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Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments
\def\spacingset#1{\renewcommand{\baselinestretch}
{#1}\small\normalsize} \spacingset{1}
\title{\bf Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments}
\author{Rahul Singh\thanks{
Rahul Singh is a PhD Candidate in Economics and Statistics, MIT Department of Economics, Cambridge MA 02143 (e-mail: \url{[email removed]}). I am particularly grateful to Olivia Foley for sharing medical expertise that is critical for the empirical application. I thank Alberto Abadie, Xiaohong Chen, Victor Chernozhukov, Ben Deaner, Anna Mikusheva, Whitney Newey, and Vasilis Syrgkanis for helpful comments. I thank Douglas Almond, Kenneth Chay, and David Lee for building the data set, as well as Matias Cattaneo for sharing it with their permission. I am grateful to the Jerry A Hausman Graduate Dissertation Fellowship for financial support.
}\hspace{.2cm}\\
MIT Economics}
\date{Original draft: December 18, 2020. This draft: March 23, 2023.}
\maketitle
\bigskip
\begin{abstract}
Negative control is a strategy for learning the causal relationship between treatment and outcome in the presence of unmeasured confounding. The treatment effect can nonetheless be identified if two auxiliary variables are available: a negative control treatment (which has no effect on the actual outcome), and a negative control outcome (which is not affected by the actual treatment). These auxiliary variables can also be viewed as proxies for a traditional set of control variables, and they bear resemblance to instrumental variables. I propose a family of algorithms based on kernel ridge regression for learning nonparametric treatment effects with negative controls. Examples include dose response curves, dose response curves with distribution shift, and heterogeneous treatment effects. Data may be discrete or continuous, and low, high, or infinite dimensional. I prove uniform consistency and provide finite sample rates of convergence. I estimate the dose response curve of cigarette smoking on infant birth weight adjusting for unobserved confounding due to household income, using a data set of singleton births in the state of Pennsylvania between 1989 and 1991.
\end{abstract}
\noindent
{\it Keywords:} potential outcome, reproducing kernel Hilbert space, dose response
\vfill
\newpage
\spacingset{1.5}
\section{Introduction}\label{section:intro}
\textit{Selection on observables} is the popular assumption in causal inference that the assignment of treatment $D$ is as good as random after conditioning on covariates $X$. It is a strong causal assumption which is often violated even in laboratory settings. Negative controls, widely used in laboratory science, guard against unobserved confounding. The idea is to check for spurious relationships that would only be nonzero in the presence of an unobserved confounder $U$--an approach sometimes called \textit{falsification} or \textit{specificity} testing. Consider two auxiliary variables: a negative control treatment $Z$ (which a priori has no effect on the actual outcome $Y$), and a negative control outcome $W$ (which a priori is not affected by the actual treatment $D$). \cite{miao2018confounding} and \cite{deaner2018nonparametric} carefully formalize a learning problem in which negative controls $(Z,W)$ can not only check for the presence of unobserved confounding $U$, but also recover the causal relationship of interest.
As a concrete example, consider the empirical strategy of \cite{lousdal2020negative}. The goal is to measure the effect of mammography screening $D$ on death from breast cancer $Y$. The set of covariates $X$ includes marriage status, number of children, age at first birth, years of education, annual income, and hormone drug use. The authors used negative controls to document that, even after taking into account the covariates $X$, unobserved confounding $U$ drives spurious correlations. Specifically, dental care participation $Z$ decreases the likelihood of death from breast cancer $Y$ in the dataset. Mammography screening $D$ decreases the likelihood of death from other causes $W$ in the dataset. The authors conclude that unobserved confounding contaminates treatment effect estimation in this setting.
In the present work, I propose a family of nonparametric algorithms based on kernel ridge regression that use negative controls to not only detect but also adjust for unobserved confounding. I consider treatment effects of the population, of subpopulations, and of alternative populations with alternative covariate distributions. Moreover, I allow for treatments, covariates, and negative controls that may be discrete or continuous, and low, high, or infinite dimensional. Due to the intuitive nature of negative controls, I use such terminology throughout the paper. In recent work, \cite{tchetgen2020introduction} refer to negative controls as proxy variables in order to emphasize that they may arise in not only experimental but also observational settings. Due to the formal resemblance between negative controls and instrumental variables, the new statistical results I provide also apply to nonparametric instrumental variable regression (NPIV). Altogether, I provide conceptual, algorithmic, and statistical contributions.
{\bf Conceptual.} I unify a variety of learning problems with unobserved confounding into one general nonparametric learning problem. In semiparametric causal inference, treatment $D$ is restricted to be binary. I consider nonparametric causal inference, allowing the treatment $D$ to be binary, discrete, or continuous. It appears that this is the first work on \textcolor{black}{conditional} dose response curves and heterogeneous treatment effects using negative controls. \textcolor{black}{See Section~\ref{sec:related} for a discussion of related work on related estimands, e.g. \cite{mastouri2021proximal,kallus2021causal,ghassami2021minimax}}. I provide a template for future epidemiology research to estimate dose response curves and heterogeneous treatment effects from medical records despite unobserved confounding.
{\bf Algorithmic.}
I propose a family of novel estimators with closed form solutions that are straightforward to implement by matrix operations. To do so, I assume that the true causal relationship is a function in a reproducing kernel Hilbert space (RKHS), which is a popular nonparametric setting in machine learning. The hyperparameters are ridge regression penalties and kernel hyperparameters. For the former, I derive the closed form solution for leave-one-out cross validation. The latter have well known heuristics. I evaluate the estimators in simulations against alternative estimators that ignore unobserved confounding.
{\bf Statistical.} I prove uniform ($\sup$ norm) consistency with finite sample rates. A uniform guarantee encodes caution about worst case scenarios when informing policy decisions. The finite sample rates of convergence do not directly depend on the data dimension but rather the smoothness of the true causal relationship. An important intermediate result is finite sample analysis of NPIV in $\sup$ norm. Of independent interest, I relate assumptions required in ill posed inverse problems--existence and completeness--to the RKHS setting. This characterization appears to be absent from previous work on NPIV in the RKHS.
