta^{*}),0\} -\max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\nonumber\\ &\qquad\leq \max\{L_{0}\pi_{k}(z',y_{-k}';\theta)-L_{0}\pi_{k}(z',y_{-k}';\theta^{*}),0\} \nonumber\\ &\qquad\leq L_{0} |\pi_{k}(z',y_{-k}';\theta^{*})-\pi_{k}(z',y_{-k}';\theta)|\nonumber\\ &\qquad\leq L_{0} L_{k} ||\theta - \theta^{*}||. \end{align} From here we can deduce that Assumption (ref)(ii) is satisfied for any $\delta>0$ with $C_{2}=L_{0}L$ where $L= \min_{k} L_{k}$. To verify Assumption (ref)(i), we will first introduce the following Lemma and provide a sketch of its proof: \begin{lemma} Consider the simultaneous discrete choice environment of Example (ref), but with the new moment conditions (ref) and (ref) in place of (ref) and (ref). Now fix some value $\theta \in \Theta$. If there exists a random variable $U$ with distribution $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$ satisfying: \begin{align} P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right) -0.5 \leq \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}, \\ 0.5 - P \left(U_{k}\leq\pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right) \leq \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}, \end{align} for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$, then $\theta \in \Theta^{*}$. \end{lemma} \begin{remark} Note that, precisely because of the result in this Lemma, the new moment conditions implied by (ref) and (ref) satisfy the no-backtracking principle from Remark (ref). Indeed, this Lemma shows that (ref) and (ref) are sufficient to characterize the identified set. Since these moment conditions do not depend on the counterfactual $\gamma$ of interest, the no-backtracking principle is satisfied. \end{remark} \begin{proof}[Proof] Note by assumption there exists a random variable $U$ with distribution $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$ satisfying (ref) and (ref) for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$. Take $\widetilde{U}$ to be a random vector satisfying: \begin{align*} P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) = P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right), \end{align*} for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$, so that $\widetilde{U}$ satisfies (ref) and (ref). We must show that we can fix probabilities of the form $P(\widetilde{U}_{k} \leq \pi_{k}(z',y_{-k}';\theta)|Z_{k}=z,Y_{-k}=y_{-k})$ for $(z',y_{-k}')\neq (z,y_{-k})$ in a way that satisfies the remaining constraints from (ref) and (ref), as well as the constraints: \begin{align*} &P(U_{k} \leq \pi_{k}(z',y_{-k}';\theta)|Z_{k}=z,Y_{-k}=y_{-k}) \leq P(U_{k} \leq \pi_{k}(z,y_{-k};\theta)|Z_{k}=z,Y_{-k}=y_{-k}), \end{align*} if $\pi_{k}(z',y_{-k}';\theta) \leq \pi_{k}(z,y_{-k};\theta)$, and: \begin{align*} P(U_{k} \leq \pi_{k}(z,y_{-k};\theta)|Z_{k}=z,Y_{-k}=y_{-k}) \leq P(U_{k} \leq \pi_{k}(z',y_{-k}';\theta)|Z_{k}=z,Y_{-k}=y_{-k}), \end{align*} if $\pi_{k}(z,y_{-k};\theta) \leq \pi_{k}(z',y_{-k}';\theta)$. However, such an allocation of probability is clearly always possible. \end{proof} The contrapositive of this result says that if $\theta \notin \Theta^{*}$, then there is no random variable $U$ with distribution $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$ satisfying (ref) and (ref); in other words, if $\theta \notin \Theta^{*}$, then every distribution $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$ violates either (ref) or (ref). Thus, the Lemma suggests that when analysing violations of the moment conditions in order to verify Assumption (ref)(i), it suffices to focus on the moment conditions (ref) and (ref). Finally, there is an important property that will be utilized repeatedly when verifying Assumption (ref)(i): for any $P_{U|Y,Z}\in \mathcal{P}_{U|Y,Z}(\theta)$ and any $P_{U'|Y,Z}\in \mathcal{P}_{U|Y,Z}(\theta')$, we must have: \begin{align} P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} \right) = P \left(U_{k}'\leq \pi_{k}(z,y_{-k};\theta') | Z_{k}=z, Y_{-k}=y_{-k} \right) \end{align} for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$. Indeed, this property follows from the fact that both $P_{U|Y,Z}$ and $P_{U'|Y,Z}$ satisfy the support restrictions for the simultaneous discrete choice model at $\theta$ and $\theta'$, respectively, and thus they must both rationalize the same observed conditional choice probabilities. Now we are prepared to verify Assumption (ref)(i). First fix some value of $\theta \notin \Theta$. If $\mathcal{P}_{U|Y,Z}(\theta)$ is empty, then Assumption (ref)(i) is satisfied for any $C_{1},\delta>0$. Thus, we will focus attention on the non-trivial case where $\mathcal{P}_{U|Y,Z}(\theta)$ is non-empty. Note that if $P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})=0.5$ for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$, then (ref) and (ref) will be satisfied for any $P_{U|Y,Z} \in \mathcal{P}_{U|Y,Z}(\theta)$. By Lemma (ref) this implies $\theta \in \Theta^{*}$, contradicting the fact that $\theta \notin \Theta$. We conclude that if $P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})=0.5$ for $k=1,\ldots,K$ and for every $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$ then $\theta \notin \Theta^{*}$ implies $\mathcal{P}_{U|Y,Z}(\theta)$ is empty, a case we have ruled out. Thus, we will take as a starting point that there exists at least one $k$ and one pair $(z,y_{-k}) \in \mathcal{Z} \times \mathcal{Y}^{K-1}$ such that $P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})\neq 0.5$. Now define: \begin{align} \tau:= \min_{k}\min_{(z,y_{-k})} |0.5 - P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})| && s.t. &&|0.5 - P(Y_{k}=1|Z=z,Y_{-k}=y_{-k})|>0. \end{align} By assumption and by construction we have $\tau>0$. We now consider violations of the moment conditions (ref) and (ref) in turn. First, consider a violation of (ref). In particular, for our fixed value of $\theta \notin \Theta$ suppose: \begin{align} P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -0.5 > \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}, \end{align} for some $k$ and $(z,y_{-k})$ pair, where $\widetilde{U}_{k}$ is a subvector of $\widetilde{U}$ whose distribution is a member of $\mathcal{P}_{U|Y,Z}(\theta)$. Furthermore, let $\theta^{*} \in \Theta^{*}$ be the element of $\Theta^{*}$ closest to $\theta$ (such an element exists since $\Theta^{*}$ will be closed, which follows from continuity of the payoff functions). There are four cases to consider: \begin{enumerate} • $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ and $\pi_{k}(z,y_{-k};\theta)\leq 0$. Then we have: \begin{align} \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\} = 0. \end{align} However, since $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ it must be that: \begin{align*} 0.5 \geq P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) = P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ), \end{align*} where we have used property (ref) and the fact that $\theta^{*}$ satisfies both (ref) and (ref). But then this implies: \begin{align} P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - 0.5 \leq 0. \end{align} Combining (ref) and (ref) contradicts the assumption of (ref). Thus, this case is not possible under the assumption of (ref). • $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ and $\pi_{k}(z,y_{-k};\theta)> 0$. Then we have: \begin{align} \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\} = L_{0}\pi_{k}(z,y_{-k};\theta). \end{align} However, since $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ then it must be that: \begin{align*} 0.5 \geq P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) = P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ), \end{align*} where we have used property (ref) and the fact that $\theta^{*}$ satisfies both (ref) and (ref). But then this implies: \begin{align} P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k}) - 0.5 \leq 0. \end{align} Combining (ref) and (ref) contradicts the assumption of (ref). Thus, this case is not possible under the assumption of (ref). • $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ and $\pi_{k}(z,y_{-k};\theta)\leq 0$. Then we have: \begin{align*} \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}=0. \end{align*} Then: \begin{align*} &P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -0.5 -\max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &=P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -0.5\\ &\geq \tau, \end{align*} where the last line follows from the fact that $P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} )-0.5>0$ by assumption of (ref) and the fact $\pi_{k}(z,y_{-k};\theta)\leq 0$, and by the definition of $\tau$ from (ref). • $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ and $\pi_{k}(z,y_{-k};\theta)> 0$. First note that by assumption we have: \begin{align*} &P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} )-0.5 - \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &>0\\ &\geq P (U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} )-0.5 - \max\{L_{0}\pi_{k}(z,y_{-k};\theta^{*}),0\}. \end{align*} Using (ref) and the fact that $\pi_{k}(z,y_{-k};\theta^{*}) >0$ and $\pi_{k}(z,y_{-k};\theta)> 0$, this implies $\pi_{k}(z,y_{-k};\theta^{*}) > \pi_{k}(z,y_{-k};\theta)$. Now let $\theta'$ be a convex combination of $\theta^{*}$ and $\theta$ satisfying: \begin{align*} P(U_{k}'\leq \pi_{k}(z,y_{-k};\theta') | Z_{k}=z, Y_{-k}=y_{-k} ) - 0.5 - L_{0}\pi_{k}(z,y_{-k};\theta')=0, \end{align*} for some selection $U_{k}'$. Such an element always exists by linearity of $\pi_{k}$. Then: \begin{align*} &P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - 0.5 - \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &=P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - 0.5 - \max\{L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &\qquad+ L_{0}\pi_{k}(z,y_{-k};\theta') - L_{0}\pi_{k}(z,y_{-k};\theta')\\ &= L_{0}\pi_{k}(z,y_{-k};\theta') - L_{0}\pi_{k}(z,y_{-k};\theta)\\ &= L_{0}|\pi_{k}(z,y_{-k};\theta') - \pi_{k}(z,y_{-k};\theta)|\\ &\geq L_{0}L_{k}'||\theta'-\theta||\\ &\geq L_{0}L_{k}'||\theta^{*}-\theta||. \end{align*} In the third last line we used the fact that $\pi_{k}(z,y_{-k};\theta^{*}) > \pi_{k}(z,y_{-k};\theta)$. In the second last line we have used the reverse Lipschitz condition, and in the final line we have used the fact that $\theta'$ lies between $\theta$ and $\theta^{*}$, by virtue of being a convex combination of these elements. \end{enumerate} Next, consider a violation of (ref). In particular, for our fixed $\theta \notin \Theta$ suppose: \begin{align} 0.5 - P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) > \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}, \end{align} for some $k$ and $(z,y_{-k})$ pair, where $\widetilde{U}_{k}$ is a subvector of $\widetilde{U}$ whose distribution is is a member of $\mathcal{P}_{U|Y,Z}(\theta)$. Again, let $\theta^{*} \in \Theta^{*}$ be the element of $\Theta^{*}$ closest to $\theta$. There are again four cases to consider: \begin{enumerate} • $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ and $\pi_{k}(z,y_{-k};\theta)\leq 0$. First note that by assumption we have: \begin{align*} &0.5 - P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &>0\\ &\geq 0.5 - P (U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} ) - \max\{-L_{0}\pi_{k}(z,y_{-k};\theta^{*}),0\}. \end{align*} Using (ref) and the fact that $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ and $\pi_{k}(z,y_{-k};\theta)\leq 0$, this implies $\pi_{k}(z,y_{-k};\theta^{*}) < \pi_{k}(z,y_{-k};\theta)$. Now let $\theta'$ be a convex combination of $\theta^{*}$ and $\theta$ satisfying: \begin{align*} 0.5- P(U_{k}'\leq \pi_{k}(z,y_{-k};\theta') | Z_{k}=z, Y_{-k}=y_{-k} ) + L_{0}\pi_{k}(z,y_{-k};\theta')=0, \end{align*} for some selection $U_{k}'$. Such an element always exists by linearity of $\pi_{k}$. Then: \begin{align*} &0.5- P(U_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &=0.5- P(U_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) - \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &\qquad+ L_{0}\pi_{k}(z,y_{-k};\theta') - L_{0}\pi_{k}(z,y_{-k};\theta')\\ &= L_{0}\pi_{k}(z,y_{-k};\theta) - L_{0}\pi_{k}(z,y_{-k};\theta')\\ &= L_{0}|\pi_{k}(z,y_{-k};\theta) - \pi_{k}(z,y_{-k};\theta')|\\ &\geq L_{0}L_{k}'||\theta-\theta'||\\ &\geq L_{0}L_{k}'||\theta-\theta^{*}||. \end{align*} In the third last line we used the fact that $\pi_{k}(z,y_{-k};\theta^{*}) < \pi_{k}(z,y_{-k};\theta)$. In the second last line we have used the reverse Lipschitz condition, and in the final line we have used the fact that $\theta'$ lies between $\theta$ and $\theta^{*}$, by virtue of being a convex combination of these elements. • $\pi_{k}(z,y_{-k};\theta^{*}) \leq 0$ and $\pi_{k}(z,y_{-k};\theta)> 0$. Then we have: \begin{align*} \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}=0. \end{align*} Then: \begin{align*} & 0.5 - P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) -\max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\}\\ &=0.5 - P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} )\\ &\geq \tau, \end{align*} where the last line follows from the fact that $0.5 - P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} )>0$ by assumption of (ref) and the fact $\pi_{k}(z,y_{-k};\theta)> 0$, and by the definition of $\tau$ from (ref). • $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ and $\pi_{k}(z,y_{-k};\theta)\leq 0$. Then we have: \begin{align} \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\} = -L_{0}\pi_{k}(z,y_{-k};\theta). \end{align} However, since $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ then it must be that: \begin{align*} 0.5 \leq P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) = P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ), \end{align*} where we have used property (ref) and the fact that $\theta^{*}$ satisfies both (ref) and (ref). But then this implies: \begin{align} 0.5 - P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) \leq 0. \end{align} Combining (ref) and (ref) contradicts the assumption of (ref). Thus, this case is not possible under the assumption of (ref). • $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ and $\pi_{k}(z,y_{-k};\theta)> 0$. Then we have: \begin{align} \max\{-L_{0}\pi_{k}(z,y_{-k};\theta),0\} = 0. \end{align} However, since $\pi_{k}(z,y_{-k};\theta^{*}) > 0$ then it must be that: \begin{align*} 0.5 \leq P \left(U_{k}\leq \pi_{k}(z,y_{-k};\theta^{*}) | Z_{k}=z, Y_{-k}=y_{-k} \right) = P (\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ), \end{align*} where we have used property (ref) and the fact that $\theta^{*}$ satisfies both (ref) and (ref). But then this implies: \begin{align} 0.5 - P(\widetilde{U}_{k}\leq \pi_{k}(z,y_{-k};\theta) | Z_{k}=z, Y_{-k}=y_{-k} ) \leq 0. \end{align} Combining (ref) and (ref) contradicts the assumption of (ref). Thus, this case is not possible under the assumption of (ref). \end{enumerate} Combining everything, we conclude that Assumption (ref) holds with $C_{1} = L_{0} L'$ and $\delta = \tau/(L_{0} L')$, where $L' = \min_{k} L_{k}'$. \subsubsection{Verification of Learnability} By the assumed linearity of $\pi_{k}$ with respect to $\theta$, and since $\pi_{k}$ depends only on the subvector $\theta_{k}$ of $\theta$, the function $(u,\theta) \mapsto \pi_{k}(\gamma(z,y_{-k});\theta) - u$ is a hyperplane in $\mathbb{R}^{d_{k}}$ for each $(z,y_{-k})$, where $d_{k}$ is the dimension of $\theta_{k}$. By Lemma 2.6.15 in van1996weak, for example, $\Phi$ is a Vapnik-Chervonenkis (VC) class with VC dimension at most $d_{k}+2$. Furthermore, recalle that $\Phi$ can be taken to be uniformly bounded in absolute value by $1$. Using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \Phi, ||\cdot||_{Q,2} \right) = O(1), \end{align*} so that $\Phi$ easily satisfies the entropy growth condition. Now let $j$ index a generic moment function: \begin{align*} m_{j}(Y_{-k},Z,U,\theta) = \left(\mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\} - \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\} - 0.5\right)\mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\}, \end{align*} and let $\mathcal{M}_{j}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j}= \left\{ m_{j}(\cdot,u,\theta):\mathcal{Y}\times\mathcal{Z} \to \mathbb{R} : (u,\theta) \in \mathcal{U}\times\Theta \right\}. \end{align*} Note that the values $(z',y_{-k}')$ are not arguments of the function, but instead are associated with the index $j$. Since $\pi_{k}$ takes values in the interval $[-1,1]$, the class $\mathcal{M}_{j}$ is uniformly bounded. We claim that there exists no set of size $2$ shattered by $\mathcal{M}_{j}$, implying $\mathcal{M}_{j}$ is a VC-subgraph class. We will prove this by way of contradiction. In particular, suppose that there exists two points $(y_{1},z_{1})$ and $(y_{2},z_{2})$, and values $t_{1},t_{2} \in \mathbb{R}$ such that: \begin{align} \left| \left\{\begin{bmatrix}\mathbbm{1}\{m_{j}(y_{1},z_{1},u,\theta) \geq t_{1}\}\\ \mathbbm{1}\{m_{j}(y_{2},z_{2},u,\theta) \geq t_{2}\}\end{bmatrix} : (u,\theta) \in \mathcal{U} \times \Theta \right\}\right| = 4. \end{align} In other words, we suppose the set $\{(y_{1},z_{1}),(y_{2},z_{2})\}$ is shattered by $\mathcal{M}_{j}$, and that $t_{1},t_{2} \in \mathbb{R}$ witness the shattering. We have: \begin{align*} m_{j}(y_{1},z_{1},u,\theta) = \left(\mathbbm{1}\{u_{k}\leq \pi_{k}(z',y_{-k}';\theta)\} - \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\} - 0.5\right)\mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\},\\ m_{j}(y_{2},z_{2},u,\theta) = \left(\mathbbm{1}\{u_{k}\leq \pi_{k}(z',y_{-k}';\theta)\} - \max\{L_{0}\pi_{k}(z',y_{-k}';\theta),0\} - 0.5\right)\mathbbm{1}\{z_{2,k}=z, y_{2,-k}=y_{-k}\}. \end{align*} Now consider two cases: \begin{enumerate} • $\mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\}=\mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\}$: In this case the two functions $m_{j}(y_{1},z_{1},u,\theta)$ and $m_{j}(y_{2},z_{2},u,\theta)$ are identical for all $(u,\theta) \in \mathcal{U} \times \Theta$. This means (ref) is impossible, since at least one of the vectors $(1,0)$ and $(0,1)$ cannot be picked out by $\mathcal{M}_{j}$. • $\mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\}\neq \mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\}$: In this case at least one of the functions $m_{j}(y_{1},z_{1},u,\theta)$ or $m_{j}(y_{2},z_{2},u,\theta)$ is the zero function. Again, this means (ref) is impossible. For example, if $m_{j}(y_{1},z_{1},u,\theta)$ is the zero function, then it is impossible for $\mathcal{M}_{j}$ to pick out both $(0,0)$ and $(1,0)$ or both $(0,1)$ and $(1,1)$. \end{enumerate} Since $(y_{1},z_{1})$ and $(y_{2},z_{2})$ were arbitrary, we conclude that there exists no set of size $2$ shattered by $\mathcal{M}_{j}$. This implies that $\mathcal{M}_{j}$ is a VC-subgraph class, and using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \mathcal{M}_{j}, ||\cdot||_{Q,2} \right) = O(1). \end{align*} Thus, $\mathcal{M}_{j}$ easily satisfies the entropy growth condition. Finally, let $j'$ index a generic moment function: \begin{align*} m_{j}(Y_{-k},Z,U,\theta) = \left(0.5 - \mathbbm{1}\{U_{k}\leq \pi_{k}(z',y_{-k}';\theta)\}-\max\{-L_{0}\pi_{k}(z',y_{-k}';\theta),0\}\right) \mathbbm{1}\{Z_{k}=z, Y_{-k}=y_{-k}\}, \end{align*} and let $\mathcal{M}_{j'}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j'}= \left\{ m_{j'}(\cdot,u,\theta):\mathcal{Y}\times\mathcal{Z} \to \mathbb{R} : (u,\theta) \in \mathcal{U}\times\Theta \right\}. \end{align*} A nearly identical argument as for $\mathcal{M}_{j}$ reveals that $\mathcal{M}_{j'}$ is a VC-subgraph class and thus trivially satisfies the entropy growth condition. We conclude using Theorem (ref)(ii) that our class of