To illustrate how the proposed estimators are useful, I conduct a case study. Estimating the effect of cigarette smoking on infant birth weight is challenging for several reasons. First, pregnant women are classified as a vulnerable population, so they are typically excluded from clinical trials; observational data are the only option. Second, pregnancy induces many physiological changes, so medical knowledge predicts different dose response curves for women who are pregnant compared to women who are not pregnant. Third, medical records exclude an \textit{unobserved confounder} known to be crucial for maternal-fetal health: household income. I argue that medical records include variables that satisfy the properties of \textit{negative controls} for unobserved income, and discuss what issues may arise if there are additional unobserved confounders. I provide preliminary results and outline directions for future work on this important topic.
The structure of the paper is as follows. Section~\ref{sec:related} describes related work. Section~\ref{sec:problem} formalizes the learning problem. Section~\ref{section:algorithm} proposes the new algorithms. Section~\ref{sec:consistency} proves uniform consistency. Section~\ref{section:experiments} conducts simulation experiments and estimates the dose response curve of cigarette smoking on infant birth weight, adjusting for unobserved confounding due to household income. Section~\ref{sec:conclusion} concludes.
\section{Related work}\label{sec:related}
I view dose response curves and heterogeneous treatment effects as reweightings of a structural function defined by an ill posed inverse problem. As such, I extend the \textit{partial means} framework \cite{newey1994kernel}. Existing work on partial means considers consumer surplus \cite{newey1994kernel} and certain causal parameters \cite{singh2020kernel} to be reweightings of a \textit{regression function}. By contrast, I consider causal parameters that are reweightings of a \textit{structural function}. My uniform analysis therefore generalizes uniform analysis in previous work. To express causal parameters in this way, I generalize identification theorems for treatment effects that use negative controls \cite{miao2018identifying,miao2018confounding,deaner2018nonparametric,tchetgen2020introduction}.
Early work on negative controls emphasized their role in \textit{detection} of unobserved confounding. As early as the 1950s, epidemiologists proposed the principle of causal specificity as a diagnostic tool \cite{berkson1958smoking,yerushalmy1959methodology,ab1965environment}. Subsequent work formalized these concepts \cite{rosenbaum1989role,weiss2002can,lipsitch2010negative}. A more recent literature emphasizes the role of negative controls in \textit{adjustment} for unobserved confounding. Many papers eliminate
the bias from unobserved confounding by imposing additional structure: linearity and normality \cite{gagnon2012using,wang2017confounder}; joint normality \cite{kuroki2014measurement}; rank preservation of individual potential outcomes \cite{tchetgen2014control}; or monotonicity of confounding effects \cite{sofer2016negative}. I generalize identification results that relax such additional structure.
In econometrics, closely related strategies adjust for unobserved confounding in dynamic settings: \textit{difference-in-difference} \cite{card1990impact,meyer1995natural,abadie2005semiparametric}, and \textit{panel proxy control} \cite{deaner2018nonparametric}. Traditional difference-in-difference analysis requires strong assumptions such as linearity and additive separability of confounding. \cite{athey2006identification} present a more general approach, called \textit{changes-in-changes}, articulated in terms of a nonseparable, nonlinear structural model. A key assumption is monotonicity of confounding effects. Importantly, the model I present allows nonlinearity \textit{without} requiring monotonicity of confounding effects. The panel proxy control approach is also articulated in terms of a nonseparable, nonlinear structural model, and its static special case closely resembles the negative control model. \cite{deaner2018nonparametric} presents a series estimator as well as innovative strategies to handle ill posedness and completeness for both static and dynamic settings. It is straightforward to use the techniques developed in this paper to derive an RKHS estimator for the panel proxy setting. See \cite{sofer2016negative} and \cite{deaner2018nonparametric} for explicit comparisons of negative control with difference-in-difference and panel proxy control, respectively.
As previewed above, the causal parameters studied in this work are reweightings of a structural function called a confounding bridge, which closely resembles a nonparametric instrumental variable regression (NPIV). NPIV has a rich literature, including the seminal works of \cite{newey2003instrumental,hall2005nonparametric,blundell2007semi,darolles2011nonparametric,chen2011rate,chen2012estimation}, among others. The kernel ridge regression approach in this work employs RKHS-norm Tikhonov regularization over an infinite dimensional RKHS with a low effective dimension. See e.g. \cite{darolles2011nonparametric,hall2005nonparametric,horowitz2005nonparametric,carrasco2007linear, chen2012estimation}, and references therein, for a rich variety NPIV estimators that employ various types of Tikhonov regularizations over various infinite dimensional function spaces. In Appendix~\ref{sec:source}, I compare my approximation assumptions to the approximation assumptions employed in this literature, building on the discussion of \cite{chen2011rate}.
I contribute to a growing literature that adapts RKHS methods to treatment effect estimation. \cite{nie2017quasi} propose an RKHS estimator of heterogeneous treatment effects under selection on observables and prove mean square error rates. I pursue a more general definition of heterogeneous treatment effects conditional on some interpretable subvector $V\subset X$ \cite{abrevaya2015estimating} and allow for unobserved confounding. \cite{singh2019kernel} present an RKHS approach for nonparametric instrumental variable regression and prove projected mean square error rates in the sense of \cite{ai2003efficient}. \cite{singh2020kernel} present an RKHS approach for treatment effects identified by selection on observables and prove uniform rates. I unify the RKHS constructions in both works in order to handle both ill posedness and reweighting. My work is complementary in that I consider a new causal setting. My uniform analysis applies to not only negative control treatment effects but also nonparametric instrumental variable regression, providing alternative results under the same assumptions as \cite{singh2019kernel}. I build on fundamental statistical contributions from \cite{smale2005shannon,smale2007learning,fischer2017sobolev}.
\textcolor{black}{
This draft subsumes \cite{singh2020kernel_original}.
Several other works have proposed alternative RKHS estimators for the negative control setting. Independently and contemporaneously to \cite{singh2020kernel_original},
\cite{mastouri2021proximal} propose estimators for the dose response curve. \cite{mastouri2021proximal} formulate a kernel two stage regression approach and a kernel moment restriction approach, and formalize connections between them. \cite{mastouri2021proximal} analyze excess risk of a surrogate loss for the former, and consistency for the latter. Excess risk of a surrogate loss corresponds to projected mean square error for the confounding bridge.
See Sections~\ref{section:algorithm} and~\ref{sec:consistency} for further comparisons. \cite{kallus2021causal,ghassami2021minimax} study the semiparametric problem rather than the nonparametric problem considered here. Both works propose doubly robust estimators that combine nuisance functions estimated by a minimax procedure. \cite{chernozhukov2021simple} provide abstract conditions to translate learning theory rates into semiparametric inference when treatment is binary. I summarize the connection to semiparametrics in Appendix~\ref{sec:semi}.}
\textcolor{black}{The emphasis of this work is uniform consistency for the nonparametric case.
The main theoretical results of this paper are (i) uniform consistency of the confounding bridge and (ii) uniform consistency of causal functions estimated using a kernel two stage regression approach.
Uniform nonparametric inference remains an open question for future research.}
\section{Learning problem}\label{sec:problem}
\subsection{Treatment effects}
Treatment effects are statements about counterfactual outcomes given hypothetical interventions. Though we observe outcome $Y$, we seek to infer means of counterfactual outcomes $\{Y^{(d)}\}$, where $Y^{(d)}$ is the potential outcome given the hypothetical intervention $D=d$. The treatment effect literature aims to measure a rich variety of treatment effects, which I quote from \cite[Definition 3.1]{singh2020kernel}.
\begin{definition}[Treatment effects]\label{def:TE}
I define the following treatment effects.
\begin{enumerate}
\item Dose response: $\theta_0^{ATE}(d):=\mathbb{E}[Y^{(d)}]$ is the counterfactual mean outcome given intervention $D=d$ for the entire population.
\item Dose response with distribution shift: $ \theta_0^{DS}(d,\tilde{\mathbb{P}}):=\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(d)}]$ is the counterfactual mean outcome given intervention $D=d$ for an alternative population with data distribution $\tilde{\mathbb{P}}$ (elaborated in Assumption~\ref{assumption:covariate}).
\item Conditional dose response: $ \theta_0^{ATT}(d,d'):=\mathbb{E}[Y^{(d')}|D=d]$ is the counterfactual mean outcome given intervention $D=d'$ for the subpopulation who actually received treatment $D=d$.
\item Heterogeneous treatment effect: $\theta_0^{CATE}(d,v):=\mathbb{E}[Y^{(d)}|V=v]$ is the counterfactual mean outcome given intervention $D=d$ for the subpopulation with covariate value $V=v$.
\end{enumerate}
\end{definition}
The superscipt of each nonparametic treatment effect corresponds to its semiparametric analogue. If treatment is binary, then average treatment effect (ATE) is $\mathbb{E}[Y^{(1)}-Y^{(0)}]$; average treatment effect with distribution shift (DS) is $\mathbb{E}_{\tilde{\mathbb{P}}}[Y^{(1)}-Y^{(0)}]$; average treatment on the treated (ATT) is $\mathbb{E}[Y^{(1)}-Y^{(0)}|D=1]$; and conditional average treatment effect (CATE) is $\mathbb{E}[Y^{(1)}-Y^{(0)}|V=v]$. Rather than differences of potential outcomes indexed by binary treatment, I analyze potential outcomes indexed by discrete or continuous treatment.
$\theta_0^{ATE}(d)$ has many names: dose response curve, continuous treatment effect, and average structural function. If treatment is binary, then $\theta_0^{ATE}(d)$ is a vector in $\mathbb{R}^2$ and the learning problem is semiparametric. If the treatment is discrete or continuous, then $\theta_0^{ATE}(d)$ is a function and the learning problem is nonparametric. $\theta_0^{DS}(d,\tilde{\mathbb{P}})$ is a closely related variant that handles the scenario where the covariate distribution has shifted. This variant may be called distribution shift, covariate shift, policy effect, or transfer learning.
Both $\theta_0^{ATT}(d,d')$ and $\theta_0^{CATE}(d,v)$ involve conditioning on a particular subpopulation. If treatment is binary, then $\theta_0^{ATT}(d,d')$ is a matrix in $\mathbb{R}^{2\times 2}$ and the learning problem is semiparametric. If the treatment is discrete or continuous, then $\theta_0^{ATT}(d,d')$ is a surface and the learning problem is nonparametric. Likewise for $\theta_0^{CATE}(d,v)$. $\theta_0^{ATT}(d,d')$ is called the conditional dose response, and $\theta_0^{CATE}(d,v)$ is called the heterogeneous treatment effect. The possibility for $D$ to be discrete or continuous and for $V$ to be a particular covariate, rather than the full set of covariates required for identification, is more general than the typical heterogeneous treatment effect \cite{nie2017quasi}. For $\theta_0^{CATE}$, I slightly abuse notation by denoting the complete set of identifying covariates as $(V,X)$.
\subsection{Negative control identification}
In pioneering work, \cite{tchetgen2020introduction} propose a potential outcome model in which treatment effects can be measured from outcomes $Y$, treatments $D$, and covariates $(V,X)$ despite unobserved confounding $U$. The technique involves two auxiliary variables: negative control treatment $Z$, and negative control outcome $W$. In this model, potential outcomes $\{Y^{(d,z)}\}$ and potential negative control outcomes $\{W^{(d,z)}\}$ are initially indexed by both the treatment value $D=d$ and the negative control treatment value $Z=z$. The identification strategy requires prior knowledge of how the unobserved confounder, which may be a vector, relates to the observed variables. The validity of negative controls as articulated in Assumptions~\ref{assumption:negative} and~\ref{assumption:solution} is \textit{relative} to a conjectured unobserved confounder.
\begin{assumption}[Negative controls]\label{assumption:negative}
Assume
\begin{enumerate}
\item No interference: if $D=d$ and $Z=z$ then $Y=Y^{(d,z)}$ and $W=W^{(d,z)}$.
\item Latent exchangeability: $\{Y^{(d,z)}\},\{W^{(d,z)}\} \raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} D,Z| U,X $.
\item Overlap: if $f(u,x)>0$ then $f(d,z|u,x)>0$, where $f(u,x)$ and $f(d,z|u,x)$ are densities.
\item Negative control treatment and outcome: $Y^{(d,z)}=Y^{(d)}$ and $W^{(d,z)}=W$.
\end{enumerate}
For $\theta_0^{CATE}$, replace $X$ with $(V,X)$.
\end{assumption}
No interference is also called consistency or the stable unit treatment value assumption in causal inference, and it rules out network effects. Latent exchangeability states that conditional on covariates $X$ and unobserved confounder $U$, treatment assignment and negative control treatment assignment are as good as random. Latent exchangeability relaxes the classic assumption of conditional exchangeability in which $U=\varnothing$, i.e. in which there is no unobserved confounder. Overlap ensures that there is no confounder-covariate stratum such that treatment and negative control treatment have a restricted support; for any stratum, any value of treatment or negative control treatment can occur.