policies $\Gamma$ is PAMPAC learnable with a rate of convergence of $O(n^{-1/2})$. \subsection{Example (ref): Program Evaluation} \subsubsection{Verification of Assumptions (ref), (ref) and (ref)} We will now proceed to verify Assumption (ref), (ref) and (ref). First note that Assumption (ref) is trivially satisfied, since the probability space $(\Omega,\mathfrak{A},P)$ is complete, $\mathcal{U}$ is a compact subset of euclidean space, and $\Theta$ is a Polish space; in particular, since $\mathcal{Z}$ (and thus also $\mathcal{X}$) is finite, $\mathcal{G}$ can be considered as the set of all positive measurable functions $g:\mathcal{Z}\to [0,1]$, in which case each $g \in \mathcal{G}$ has an equivalent representation as a vector in $[0,1]^{|\mathcal{Z}|}$. The same logic applies to each $t \in \mathcal{T}$. Next, let us recall the multifunction: \begin{align} \bm G^{-} \left( Y,D,Z,\theta \right):= \text{cl}\left\{ (U_{0},U_{1},U) \in \mathcal{U} : \begin{array}{l} Y = U_{0}(1-D) + U_{1}D, \\ D = \mathbbm{1}\{g(Z) \geq U\} \end{array} \right\}. \end{align} Close inspection of this multifunction shows that: \begin{align} \bm G^{-} \left( y,d,z,\theta \right)=\begin{cases} \{ y \} \times [\underline{Y},\overline{Y}] \times [g(z),1], &\text{ if } d=0,\\ [\underline{Y},\overline{Y}] \times \{ y \} \times [0,g(z)], &\text{ if } d=1. \end{cases} \end{align} Now for any $(u_{0},u_{1},u)\in \mathcal{U}$ we have: \begin{align} &d((u_{0},u_{1},u),\bm G^{-} \left( Y,D,Z,\theta \right))\nonumber\\ &= D \max\{|u_{0} - Y|,g(Z)-u \} + (1-D)\max\{|u_{1} - Y|,Z-g(z) \}. \end{align} Since $g \in \mathcal{G}$ is measurable by definition, from here it is easily verified that the distance above is measurable with respect to $\mathfrak{B}(\mathcal{Y})\otimes\mathfrak{B}(\mathcal{D})\otimes\mathfrak{B}(\mathcal{Z})$. Since $(u_{0},u_{1},u) \in \mathcal{U}$ was arbitrary, by the result of himmelberg1975measurable (see also Theorem 1.3.3 in molchanov2017theory) this implies that $\bm G^{-}$ is an Effros-measurable multifunction, as desired. Modulo changes in notation, it is easily seen that the conditional distribution of the vector $(U_{0},U_{1},U)$ given $(Y,D,Z)$ satisfies (ref) in Assumption (ref) using the multifunction in (ref) with $g(\cdot)=g_{0}(\cdot)$. Finally, note that all of the moment functions in the moment conditions (ref) - (ref) are measurable and bounded by $1$, and the moment functions from the moment conditions in (ref) and (ref) are measurable and bounded by $\max\{|\underline{Y}|,|\overline{Y}|\}$. Turning to the counterfactual domain, recall the multifunction: \begin{align} \bm G^\star(Z,U_{0},U_{1},U,\theta,\gamma):=\left\{ (Y_{\gamma}^\star,D_{\gamma}^\star) \in \mathcal{Y} \times \{0,1\} : \begin{array}{l} Y_{\gamma}^\star = U_{0}(1-D_{\gamma}^\star) + U_{1}D_{\gamma}^\star, \\ D_{\gamma}^\star = \mathbbm{1}\{g(\gamma(Z)) \geq U\} \end{array} \right\}. \end{align} Note here we take $\mathcal{Y}^\star = \mathcal{Y}$, although this is not necessary. Furthermore, close inspection of this multifunction shows that: \begin{align} \bm G^\star(z,u_{0},u_{1},u,\theta,\gamma)=\begin{cases} (u_{1},1), &\text{ if } u \leq g(\gamma(z)),\\ (u_{0},0), &\text{ if } g(\gamma(z)) < u. \end{cases} \end{align} In this case, the counterfactual map in (ref) is single-valued. In this case, Effros measurability is equivalent to the usual notion of measurability for functions, and measurability of $\bm G^\star$ follows from familiar arguments after noting that both $g$ and $\gamma$ are measurable functions. Finally, modulo changes in notation, it is easily seen that the conditional distribution of the vector $(Y_{\gamma}^\star,D_{\gamma}^\star)$ given $(Y,D,Z,U_{0},U_{1},U)$ satisfies (ref) in Assumption (ref) using the multifunction in (ref) with $g(\cdot)=g_{0}(\cdot)$. \subsubsection{Verification of Assumption (ref)} First we focus on (ref) - (ref).\todolt{There is some question for this assumption when the moment conditions do not depend on $\theta$...} Since these moments do not depend on $t \in \mathcal{T}$, to verify Assumption (ref) it suffices to focus on the parameter $g \in \mathcal{G}$. From the moment conditions (ref) and (ref) we have: \begin{align} g(z_{0},x) = P(D=1|Z=z_{0},X=x) \iff \begin{cases} \E[\left(D - g_{0}(z_{0},x)\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] \leq 0 \\ \E[\left(g_{0}(z_{0},x)-D\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}] \leq 0 \end{cases}, \end{align} and from (ref) and (ref) we have: \begin{align} g_{0}(z_{0},x) = P(U \leq g_{0}(z_{0},x)| X=x) \iff \begin{cases} \E[\left(\mathbbm{1}\{U \leq g_{0}(z_{0},x)\} - g_{0}(z_{0},x)\right)\mathbbm{1}\{X=x\}]\leq 0\\ \E[\left(g_{0}(z_{0},x) - \mathbbm{1}\{U \leq g_{0}(z_{0},x)\}\right)\mathbbm{1}\{X=x\}]\leq 0 \end{cases}, \end{align} For notational simplicity, let $g_{0}(z) := g_{0}(z_{0},x)$ for $z=(z_{0},x)$. From (ref) we see that $g_{0}(z)$ is point-identified. Define: \begin{align*} \mathcal{G}^{*} = \{ g : (g,t) \in \Theta^{*} \text{ for some $t\in \mathcal{T}$}\}. \end{align*} Then point-identification of $g_{0}$ implies that $\mathcal{G}^{*}$ is a singleton, and that for any $g \in \mathcal{G}$: \begin{align*} d(g,\mathcal{G}^{*}) = \max_{z \in \mathcal{Z}} |g(z) - g_{0}(z)|. \end{align*} From here it is straightforward to use conditions (ref) and (ref) to argue that part (i) of Assumption (ref) is satisfied with $C_{1}=1$ for any $\delta>0$. In particular, suppose $g \notin \mathcal{G}^{*}$, and that $z^{*} \in \mathcal{Z}$ satisfies: \begin{align*} d(g,\mathcal{G}^{*}) = \max_{z \in \mathcal{Z}} |g(z) - g_{0}(z)|=|g(z^{*}) - g_{0}(z^{*})|. \end{align*} Without loss of generality, suppose that $g(z^{*})>g_{0}(z^{*})$. Then from (ref) we have: \begin{align*} \E[\left(g_{0}(z^{*})-D\right)\mathbbm{1}\{Z=z^{*}\}]= 0<\E[\left(g(z^{*})-D\right)\mathbbm{1}\{Z=z^{*}\}]. \end{align*} Thus: \begin{align*} \E[\left(g(z^{*})-D\right)\mathbbm{1}\{Z=z^{*}\}] &= \E[\left(g(z^{*})-D\right)\mathbbm{1}\{Z=z^{*}\}] - \E[\left(g_{0}(z^{*})-D\right)\mathbbm{1}\{Z=z^{*}\}]\\ &= g(z^{*}) - g_{0}(z^{*})\\ &= |g(z^{*}) - g_{0}(z^{*})|\\ &= d(g,\mathcal{G}^{*}). \end{align*} Now to complete the verification of part (i) of Assumption (ref) we turn to (ref) - (ref), which can be written as: \begin{align} \E\left[t(z_{0},x)-\mathbbm{1}\{Z=z_{0},X=x\}\right] &= 0,\qquad\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X}, \end{align} and: \begin{align} \E\left[U_{d}\left(\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{X=x\}t(z_{0},x) \right)\right] &\leq 0, \,\,\forall z_{0} \in \mathcal{Z}_{0},\, x \in \mathcal{X},\, d\in \{0,1\}. \end{align} Since these moments do not depend on $g \in \mathcal{G}$, to verify Assumption (ref) for these moments it suffices to focus on the parameter $t \in \mathcal{T}$. Now define: \begin{align*} \mathcal{T}^{*} = \{ t : (g,t) \in \Theta^{*} \text{ for some $g\in \mathcal{G}$}\}. \end{align*} From (ref) it is clear that $t_{0}$ is also point identified. Since $g_{0}$ is also point identified we have $\Theta^{*} = \{g_{0}\}\times \{t_{0}\}$. Because of this, we claim that it suffices to focus on the conditions from (ref); indeed, $t \notin \mathcal{T}^{*} \iff t\neq t_{0}$ implies that $t \notin \mathcal{T}^{*}$ if and only if (ref) is violated. Now consider any $t \notin \mathcal{T}^{*}$ and let $(z_{0}^{*},x^{*})$ satisfy: \begin{align*} (z_{0}^{*},x^{*}) = \argmax_{z_{0},x} |t(z_{0},x) - t_{0}(z_{0},x)|. \end{align*} Without loss of generality we can suppose $t(z_{0},x) > t_{0}(z_{0},x)$. Then: \begin{align*} \E\left[t(z_{0}^{*},x^{*})-\mathbbm{1}\{Z=z_{0}^{*},X=x^{*}\}\right] &= \E\left[t(z_{0}^{*},x^{*})-\mathbbm{1}\{Z=z_{0}^{*},X=x^{*}\}\right] - \E\left[t_{0}(z_{0}^{*},x^{*})-\mathbbm{1}\{Z=z_{0}^{*},X=x^{*}\}\right] \\ &=t(z_{0}^{*},x^{*}) -t_{0}(z_{0}^{*},x^{*})\\ &= | t(z_{0}^{*},x^{*}) -t_{0}(z_{0}^{*},x^{*})|\\ &=d(t,\mathcal{T}^{*}). \end{align*} Combining everything, if $\mathcal{J}$ indexes all the moment constraints and if $\theta \notin \Theta^{*}$ with $\theta=(g,t)$, then we know: \begin{align*} \inf_{P_{U_{0},U_{1},U|Y,D,Z} \in \mathcal{P}_{U_{0},U_{1},U|Y,D,Z}(\theta) } \max_{j \in \mathcal{J}} | \E[m_{j}(y,d,z,u_{0},u_{1},u,\theta)]|_{+} \geq \max\{d(g,\mathcal{G}^{*}), d(t,\mathcal{T}^{*})\} \geq d(\theta,\Theta^{*}). \end{align*} Conclude that Assumption (ref) is satisfied with $C_{1}=1$ for any $\delta>0$.\\ For part (ii) of Assumption (ref), we claim that we can set $C_{2} = 1$. To show why, we will apply Lemma (ref) to our environment. First note that $\varphi$ is the identity function when we are interested in $\E_{P}[Y_{\gamma}^\star]$. Thus $L_{\varphi}=1$ in Lemma (ref). Next, note from the definition of our support restrictions $\bm G^{-}$ and $\bm G^\star$ we can deduce that: \begin{align} d((u_{0},u_{1},u),\bm G^{-}(y,d,z,\theta)) &= \begin{cases} \max\{|u_{0}-y|,|g(z)-u|_{+}\}, &\text{ if }d=0,\\ \max\{|u_{1}-y|,|u-g(z)|_{+}\}, &\text{ if }d=1. \end{cases}\\\nonumber\\ d((y^\star,d^\star),\bm G^{\star}(y,d,z,u_{0},u_{1},u,\theta,\gamma)) &= \begin{cases} \max\{|u_{0}-y|,|g(z)-u|_{+}\}, &\text{ if }u> g(\gamma(z)),\\ \max\{|u_{1}-y|,|u-g(z)|_{+}\}, &\text{ if }u\leq g(\gamma(z)). \end{cases} \end{align} We now define the sets $\Theta^{-}$ and $\Theta^{\star}$ given in Lemma (ref) in the context of this example: \begin{align} \Theta^{-}(y,d,z,u_{0},u_{1},u)\cap \Theta_{\delta}^{*} &:= \left\{\theta \in \Theta_{\delta}^{*} : (u_{0},u_{1},u) \in \bm G^{-}(y,d,z,\theta) \right\} &&\nonumber\\\nonumber\\ &=\begin{cases} \{\theta\in \Theta_{\delta}^{*} : g(z)\in [0,u]\}, &\text{ if }d=0 \text{ and } u_{0}=y,\\ \{\theta\in \Theta_{\delta}^{*} : g(z)\in [u,1]\}, &\text{ if }d=1 \text{ and } u_{1}=y,\\ \emptyset, &\text{ otherwise,} \end{cases}&&\\\nonumber\\ \Theta^{\star}(v,\gamma)\cap \Theta_{\delta}^{*} &:= \left\{\Theta_{\delta}^{*} \in \Theta : (y^\star,d^\star) \in \bm G^{\star}(y,d,z,u_{0},u_{1},u,\theta,\gamma) \right\}\nonumber &&\\ \nonumber\\ &=\begin{cases} \{\theta \in \Theta_{\delta}^{*} : g(\gamma(z)) \in [0,u]\}, &\text{ if }d^{*}=0 \text{ and } y^\star=u_{0},\\ \{\theta \in \Theta_{\delta}^{*} : g(\gamma(z)) \in [u,1]\}, &\text{ if }d^{*}=1 \text{ and } y^\star=u_{1},\\ \emptyset, &\text{ otherwise.} \end{cases}&& \end{align} With these definitions, we have for any $\theta \in \Theta_{\delta}^{*}$: \begin{align} d(\theta,\Theta^{-}(y,d,z,u_{0},u_{1},u)\cap \Theta_{\delta}^{*}) &=\begin{cases} |g(z)-u|_{+}, &\text{ if }d=0 \text{ and } u_{0}=y,\\ |u-g(z)|_{+}, &\text{ if }d=1 \text{ and } u_{1}=y,\\ +\infty, &\text{ otherwise,} \end{cases}\\\nonumber\\ d(\theta,\Theta^{\star}(v,\gamma)\cap \Theta_{\delta}^{*})&=\begin{cases} |g(\gamma(z))-u|_{+}, &\text{ if }d^{*}=0 \text{ and } y^\star=u_{0},\\ |u-g(\gamma(z))|_{+}, &\text{ if }d^{*}=1 \text{ and } y^\star=u_{1},\\ \emptyset, &\text{ otherwise.} \end{cases} \end{align} Combining (ref) with (ref) we can verify condition (ref) with $\ell_{1}=1$. Furthermore, by combining (ref) with (ref) we can verify condition (ref) with $\ell_{2}=1$. Applying Lemma (ref) then yields the choice $C_{2} = L_{\varphi} \max\{ \ell_{1},\ell_{2}\} = 1$, as claimed above. Note also that this value of $C_{2}$ works for any $\delta>0$. It thus suffices to set $\mu^{*}=1$ in Theorem (ref). Also, recall the moment functions for this example from equations (ref) - (ref). The Theorem then states that the lower and upper bounds on the closed convex hull of the identified set for $\E[Y_{\gamma}^\star]$ can be computed as the solutions to the problems (ref) and (ref). Intuitively, under the assumptions of the Theorem the infimum over $\theta \in \Theta$ and supremum over $\theta \in \Theta$ in problems (ref) and (ref) will be obtained at the value $\theta_{0} \in \Theta$. \subsubsection{Verification of Learnability} We claim that $\Phi$ is a VC class with VC index of at most $|\mathcal{Z}|+1$. To prove this, we must show that there exists no set of points $\mathcal{Z}_{n}=\{z_{1},\ldots,z_{n}\}$ with $n=|\mathcal{Z}|+1$ shattered by $\Phi$. Let $t_{1},\ldots,t_{n}$ be arbitrary real numbers. Now define the set: \begin{align*} B:= \left\{\begin{bmatrix}\mathbbm{1}\left\{\,\, \mathbbm{1}\{g(\gamma(z_{1})) \geq u \}(u_{1}-u_{0}) + u_{0}\geq t_{1} \,\,\right\} \\ \mathbbm{1}\left\{\,\,\mathbbm{1}\{g(\gamma(z_{2})) \geq u \}(u_{1}-u_{0}) + u_{0}\geq t_{2} \,\,\right\} \\ \vdots \\ \mathbbm{1}\left\{\,\,\mathbbm{1}\{g(\gamma(z_{n})) \geq u \}(u_{1}-u_{0}) + u_{0}\geq t_{n}\,\,\right\} \end{bmatrix} : (u_{0},u_{1},u,\theta) \in \mathcal{U} \times \Theta \right\}. \end{align*} If $B$ contains the vector $b\in \{0,1\}^{n}$, then we say that $\Phi$ “picks out” $b$. It suffices to show that there always exists at least one vector $b \in \{0,1\}^{n}$ that $\Phi$ fails to pick out. Since $n>|\mathcal{Z}|$, there exists at least one $z \in \mathcal{Z}$ that appears twice in the set $\mathcal{Z}_{n}$. Thus there is some $i,j \in \{1,\ldots,n\}$ such that $z_{i}=z_{j}$. Then regardless of the values of $(u_{0},u_{1},u,\theta)$ we will always have: \begin{align*} \mathbbm{1}\{g(\gamma(z_{i})) \geq u \}(u_{1}-u_{0}) + u_{0} = \mathbbm{1}\{g(\gamma(z_{j})) \geq u \}(u_{1}-u_{0}) + u_{0}. \end{align*} We then have: \begin{enumerate} • If $t_{i} = t_{j}$ then $\Phi$ fails to pick out any vector $b \in \{0,1\}^{n}$ with $b_{i}=0$ and $b_{j}=1$. • If $t_{i} < t_{j}$ then $\Phi$ fails to pick out any vector $b \in \{0,1\}^{n}$ with $b_{i}=0$ and $b_{j}=1$. • If $t_{j} < t_{i}$ then $\Phi$ fails to pick out any vector $b \in \{0,1\}^{n}$ with $b_{i}=1$ and $b_{j}=0$. \end{enumerate} Since this covers all possibilities for $t_{i},t_{j} \in \mathbb{R}$, we conclude that there always exists at least one binary vector that $\Phi$ fails to pick out, and thus $\Phi$ shatters no set of size $n = |\mathcal{Z}|+1$. Now using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \Phi, ||\cdot||_{Q,2} \right) = O(1), \end{align*} so that $\Phi$ easily satisfies the entropy growth condition. Now let $j$ index a generic moment function: \begin{align*} m_{j}(D,Z,\theta) = \left(D - g(z_{0},x)\right)\mathbbm{1}\{Z_{0}=z_{0},X=x\}, \end{align*} and let $\mathcal{M}_{j}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j}= \left\{ m_{j}(\cdot,\theta):\{0,1\}\times \mathcal{Z} \to \mathbb{R} : \theta \in \Theta \right\}. \end{align*} Note this class indexes the moment functions from the moment conditions (ref). Also note that $(z_{0},x)$ are not arguments of the moment function, but are instead are associated with the index $j$. We claim that there exists no set of size $3$ shattered by $\mathcal{M}_{j}$, implying $\mathcal{M}_{j}$ is a VC-subgraph class. We will prove this by way of contradiction. In particular, suppose that there exists three points $(d_{1},z_{1})$, $(d_{2},z_{2})$, and $(d_{3},z_{3})$ and values $t_{1},t_{2}, t_{3} \in \mathbb{R}$ such that: \begin{align} \left| \left\{\begin{bmatrix}\mathbbm{1}\{m_{j}(d_{1},z_{1},\theta) \geq t_{1}\}\\ \mathbbm{1}\{m_{j}(d_{2},z_{2},\theta) \geq t_{2}\}\\ \mathbbm{1}\{m_{j}(d_{3},z_{3},\theta) \geq t_{3}\}\end{bmatrix} : \theta \in \Theta \right\}\right| = 8. \end{align} In other words, we suppose the set $\{(d_{1},z_{1}),(d_{2},z_{2}),(d_{3},z_{3})\}$ is shattered by $\mathcal{M}_{j}$, and that $t_{1},t_{2},t_{3} \in \mathbb{R}$ witness the shattering. We have: \begin{align*} m_{j}(d_{1},z_{1},\theta) &=\left(d_{1} - g(z_{0},x)\right)\mathbbm{1}\{z_{1,0}=z_{0},x_1=x\},\\ m_{j}(d_{2},z_{2},\theta) &=\left(d_{2} - g(z_{0},x)\right)\mathbbm{1}\{z_{2,0}=z_{0},x_{2}=x\}\\ m_{j}(d_{3},z_{3},\theta) &=\left(d_{3} - g(z_{0},x)\right)\mathbbm{1}\{z_{3,0}=z_{0},x_{3}=x\}. \end{align*} Now consider two cases: \begin{enumerate} • $\mathbbm{1}\{z_{1,0}=z_{0},x_1=x\}=\mathbbm{1}\{z_{2,0}=z_{0},x_{2}=x\}=\mathbbm{1}\{z_{3,0}=z_{0},x_{3}=x\}=1$: Note that since $d_{i} \in \{0,1\}$, at least two functions $m_{j}(d_{1},z_{1},\theta)$, $m_{j}(d_{2},z_{2},\theta)$ and $m_{j}(d_{3},z_{3},\theta)$ are identical for all $\theta \in \Theta$. This means (ref) is impossible. For instance, suppose that $m_{j}(d_{1},z_{1},\theta)= m_{j}(d_{2},z_{2},\theta)$. Then at least one of the vectors $(1,0,0)$ or $(0,1,0)$ cannot be picked out by $\mathcal{M}_{j}$. • Either $\mathbbm{1}\{z_{1,k}=z, y_{1,-k}=y_{-k}\}=0$ or $\mathbbm{1}\{z_{2,k}=z, y_{2,-k}=y_{-k}\}=0$ or $\mathbbm{1}\{z_{3,k}=z, y_{3,-k}=y_{-k}\}=0$: In this case at least one of the functions $m_{j}(d_{1},z_{1},\theta)$, $m_{j}(d_{2},z_{2},\theta)$ or $m_{j}(d_{3},z_{3},\theta)$ is equal to zero for all $\theta \in \Theta$. Again, this means (ref) is impossible. For example, if $m_{j}(d_{1},z_{1},\theta)$ is the zero function, then it is impossible for $\mathcal{M}_{j}$ to pick out both $(0,0,0)$ and $(1,0,0)$. \end{enumerate} Since $(d_{1},z_{1})$, $(d_{2},z_{2})$, and $(d_{3},z_{3})$ were arbitrary, we conclude that there exists no set of size $3$ shattered by $\mathcal{M}_{j}$. This implies that $\mathcal{M}_{j}$ is a VC-subgraph class, and using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \mathcal{M}_{j}, ||\cdot||_{Q,2} \right) = O(1). \end{align*} Thus, $\mathcal{M}_{j}$ easily satisfies the entropy growth condition. Given the relation between the moment functions from (ref) and (ref), a nearly identical analysis holds for the moment functions from the moment conditions (ref). Now let $j'$ index a generic moment function: \begin{align*} m_{j'}(X,U,\theta) = \left(\mathbbm{1}\{U \leq g(z_{0},x)\} - g(z_{0},x)\right)\mathbbm{1}\{X=x\}, \end{align*} and let $\mathcal{M}_{j'}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j'}= \left\{ m_{j'}(\cdot,u,\theta):\mathcal{X} \to \mathbb{R} : (u,\theta) \in \mathcal{U}\times\Theta \right\}. \end{align*} Note this class indexes the moment functions from the moment conditions (ref). Also note that $(z_{0},x)$ are not arguments of the moment function, but are instead are associated with the index $j'$. We claim that there exists no set of size $2$ shattered by $\mathcal{M}_{j'}$, implying $\mathcal{M}_{j'}$ is a VC-subgraph class. We will prove this by way of contradiction. In particular, suppose that there exists two points $x_{1}$ and $x_{2}$, and values $t_{1},t_{2} \in \mathbb{R}$ such that: \begin{align} \left| \left\{\begin{bmatrix}\mathbbm{1}\{m_{j'}(x_{1},u,\theta) \geq t_{1}\}\\ \mathbbm{1}\{m_{j'}(x_{2},u,\theta) \geq t_{2}\}\end{bmatrix} : (u,\theta) \in \mathcal{U} \times \Theta \right\}\right| = 4. \end{align} In other words, we suppose the set $\{x_{1},x_{2}\}$ is shattered by $\mathcal{M}_{j'}$, and that $t_{1},t_{2} \in \mathbb{R}$ witness the shattering. We have: \begin{align*} m_{j'}(x_{1},u,\theta) =\left(\mathbbm{1}\{u \leq g(z_{0},x)\} - g(z_{0},x)\right)\mathbbm{1}\{x_{1}=x\},\\ m_{j'}(x_{2},u,\theta) =\left(\mathbbm{1}\{u \leq g(z_{0},x)\} - g(z_{0},x)\right)\mathbbm{1}\{x_{2}=x\}. \end{align*} Now consider two cases: \begin{enumerate} • $\mathbbm{1}\{x_{1}=x\}=\mathbbm{1}\{x_{2}=x\}=1$: Then the two functions $m_{j'}(x_{1},u,\theta)$ and $m_{j'}(x_{2},u,\theta)$ are identical for all $(u,\theta)\in \mathcal{U} \times \Theta$. This means (ref) is impossible, since at least one of the vectors $(1,0)$ and $(0,1)$ cannot be picked out by $\mathcal{M}_{j'}$. • Either $\mathbbm{1}\{x_{1}=x\}=0$ or $\mathbbm{1}\{x_{2}=x\}=0$: In this case at least one of the functions $m_{j'}(x_{1},u,\theta)$ or $m_{j'}(x_{2},u,\theta)$ is the zero function. Again, this means (ref) is impossible. For example, if $m_{j'}(x_{1},u,\theta)$ is the zero function, then it is impossible for $\mathcal{M}_{j'}$ to pick out both $(0,0)$ and $(1,0)$. \end{enumerate} Since $x_{1}$ and $x_{2}$ were arbitrary, we conclude that there exists no set of size $2$ shattered by $\mathcal{M}_{j'}$. This implies that $\mathcal{M}_{j'}$ is a VC-subgraph class, and using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \mathcal{M}_{j'}, ||\cdot||_{Q,2} \right) = O(1). \end{align*} Thus, $\mathcal{M}_{j'}$ easily satisfies the entropy growth condition. Given the relation between the moment functions from (ref) and (ref), a nearly identical analysis holds for the moment functions from the moment conditions (ref). Now let $j''$ index a generic moment function: \begin{align*} m_{j”}(Z,\theta) = t(z_{0},x)-\mathbbm{1}\{Z_{0}=z_{0},X=x)\} , \end{align*} and let $\mathcal{M}_{j''}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j”}= \left\{ m_{j”}(\cdot,\theta):\mathcal{Z} \to \mathbb{R} : \theta \in \Theta \right\}. \end{align*} Note this class indexes the moment functions from the moment conditions (ref). Also note that $(z_{0},x)$ are not arguments of the moment function, but are instead are associated with the index $j''$. We claim that there exists no set of size $3$ shattered by $\mathcal{M}_{j''}$, implying $\mathcal{M}_{j''}$ is a VC-subgraph class. To see this, note that for any three points $\{z_{1},z_{2},z_{3}\}$ we have: \begin{align*} m_{j”}(z_{1},\theta) =t(z_{0},x) - \mathbbm{1}\{z_{1,0}=z_{0},x_{1}=x)\},\\ m_{j”}(z_{2},\theta) =t(z_{0},x) - \mathbbm{1}\{z_{2,0}=z_{0},x_{2}=x)\},\\ m_{j”}(z_{3},\theta) =t(z_{0},x) - \mathbbm{1}\{z_{2,0}=z_{0},x_{2}=x)\}. \end{align*} The conclusion follows from the fact that two of these moment functions must always be the same. This implies that $\mathcal{M}_{j''}$ is a VC-subgraph class, and using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \mathcal{M}_{j”}, ||\cdot||_{Q,2} \right) = O(1). \end{align*} Thus, $\mathcal{M}_{j''}$ easily satisfies the entropy growth condition. Given the relation between the moment functions from (ref) and (ref), a nearly identical analysis holds for the moment functions from the moment conditions (ref). Finally, let $j'''$ index a generic moment function: \begin{align*} m_{j”'}(Z,U_{d},\theta)=U_{d}\left(\mathbbm{1}\{Z=z_{0},X=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{X=x\}t(z_{0},x) \right). \end{align*} and let $\mathcal{M}_{j''}$ be the associated class of functions: \begin{align*} \mathcal{M}_{j”'}= \left\{ m_{j”'}(\cdot,u_{d},\theta):\mathcal{Z} \to \mathbb{R} : (u_{d},\theta) \in [\underline{Y},\overline{Y}]\times \Theta \right\}. \end{align*} Note this class indexes the moment functions from the moment conditions (ref). Also note that $(z_{0},x)$ are not arguments of the moment function, but are instead are associated with the index $j'''$. We claim that there exists no set of size $5$ shattered by $\mathcal{M}_{j'''}$, implying $\mathcal{M}_{j'''}$ is a VC-subgraph class. To see this, note that for any five points $\{z_{1},z_{2},z_{3},z_{4},z_{5}\}$ we have: \begin{align*} m_{j”'}(z_{1},u_{d},\theta) =u_{d}\left(\mathbbm{1}\{z_{1,0}=z_{0},x_{1}=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{x_{1}=x\}t(z_{0},x) \right),\\ m_{j”'}(z_{2},u_{d},\theta) =u_{d}\left(\mathbbm{1}\{z_{2,0}=z_{0},x_{2}=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{x_{2}=x\}t(z_{0},x) \right),\\ m_{j”'}(z_{3},u_{d},\theta) =u_{d}\left(\mathbbm{1}\{z_{3,0}=z_{0},x_{3}=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{x_{3}=x\}t(z_{0},x) \right),\\ m_{j”'}(z_{4},u_{d},\theta) =u_{d}\left(\mathbbm{1}\{z_{4,0}=z_{0},x_{4}=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{x_{4}=x\}t(z_{0},x) \right),\\ m_{j”'}(z_{5},u_{d},\theta) =u_{d}\left(\mathbbm{1}\{z_{5,0}=z_{0},x_{5}=x\}\sum_{z_{0} \in \mathcal{Z}_{0}} t(z_{0},x)- \mathbbm{1}\{x_{5}=x\}t(z_{0},x) \right). \end{align*} The conclusion follows from the fact that two of these moment functions must always be identical for all $\theta$. This implies that $\mathcal{M}_{j'''}$ is a VC-subgraph class, and using, for example, Theorem 2.6.7 in van1996weak, we can deduce: \begin{align*} \sup_{Q \in \mathcal{Q}_{n}} \log N \left( \varepsilon, \mathcal{M}_{j”'}, ||\cdot||_{Q,2} \right) = O(1). \end{align*} Thus, $\mathcal{M}_{j'''}$ easily satisfies the entropy growth condition. Given the relation between the moment functions from (ref) and (ref), a nearly identical analysis holds for the moment functions from the moment conditions (ref). Combining everything and applying Theorem (ref)(ii), we thus have that the policy space $\Gamma$ for this problem is learnable. \begin{comment} \section{Results Based on Talagrand's Inequality} \subsection{A Data-Dependent Version of Talagrand's Inequality} Sharper finite sample results can be obtain from an application of Talagrand's inequality for the concentration of the supremum of an empirical process around it's mean. To introduce these results, and to save on notation, we will consider a collection of measurable functions $\mathcal{F}$ with typical element $f: \mathcal{X} \subseteq \mathbb{R}^{d} \to \mathbb{R}$. We assume throughout that: \begin{align*} \overline{F}:= \sup_{f \in \mathcal{F}}|f| <\infty. \end{align*} For simplicity we will abstract away from measurability issues, for example by taking $\mathcal{F}$ to be countable. Furthermore, let $Pf = \E_{P}[f(X)]$ and $\P_{n}f = n^{-1} \sum_{i=1}^{n} f(x_{i})$, and denote: \begin{align*} ||\P_{n}-P||_{\mathcal{F}}:= \sup_{f \in \mathcal{F}} |(\P_{n}-P)f|, &&||\mathfrak{R}_{n}||_{\mathcal{F}}:= \sup_{f \in \mathcal{F}} \left|\frac{1}{n} \sum_{i=1}^{n} \xi_{i} f(x_{i}) \right|. \end{align*} Finally, denote: \begin{align*} \sigma_{P}^{2}\left( \mathcal{F} \right) := \sup_{f \in \mathcal{F}} \left( Pf^{2} - (Pf)^{2} \right). \end{align*} A significant amount of work has been done to uncover the sharpest possible constants in Talagrand's inequality, and to the best of our knowledge, the following inequality by bousquet2002bennett is the sharpest known: \begin{align*} \text{Pr}_{P} \left( ||\P_{n}-P||_{\mathcal{F}} \geq \E ||\P_{n}-P||_{\mathcal{F}} + \sqrt{2 \frac{t}{n}\left( \sigma_{P}^{2}(\mathcal{F}) + 2 \E|| \P_{n} - P||_{\mathcal{F}} \right)} +\frac{t}{3n} \right) \leq e^{-t} \end{align*} Now recall the symmetrization inequality: $\E ||\P_{n}-P||_{\mathcal{F}} \leq 2\E ||\mathfrak{R}_{n}||_{\mathcal{F}}$. Applying this to Bousquet's inequality gives: \begin{align*} ||\P_{n}-P||_{\mathcal{F}} \leq 2\E ||\mathfrak{R}_{n}||_{\mathcal{F}} + \sqrt{2 \frac{t}{n}\left( \sigma_{P}^{2}(\mathcal{F}) + 4\E ||\mathfrak{R}_{n}||_{\mathcal{F}} \right)} +\frac{t}{3n} \end{align*} on some event $E$ with probability at least $1-e^{-t}$. From here, it is clear that in order to use this concentration inequality in practice we will require data-dependent bounds on $\sigma_{P}^{2}(\mathcal{F})$ and $\E ||\mathfrak{R}_{n}||_{\mathcal{F}}$. To do so, we will state and prove a result provided in koltchinskii2011oracle (Theorem 4.7, p. 71). Indeed, we