In a graphical causal model, there could be many sets of observed variables that could serve as covariates $X$ and many sets of unobserved variables that could serve as the unobserved confounder $U$ based on these initial criteria. The subsequent criteria provide guidance in how to choose $(X,U)$. We will see that the set of covariates $X$ should be chosen to block as much unobserved confounding as possible, because the variation in unobserved confounding that remains must be tied to variation in negative controls.
The negative control treatment condition imposes that the negative control treatment $Z$ only affects the outcome $Y$ via actual treatment $D$. It is identical to the exclusion restriction assumed for instrumental variables \cite{angrist1996identification}. The negative control outcome condition imposes that the negative control outcome $W$ is unaffected by the treatment $D$ and negative control treatment $Z$. It is an even stronger exclusion restriction. Altogether, Assumption~\ref{assumption:negative} formalizes the intuition that \textit{if there are spurious correlations then there is unobserved confounding}. It also implies $Y\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} Z | D,U,X$ and $W\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} D,Z | U,X$, which are weaker conditions used in the identification argument \cite{miao2018identifying}.
\begin{wrapfigure}{R}{0.33\textwidth}
\vspace{-10pt}
\begin{center}
\begin{adjustbox}{width=.33\textwidth}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
semithick]
\tikzstyle{every state}=[draw=black,text=black]
\node[state] (x) [fill=gray] {$X$};
\node[state] (w) [right of=x, fill=gray] {$W$};
\node[state] (z) [left of=x, fill=gray] {$Z$};
\node[state] (d) [below left of=x, fill=gray] {$D$};
\node[state] (y) [below right of=x, fill=gray] {$Y$};
\node[state] (u) [above of=x] {$U$};
\path (u) edge node {$ $} (z)
edge node {$ $} (x)
edge node {$ $} (w)
edge node {$ $} (d)
edge node {$ $} (y)
(x) edge node {$ $} (w)
edge node {$ $} (z)
edge node {$ $} (d)
edge node {$ $} (y)
(z) edge node {$ $} (d)
(w) edge node {$ $} (y)
(d) edge node {$ $} (y);;
\end{tikzpicture}
\end{adjustbox}
\vspace{-15pt}
\caption{Negative control DAG}
\label{dag:nc}
\end{center}
\vspace{-15pt}
\end{wrapfigure}
Figure~\ref{dag:nc} visualizes a representative directed acyclic graph (DAG). Despite access to covariates $X$, unobserved confounding $U$ has unblocked paths to treatment $D$ and outcome $Y$. In the DAG, we also see the proxy interpretation of this learning problem. Covariates $X$, negative control treatment $Z$, and negative control outcome $W$ are all imperfect proxies for a set of control variables that would block unobserved confounding. Covariates $X$ are proxies that induce treatment and outcome; negative control treatment $Z$ is a proxy that induces treatment only; and negative control outcome $W$ is a proxy that induces outcome only.
Next, I quote a high level technical condition, which I will later verify for the RKHS setting. Define the regression $\gamma_0(d,x,z):=\mathbb{E}[Y|D=d,X=x,Z=z]$.
\begin{assumption}[Confounding bridge]\label{assumption:solution}
Assume
\begin{enumerate}
\item Existence: there exists a solution $h_0$ to the operator equation $$\gamma_0(d,x,z)=\mathbb{E}[h(D,X,W)|D=d,X=x,Z=z].$$
\item Completeness: for any function $f$,
$$
\mathbb{E}[f(U)|D=d,X=x,Z=z]=0\quad \forall(d,x,z) \iff f(U)=0.
$$
\end{enumerate}
\end{assumption}
I call $h_0$ the \textit{confounding bridge}, following \cite{miao2018confounding}. Here, we see the formal resemblance to the nonparametric instrumental variable regression problem (NPIV) \cite{newey2003instrumental}. In the language of NPIV, the LHS $\gamma_0(d,x,z)$ is the \textit{reduced form}, while the RHS is a composition of a \textit{stage 1} conditional expectation operator
and \textit{stage 2} structural function $h_0$. In the language of functional analysis, the operator equation is a Fredholm integral equation of the first kind. Solving this operator equation for $h_0$ involves inverting a linear operator with infinite dimensional domain; it is an ill posed problem. Indeed, existence will require conditions on the spectrum of the conditional expectation operator formalized in Appendix~\ref{section:existence}. Completeness is a technical condition from the NPIV literature. Taking $f=f_1-f_2$, it states that the observed variables $(D,X,Z)$ have sufficiently rich variation in the sense that different functions of unobserved confounding $f_1(U)$ and $f_2(U)$ lead to different projections onto $(D,X,Z)$; if they lead to the same projections, then $f_1=f_2$.
There is a subtle yet fundamental connection between the existence of the confounding bridge and the relevance of negative controls.
\begin{proposition}[Relevance]\label{prop:relevance}
Suppose Assumption~\ref{assumption:negative} holds and $\gamma_0$ varies in $z$, i.e. there exist $(z,z')$ such that $\gamma_0(d,x,z)\neq \gamma_0(d,x,z')$.
\begin{enumerate}
\item If the negative control treatment is irrelevant to the unobserved confounder in the sense that $Z\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} U |D,X$, then no confounding bridge exists.
\item If the negative control outcome is irrelevant to the unobserved confounder in the sense that $W\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} U |X$, then no confounding bridge exists.
\end{enumerate}
\end{proposition}
See Appendix~\ref{section:existence} for the proof, as well as further discussion of the cases in which $(U,Z,W)$ are discrete or continuous. Assumption~\ref{assumption:solution} formalizes the converse intuition of Assumption~\ref{assumption:negative}: \textit{if there is unobserved confounding, then there are spurious correlations}. In order to use $(Z,W)$ to adjust for unobserved confounding $U$, it must be the case that $(Z,W)$ can detect $U$ well enough.
In practice, an analyst must collaborate with domain experts in order to assess (i) what are the sources of unobserved confounding, (ii) whether the negative control exclusion restrictions hold, and (iii) whether the negative controls are relevant. In summary, the key assumptions of negative control identification apply to settings where \textit{there are spurious correlations if and only if there is unobserved confounding}. When domain knowledge is insufficient to verify these assumptions, then the strategy of negative controls is inappropriate.
To handle $\theta_0^{DS}$, I generalize a standard assumption in transfer learning.
\begin{assumption}[Distribution shift]\label{assumption:covariate}
Assume
\begin{enumerate}
\item The difference in population distributions $\mathbb{P}$ and $\tilde{\mathbb{P}}$ is only in the marginal distribution of treatments, negative control treatments, and covariates:
$$
\tilde{\mathbb{P}}(Y,W,D,X,Z)=\mathbb{P}(Y,W|D,X,Z)\tilde{\mathbb{P}}(D,X,Z).
$$
\item $\tilde{\mathbb{P}}(D,X,Z)$ is absolutely continuous with respect to $\mathbb{P}(D,X,Z)$.
\end{enumerate}
\end{assumption}
\begin{proposition}[Invariance of confounding bridge]\label{prop:covariate}
Under Assumptions~\ref{assumption:solution} and~\ref{assumption:covariate}, the confounding bridge $h_0$ remains the same across the different populations $\mathbb{P}$ and $\tilde{\mathbb{P}}$.
\end{proposition}
See Appendix~\ref{sec:id} for the proof. It appears that Assumption~\ref{assumption:covariate} and Proposition~\ref{prop:covariate} are the first formalization of distribution shift in negative control and NPIV settings.
In summary, I place three assumptions: availability of negative controls (Assumption~\ref{assumption:negative}); existence and completeness of the confounding bridge (Assumption~\ref{assumption:solution}); and invariance of the confounding bridge for transfer learning (Assumption~\ref{assumption:covariate}). Formally, the theorem that uses these assumptions to express treatment effects in terms of data is known as an identification result. I present the main identification result below, extending the powerful insights of \cite{miao2018identifying,miao2018confounding,deaner2018nonparametric,tchetgen2020introduction} to additional treatment effects beyond $\theta_0^{ATE}(d)$.
\begin{theorem}[Identification of treatment effects]\label{theorem:id_treatment}
If Assumptions~\ref{assumption:negative} and~\ref{assumption:solution} hold then
\begin{enumerate}
\item $\theta_0^{ATE}(d)=\int h_0(d,x,w)\mathrm{d}\mathbb{P}(x,w)$.
\item If in addition Assumption~\ref{assumption:covariate} holds, then $\theta_0^{DS}(d,\tilde{\mathbb{P}})=\int h_0(d,x,w)\mathrm{d}\tilde{\mathbb{P}}(x,w)$.
\item $\theta_0^{ATT}(d,d')=\int h_0(d',x,w)\mathrm{d}\mathbb{P}(x,w|d)$.
\item $\theta_0^{CATE}(d,v)=\int h_0(d,v,x,w)\mathrm{d}\mathbb{P}(x,w|v)$.
\end{enumerate}
\end{theorem}
See Appendix~\ref{sec:id} for the proof. In Theorem~\ref{theorem:id_treatment}, we see how negative controls $(Z,W)$ allow us to adjust for unobserved confounding $U$ and to thereby recover the treatment effect of interest. Each treatment effect is a reweighting of the confounding bridge $h_0$ defined in Assumption~\ref{assumption:solution} with respect to some distribution $\mathbb{Q}$. In this sense, each treatment effect is an example of the same general nonparametric learning problem: $\int h_0(d,x,w)\mathrm{d}\mathbb{Q}$,
where
$\mathbb{Q}$ may be an unconditional distribution such as $\mathbb{P}(x,w)$ or a conditional distribution such as $\mathbb{P}(x,w|d)$.
\subsection{RKHS background}
Until this point, I have placed only causal assumptions formalized in Assumptions~\ref{assumption:negative},~\ref{assumption:solution}, and~\ref{assumption:covariate}. For computational and analytical tractability, I now place additional structure on the learning problem: I assume key quantities are elements of a reproducing kernel Hilbert space (RKHS). The RKHS is a canonical setting in machine learning, and it is a space of smooth functions that generalizes the Sobolev space. For a broad statistical audience, I organize ideas from RKHS learning theory that underpin the algorithm derivation (Section~\ref{section:algorithm}) and consistency guarantee (Section~\ref{sec:consistency}) to follow.
\textbf{Kernel and feature map.} I begin with basic kernel and feature map notation. Consider the RKHS $\mathcal{H}$ which consist of functions of the form $f:\mathcal{A}\rightarrow\mathbb{R}$, where $\mathcal{A}$ is a Polish space (defined formally below). An RKHS $\mathcal{H}$ is characterized by its feature map $\phi(a)$, which can be interpreted as the dictionary of basis functions for the RKHS in the sense that, for any $f\in\mathcal{H}$, $f(a)=\langle f,\phi(a)\rangle_{\mathcal{H}}$. The kernel $k:\mathcal{A}\times \mathcal{A}\rightarrow \mathbb{R}$ is a positive definite, symmetric, and continuous function such that $k(a,a')=\langle \phi(a),\phi(a') \rangle_{\mathcal{H}}$; it is the inner product of features, so it encodes the geometry of the RKHS. Alternatively, one may define the kernel first, then define the feature map as $\phi:\mathcal{A}\rightarrow \mathcal{H}$, $a\mapsto k(a,\cdot)$. The feature map perspective is helpful for theory, but the kernel perspective is helpful for practice, since $k(a,a')$ is a scalar that can be computed. Ultimately, I will reduce the algorithm to kernel evaluations.
\textbf{Kernel mean embedding.} We have seen how the feature map helps to evaluate a function. A related object, called the kernel mean embedding, helps to take the expectation of a function. Suppose we wish to calculate $\mathbb{E}[f(A)]$. The idea of kernel mean embedding is to write
$$
\mathbb{E}[f(A)]=\int f(a)\mathrm{d}\mathbb{P}(a) = \int \langle f,\phi(a)\rangle_{\mathcal{H}} \mathrm{d}\mathbb{P}(a) = \left\langle f,\int \phi(a)\mathrm{d}\mathbb{P}(a) \right\rangle_{\mathcal{H}} =\langle f,\mu \rangle_{\mathcal{H}}
$$
where the exchange of expectation and inner product requires weak regularity conditions (defined formally below). The object $\mu:=\int \phi(a)\mathrm{d}\mathbb{P}(a)$ is called the \textit{mean embedding} of the distribution $\mathbb{P}(a)$. A kernel is \textit{characteristic} when the mapping $\mathbb{P}(a)\mapsto \mu$ is injective. The geometry of the RKHS implies that, to calculate the expectation of a function, it suffices to take the product of the function and the mean embedding of the corresponding distribution. This idea extends to conditional distributions over a subset of the arguments of the function. I will use this idea extensively when deriving the algorithm.