have the following result which follows a nearly identical proof to Theorem 4.7 in koltchinskii2011oracle: \begin{lemma} Let $\sigma_{n}^{2}(\mathcal{F})$ be any value satisfying: \begin{align*} \sigma_{n}^{2}(\mathcal{F}) \geq \sup_{f \in \mathcal{F}} \left(\frac{1}{n} \sum_{i=1}^{n} f^{2}(x_{i}) - \left( \frac{1}{n} \sum_{i=1}^{n} f(x_{i}) \right)^{2} \right). \end{align*} Now let: \begin{align} \overline{R}_{n}(\mathcal{F},t)&:= 2||\mathfrak{R}_{n}||_{\mathcal{F}} + 2 \sqrt{\frac{2t\overline{V}_{n}(\mathcal{F},t)}{n}} +\frac{6t\overline{F}}{n},\\ \overline{V}_{n}(\mathcal{F},t)&:=2\sigma_{n}^{2}(\mathcal{F}) + 36\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + \frac{2774t\overline{F}^{2}}{3n}. \end{align} Then: \begin{align*} \inf_{P \in \mathcal{P}} \text{Pr}_{P} \left(||\P_{n}-P||_{\mathcal{F}} \leq 2\overline{R}_{n}(\mathcal{F},t) + \sqrt{\frac{2t}{n}\left( \overline{V}_{n}(\mathcal{F},t) + 4\overline{F}\,\overline{R}_{n}(\mathcal{F},t) \right)} +\frac{t\overline{F}}{3n}\right)\geq 1-2e^{-t}. \end{align*} \end{lemma} \begin{proof}[Proof of Lemma (ref)] By Bousquet's inequality (bousquet2002bennett, Theorem 2.3), on some event $F$ with probability at least $1-e^{-t}$ we have: \begin{align} ||\P_{n}-P||_{\mathcal{F}} \leq \E ||\P_{n}-P||_{\mathcal{F}} + \sqrt{2 \frac{t}{n}\left( \sigma_{P}^{2}(\mathcal{F}) + 2 \overline{F} \E|| \P_{n} - P||_{\mathcal{F}} \right)} +\frac{t\overline{F}}{3n}, \end{align} where: \begin{align} \sigma_{P}^{2}(\mathcal{F}) := \sup_{f \in \mathcal{F}} \left(P f^{2} - (Pf)^{2}\right). \end{align} By symmetrization we obtain: \begin{align*} ||\P_{n}-P||_{\mathcal{F}} \leq 2\E ||\mathfrak{R}_{n}||_{\mathcal{F}} + \sqrt{2 \frac{t}{n}\left( \sigma_{P}^{2}(\mathcal{F}) + 4\overline{F}\E ||\mathfrak{R}_{n}||_{\mathcal{F}} \right)} +\frac{t\overline{F}}{3n}. \end{align*} It remains to obtain data-dependent bounds for $\E ||\mathfrak{R}_{n}||_{\mathcal{F}}$ and $\sigma_{P}^{2}(\mathcal{F})$. However, from koltchinskii2011oracle equation (4.14), after re-scaling, we have that on some event $E$ with probability at least $1-e^{-t}$: \begin{align} \E ||\mathfrak{R}_{n}||_{\mathcal{F}}\leq 2||\mathfrak{R}_{n}||_{\mathcal{F}} + 2\sigma_{P}(\mathcal{F}) \sqrt{\frac{2t}{n}} +\frac{6t\overline{F}}{n}. \end{align} Thus, it suffices to obtain a data-dependent bound on $\sigma_{P}(\mathcal{F})$. By Bousquet's inequality (bousquet2002bennett, Theorem 2.3), on the event $F$ we have with probability at least $1-e^{-t}$: \begin{align*} \sigma_{P}^{2}(\mathcal{F}) &\leq \sup_{f \in \mathcal{F}} P f^{2}\\ &= \sup_{f \in \mathcal{F}} P_{n}f^{2} + P f^{2} - P_{n}f^{2}\\ &\leq \sup_{f \in \mathcal{F}} P_{n}f^{2} + ||P_{n} - P||_{\mathcal{F}^{2}}\\ &\leq \sigma_{n}^{2}(\mathcal{F}) + \E ||\P_{n}-P||_{\mathcal{F}^{2}} + \sqrt{2 \frac{t}{n}\left( \sigma_{P}^{2}(\mathcal{F}^{2}) + 2 \overline{F}^{2} \E|| \P_{n} - P||_{\mathcal{F}^{2}} \right)} +\frac{t\overline{F}^{2}}{3n}\\ &\leq \sigma_{n}^{2}(\mathcal{F}) + \E ||\P_{n}-P||_{\mathcal{F}^{2}} + \sqrt{2 \frac{t}{n}\left( 4\overline{F}^{2}\sigma_{P}^{2}(\mathcal{F}) + 2 \overline{F}^{2} \E|| \P_{n} - P||_{\mathcal{F}^{2}} \right)} +\frac{t\overline{F}^{2}}{3n}, \end{align*} where in the last line we used the fact that: \begin{align*} Var(f^{2})&= Var((f-Pf+Pf)f)\\ &\leq 2Var((f-Pf)f) + 2Var((Pf)f) \\ &\leq 2\E[((f-Pf)f)^{2}] + 2(Pf)^{2} Var(f) \\ &\leq 2\overline{F}^{2}Var(f) + 2(Pf)^{2} Var(f) \\ &\leq 2\overline{F}^{2}Var(f) + 2\overline{F}^{2} Var(f) \\ &=4\overline{F}^{2}Var(f), \end{align*} where the first inequality uses the inequality $(a+b)^2 \leq 2(a^{2} +b^{2})$, which follows from the inequality $2ab \leq a^{2}+b^{2}$. Now note that, when restricted to $[-\overline{F},\overline{F}]$, $x \mapsto x^{2}$ is Lipschitz with Lipschitz constant $2\overline{F}$. By symmetrization and by the contraction inequality we have: \begin{align*} \E ||\P_{n}-P||_{\mathcal{F}} \leq 2 \E ||\mathfrak{R}_{n}||_{\mathcal{F}^{2}} \leq 8 \overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}}. \end{align*} Thus: \begin{align*} \sigma_{P}^{2}(\mathcal{F})&\leq \sigma_{n}^{2}(\mathcal{F}) + 8 \overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}} + \sqrt{2 \frac{t}{n}\left( 4\overline{F}^{2}\sigma_{P}^{2}(\mathcal{F}) + 16 \overline{F}^{3} \E ||\mathfrak{R}_{n}||_{\mathcal{F}}\right)} +\frac{t\overline{F}^{2}}{3n}\\ &\leq \sigma_{n}^{2}(\mathcal{F}) + 8 \overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}} + 2\overline{F} \sigma_{P}(\mathcal{F})\sqrt{\frac{2t}{n}} + 2\sqrt{\frac{8t}{n} \overline{F}^{3} \E ||\mathfrak{R}_{n}||_{\mathcal{F}}} +\frac{t\overline{F}^{2}}{3n}\\ &\leq \sigma_{n}^{2}(\mathcal{F}) + 8\overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}} + 2\overline{F} \sigma_{P}(\mathcal{F})\sqrt{\frac{2t}{n}} + \frac{8t\overline{F}^{2}}{n}+\overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}} +\frac{t\overline{F}^{2}}{3n}\\ &\leq \sigma_{n}^{2}(\mathcal{F}) + 9\overline{F} \E ||\mathfrak{R}_{n}||_{\mathcal{F}} + 2\overline{F} \sigma_{P}(\mathcal{F})\sqrt{\frac{2t}{n}} +\frac{25t\overline{F}^{2}}{3n}. \end{align*} Applying (ref), we have on the event $E\cap F$ with probability at least $1-2e^{-t}$: \begin{align*} \sigma_{P}^{2}(\mathcal{F})&\leq \sigma_{n}^{2}(\mathcal{F}) + 9\overline{F} \left( 2||\mathfrak{R}_{n}||_{\mathcal{F}} + 2\sigma_{P}(\mathcal{F}) \sqrt{\frac{2t}{n}} +\frac{6t\overline{F}}{n} \right) + 2\overline{F} \sigma_{P}(\mathcal{F})\sqrt{\frac{2t}{n}} +\frac{25t\overline{F}^{2}}{3n}\\ &= \sigma_{n}^{2}(\mathcal{F}) + 18\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + 18\overline{F}\sigma_{P}(\mathcal{F}) \sqrt{\frac{2t}{n}} +\frac{54t\overline{F}^{2}}{n} + 2\overline{F} \sigma_{P}(\mathcal{F})\sqrt{\frac{2t}{n}} +\frac{25t\overline{F}^{2}}{3n}\\ &= \sigma_{n}^{2}(\mathcal{F}) + 18\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + 20\overline{F}\sigma_{P}(\mathcal{F}) \sqrt{\frac{2t}{n}} +\frac{187t\overline{F}^{2}}{3n} \\ &= \sigma_{n}^{2}(\mathcal{F}) + 18\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + 2 \times 20\overline{F}\frac{\sigma_{P}(\mathcal{F})}{\sqrt{2}} \sqrt{\frac{t}{n}} +\frac{187t\overline{F}^{2}}{3n} \\ &\leq \sigma_{n}^{2}(\mathcal{F}) + 18\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + \frac{1}{2}\sigma_{P}^{2}(\mathcal{F}) + \frac{400t\overline{F}^{2}}{n} +\frac{187t\overline{F}^{2}}{3n}, \end{align*} and thus: \begin{align} \sigma_{P}^{2}(\mathcal{F})&\leq 2\sigma_{n}^{2}(\mathcal{F}) + 36\overline{F}||\mathfrak{R}_{n}||_{\mathcal{F}} + \frac{2774t\overline{F}^{2}}{3n}. \end{align} The result follows from combining (ref), (ref) and (ref). \end{proof} \end{comment}