\textbf{Tensor product.} What if a function is defined over multiple variables, e.g. $f:\mathcal{A}\times \mathcal{B} \rightarrow \mathbb{R}$? A natural approach is to define an RKHS $\mathcal{H}$ for such functions as the combination of RKHSs $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$ that contain functions of the form $f_1:\mathcal{A}\rightarrow \mathbb{R}$ and $f_2:\mathcal{B}\rightarrow \mathbb{R}$, respectively. Denote the individual feature maps by $\phi_{\mathcal{A}}(a)$ and $\phi_{\mathcal{B}}(b)$, then define the tensor product feature map $\phi(a,b)=\phi_{\mathcal{A}}(a) \otimes \phi_{\mathcal{B}}(b)$ for the tensor product RKHS $\mathcal{H}$. The tensor product is a generalization of the outer product; formally, $[a\otimes b]c=a \langle b,c \rangle$. Then, for any $f\in\mathcal{H}$, $f(a,b)=\langle f,\phi_{\mathcal{A}}(a)\otimes \phi_{\mathcal{B}}(b)\rangle_{\mathcal{H}}$. It turns out that the kernel of this RKHS is simply the product of the kernels of the individual RKHSs: $k(a,b;a',b')=k_{\mathcal{A}}(a,a')\cdot k_{\mathcal{B}}(b,b')$. As such, $k(a,b;a',b')$ is a scalar that can be computed. In this work, I will extensively use tensor product constructions. For this reason, the algorithm statements will have the symbol $\odot$ for the elementwise product of objects that contain kernel evaluations.
\textbf{RKHS for operators.} So far, I have defined RKHSs for functions of one or more variables. RKHSs also exist for operators. I denote by $\mathcal{L}_2(\mathcal{H}_{\mathcal{A}},\mathcal{H}_{\mathcal{B}})$ the space of Hilbert-Schmidt operators of the form $E:\mathcal{H}_{\mathcal{A}}\rightarrow \mathcal{H}_{\mathcal{B}}$. It turns out that this space is an RKHS in its own right. The operators of interest are conditional expectation operators, which correspond to conditional mean embeddings. For example, consider the goal of calculating $\mathbb{E}[f(a)|B=b]$. As before, one can express
$$
\int f(a)\mathrm{d}\mathbb{P}(a|b) = \int \langle f,\phi_{\mathcal{A}}(a)\rangle_{\mathcal{H}_{\mathcal{A}}} \mathrm{d}\mathbb{P}(a|b) = \left\langle f,\int \phi_{\mathcal{A}}(a)\mathrm{d}\mathbb{P}(a|b) \right\rangle_{\mathcal{H}_{\mathcal{A}}} =\langle f,\mu_{a}(b) \rangle_{\mathcal{H}_{\mathcal{A}}}
$$
where $\mu_{a}(b):=\int \phi_{\mathcal{A}}(a)\mathrm{d}\mathbb{P}(a|b)$ is called the \textit{conditional mean embedding} of the distribution $\mathbb{P}(a|b)$. Observe that
$$
\mu_{a}(b)=\int \phi(a)\mathrm{d}\mathbb{P}(a|b)=[E\phi(\cdot)](b)=[E^* \phi(b)](\cdot)
$$
where $E^*:\mathcal{H}_{\mathcal{B}}\rightarrow \mathcal{H}_{\mathcal{A}}$ is the adjoint of $E: \mathcal{H}_{\mathcal{A}}\rightarrow \mathcal{H}_{\mathcal{B}}$, and both are conditional expectation operators. Formally, the operators $E:f(\cdot)\mapsto \mathbb{E}[f(A)|B=\cdot]$ and $E^*:g(\cdot)\mapsto \mathbb{E}[g(B)|A=\cdot]$ encode the same information as the conditional mean embedding $\mu_{a}(b)$. This relationship will facilitate estimation and analysis.
\textbf{Closed form solution.} The final piece of RKHS machinery necessary for the algorithm derivation (Section~\ref{section:algorithm}) is the so-called \textit{kernel trick}. We have seen how a kernel evaluation is, in the end, simply a scalar. So, if an analyst can express an algorithm exclusively in terms of kernel evaluations, then the algorithm can be easily computed. A virtue of kernel methods is that they tend to have closed form solutions in terms of kernel evaluations. Conceptually, an RKHS algorithm $\hat{f}(a)=\langle \hat{f},\phi(a)\rangle_{\mathcal{H}}$ involves a possibly nonlinear feature map $\phi(\cdot)$ applied to the data, so such an algorithm maintains computational simplicity while allowing for rich nonlinearity. Consider, for example, the kernel ridge regression
\begin{equation}\label{eq:loss}
\hat{f}=\operatorname*{\arg\!\min}_{f\in \mathcal{H}} \frac{1}{n}\sum_{i=1}^n \{y_i-\langle f,\phi(a_i) \rangle_{\mathcal{H}} \}^2+\lambda\|f\|^2_{\mathcal{H}}
\end{equation}
with regularization hyperparameter $\lambda>0$. Its closed form solution is
\begin{equation}\label{eq:sol}
\hat{f}(a)=\langle \hat{f},\phi(a)\rangle_{\mathcal{H}}=\mathbf{Y}^{\top} (\mathbf{K}_{AA}+n\lambda \mathbf{I})^{-1}\mathbf{K}_{Aa}
\end{equation}
where $\mathbf{Y}\in\mathbb{R}^n$ is the vector of outcomes with $i$-th entry $y_i$, $\mathbf{K}_{AA}\in\mathbb{R}^{n\times n}$ is the kernel matrix with $(i,j)$-th entry $k(a_i,a_j)$, and $\mathbf{K}_{Aa}\in\mathbb{R}^{n}$ is the evaluation vector with $i$-th entry $k(a_i,a)$. The algorithms I propose generalize kernel ridge regression. Sometimes, instead of regressing the outcome $Y$ on features $\phi(A)$, I will regress the outcome $Y$ on mean embeddings $\mu_a(B)$. At other times, I will regress one collection of features $\phi(A)$ on another collection of features $\phi(B)$. These generalizations haves losses and closed form solutions that generalize those of kernel ridge regression. As such, they are simple combinations of kernel matrices and evaluation vectors despite being nonparametric.
\textbf{Spectral view.} The statistical guarantees of Section~\ref{sec:consistency} require the \textit{spectral} view of the RKHS $\mathcal{H}$. The spectral view is more challenging, but it is the only way to articular RKHS learning theory. Let $\mathbb{L}_2$ denote the space of square integrable functions mapping from $\mathcal{A}$ to $\mathbb{R}$ with respect to measure $\mathbb{P}$. For a fixed kernel $k$, define the convolution operator $L:\mathbb{L}_2\rightarrow \mathbb{L}_2$, $f\mapsto \int k(a,\cdot)f(a)\mathrm{d}\mathbb{P}(a)$. By the spectral theorem, we can express the operator $L$ in terms of its countable eigenvalues $\{\eta_j\}$ and eigenfunctions $\{\varphi_j\}$: $Lf=\sum_{j=1}^{\infty} \eta_j \langle f, \varphi_j \rangle \cdot \varphi_j$. Without loss of generality, $\{\eta_j\}$ is a weakly decreasing sequence and $\{\varphi_j\}$ forms an orthonormal basis of $\mathbb{L}_2$. With this spectral notation, we are ready to formalize the sense in which the RKHS $\mathcal{H}$ is a smooth subset of $\mathbb{L}_2$. Since $\{\varphi_j\}$ forms an orthonormal basis of $\mathbb{L}_2$, any $f,g\in \mathbb{L}_2$ can be expressed as $f=\sum_{j=1}^{\infty}f_j\varphi_j$ and $g=\sum_{j=1}^{\infty}g_j\varphi_j$. By \cite[Theorem 4]{cucker2002mathematical}, $\mathbb{L}_2$ and the RKHS $\mathcal{H}$ can be explicitly represented as
\begin{align*}
\mathbb{L}_2&=\left\{f=\sum_{j=1}^{\infty}f_j\varphi_j:\; \sum_{j=1}^{\infty}f_j^2<\infty\right\},\quad \langle f,g \rangle_{\mathbb{L}_2}=\sum_{j=1}^{\infty} f_jg_j \\
\mathcal{H}&=\left\{f=\sum_{j=1}^{\infty}f_j\varphi_j:\;\sum_{j=1}^{\infty} \frac{f_j^2}{\eta_j}<\infty\right\},\quad \langle f,g \rangle_{\mathcal{H}}=\sum_{j=1}^{\infty} \frac{f_jg_j}{\eta_j}.
\end{align*}
The RKHS $\mathcal{H}$ is the subset of $\mathbb{L}_2$ for which higher order terms in the series $\{\varphi_j\}$ have a smaller contribution. In the RKHS, there is a penalty on higher order coefficients, and the magnitude of the penalty corresponds to how small the eigenvalue is.
\textbf{Main assumptions.} Finally, I articulate the main approximation assumptions of this paper. Formally, to analyze bias, I assume that a statistical target $f_0$ satisfies
\begin{equation}\label{eq:prior}
f_0\in \mathcal{H}^c:=\left\{f=\sum_{j=1}^{\infty}f_j\varphi_j:\;\sum_{j=1}^{\infty} \frac{f_j^2}{\eta^c_j}<\infty\right\},\quad c\in(1,2].
\end{equation}
For $c=1$, we see that $\mathcal{H}^1=\mathcal{H}$; I am simply assuming that $f_0$ is correctly specified as an element of the RKHS. For $c>1$, I am assuming that $f_0$ is in the interior of the RKHS. This assumption is called the source condition in statistical learning theory and econometrics \cite{smale2007learning,caponnetto2007optimal,carrasco2007linear}. As we will see, a larger value of $c$ corresponds to a smoother target $f_0$ and a faster uniform rate. I allow $c$ to be as large as $c=2$, which is the highest degree of smoothness to which kernel ridge estimators can adapt. In Appendix~\ref{sec:source}, I compare this main approximation assumption with alternative approximation assumptions in the negative control and NPIV literatures.
The second main approximation is a spectral decay assumption called the effective dimension of the basis $\{\varphi_j\}$. I quantify the effective dimension as the rate at which the eigenvalues $\{\eta_j\}$ decay. Formally, to analyze variance, I assume that there exists some constant $C$ such that each $\eta_j$ satisfies
\begin{equation}\label{eq:prior2}
\eta_j\leq C j^{-b},\quad b\geq1.
\end{equation}
\cite[Lemma 10]{fischer2017sobolev} shows that a bounded kernel $k$ satisfies this condition with $b$ that is at least one. A higher value of $b$ corresponds to a lower effective dimension, better control of the variance, and a faster rate. The limit $b\rightarrow \infty$ corresponds to an RKHS with finite dimension \cite{caponnetto2007optimal}.
\textbf{Special case: Sobolev space.} The abstract approximation conditions are easy to interpret in the context of Sobolev spaces. Denote by $\mathbb{H}_2^s$ the Sobolev space of functions of the form $f:\mathcal{A}\rightarrow \mathbb{R}$ with $\mathcal{A}\subset \mathbb{R}^p$. The parameter $s$ denotes how many derivatives of $f$ are square integrable. The Sobolev space $\mathbb{H}_2^s$ is an RKHS if and only if $s>p/2$ \cite[Theorem 132]{berlinet2011reproducing}. Its kernel is known as the \textit{Mat\'ern} kernel. Suppose we take $\mathcal{H}=\mathbb{H}_2^s$ with $s>p/2$ as the RKHS for estimation. If the true target $f_0$ is in $\mathbb{H}_2^{s_0}$, then $c=s_0/s$ \cite{fischer2017sobolev}. In the notation of~\eqref{eq:prior}, $\mathbb{H}_2^{s_0}=[\mathbb{H}_2^s]^c$. Clearly $c>1$ means that the target $f_0$ is in the interior of $\mathbb{H}_2^s$. Moreover, the effective dimension of the RKHS $\mathbb{H}_2^s$ is quantified by $b=2s/p$ \cite{fischer2017sobolev}. Rates in terms of $(b,c)$ adapt to the smoothness of $f_0$ and the effective dimension of the RKHS $\mathcal{H}$. They are invariant to dimension as long as $s_0>s>p/2$.
\section{Algorithm}\label{section:algorithm}
\subsection{RKHS construction}
I provide a new RKHS construction for negative control treatment effect estimation, generalizing and unifying the constructions in \cite{singh2019kernel,singh2020kernel}. In my construction, I define RKHSs for treatment $D$, negative controls $(Z,W)$, and covariates $(V,X)$. For example, for treatment $D$ define the RKHS $\mathcal{H}_{\mathcal{D}}$ with feature map $
\phi_{\mathcal{D}}(d)
$ and kernel $k_{\mathcal{D}}(d,d')=\langle \phi_{\mathcal{D}}(d),\phi_{\mathcal{D}}(d') \rangle_{\mathcal{H}_{\mathcal{D}}}$. Formally,
$
\phi_{\mathcal{D}}:\mathcal{D}\rightarrow \mathcal{H}_{\mathcal{D}}
$ and $k_{\mathcal{D}}:\mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R}$.
To lighten notation, I suppress subscripts when arguments are provided, e.g. I write $\phi(d)=\phi_{\mathcal{D}}(d)$.
From these individual RKHSs, I construct a tensor product RKHS $\mathcal{H}$ for the confounding bridge $h_0$. For clarity of exposition, I initially focus on the case without $V$, i.e. excluding $\theta_0^{CATE}$. I assume the confounding bridge $h_0$ is an element of the RKHS with tensor product feature map $\phi(d,x,w):=\phi(d)\otimes \phi(x)\otimes \phi(w)$, i.e. $h_0\in \mathcal{H}:=\mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{W}}$. As before, the feature map can be interpreted as the dictionary of basis functions since
$
h_0(d,x,w)=\langle h_0, \phi(d,x,w) \rangle_{\mathcal{H}}=\langle h_0, \phi(d)\otimes \phi(x)\otimes \phi(w)\rangle_{\mathcal{H}}
$. In Appendix~\ref{section:existence}, I discuss how an analogous assumption for the regression $\gamma_0$ relates to Assumption~\ref{assumption:solution}.
This tensor product RKHS construction plays a central role in deriving simple representations of treatment effects and hence deriving simple estimators, under weak regularity conditions. The RKHS aspect allows for the technique of kernel mean embedding: an analyst can reweight a function $h_0(d,x,w)$ according to some counterfactual distribution by reweighting its feature map $\phi(d,x,w)$ by that distribution. The tensor product aspect ensures separability of the different variables in the feature map; since $\phi(d,x,w)=\phi(d)\otimes \phi(x)\otimes \phi(w)$, the reweighting can apply to the specific variables $\phi(x)\otimes \phi(w)$. Using these properties, I represent the causal quantities defined in Definition~\ref{def:TE} and identified in Theorem~\ref{theorem:id_treatment} in a more tractable form. To begin, I state the regularity conditions.
\begin{assumption}[RKHS regularity conditions]\label{assumption:RKHS}
Assume
\begin{enumerate}
\item $k_{\mathcal{D}}$, $k_{\mathcal{X}}$, $k_{\mathcal{W}}$, and $k_{\mathcal{Z}}$ are continuous and bounded:
$$
\sup_{d\in\mathcal{D}}\|\phi(d)\|_{\mathcal{H}_{\mathcal{D}}}\leq \kappa_d,\; \sup_{x\in\mathcal{X}}\|\phi(x)\|_{\mathcal{H}_{\mathcal{X}}}\leq \kappa_x,\; \sup_{w\in\mathcal{W}}\|\phi(w)\|_{\mathcal{H}_{\mathcal{W}}}\leq \kappa_w,\; \sup_{z\in\mathcal{Z}}\|\phi(z)\|_{\mathcal{H}_{\mathcal{Z}}}\leq \kappa_z;
$$
\item $\phi(d)$, $\phi(x)$, $\phi(w)$, and $\phi(z)$ are measurable;
\item $k_{\mathcal{X}}$ and $k_{\mathcal{W}}$ are characteristic.
\end{enumerate}
For $\theta_0^{CATE}$, extend the stated assumptions from $X$ to $(V,X)$.
\end{assumption}
Continuity, boundedness, and measurability are weak conditions satisfied by commonly used kernels. The characteristic property is a regularity condition for embedding a distribution in an RKHS, and it is satisfied by commonly used kernels as well \cite{sriperumbudur2010relation}. Formally, $k_{\mathcal{X}}$ is characteristic if and only if, for all Borel probability measures $\mathbb{Q}$, the mapping
$\mathbb{Q}\mapsto \int \phi(x)\mathrm{d}\mathbb{Q}$ is injective. I explain the role of the characteristic property below.
\begin{theorem}[Representation via kernel mean embedding]\label{theorem:representation_treatment}
Suppose the conditions of Theorem~\ref{theorem:id_treatment} hold. Further suppose Assumption~\ref{assumption:RKHS} holds and $h_0\in\mathcal{H}$. Then
$$
\gamma_0(d,x,z)=\langle h_0,\phi(d)\otimes \phi(x)\otimes \mu_w(d,x,z) \rangle_{\mathcal{H}} \text{ where } \mu_w(d,x,z):=\int \phi(w) \mathrm{d}\mathbb{P}(w|d,x,z).
$$
Moreover
\begin{enumerate}
\item $\theta_0^{ATE}(d)=\langle h_0, \phi(d)\otimes \mu \rangle_{\mathcal{H}} $ where $\mu:=\int[\phi(x)\otimes \phi(w)]\mathrm{d}\mathbb{P}(x,w)$;
\item $\theta_0^{DS}(d,\tilde{\mathbb{P}})=\langle h_0, \phi(d)\otimes \nu \rangle_{\mathcal{H}} $ where $\nu:=\int[\phi(x)\otimes \phi(w)]\mathrm{d}\tilde{\mathbb{P}}(x,w)$;
\item $\theta_0^{ATT}(d,d')=\langle h_0, \phi(d')\otimes \mu(d)\rangle_{\mathcal{H}} $ where $\mu(d):=\int[\phi(x)\otimes \phi(w)] \mathrm{d}\mathbb{P}(x,w|d)$;
\item $\theta_0^{CATE}(d,v)=\langle h_0, \phi(d)\otimes \phi(v)\otimes \mu(v)\rangle_{\mathcal{H}} $ where $\mu(v):= \int [\phi(x)\otimes \phi(w)] \mathrm{d}\mathbb{P}(x,w|v)$.
\end{enumerate}
\end{theorem}
I present the proof in Appendix~\ref{section:alg_deriv}. Whereas the expressions in Theorem~\ref{theorem:id_treatment} are reweightings of the confounding bridge $h_0$, the expressions in Theorem~\ref{theorem:representation_treatment} are inner products of $h_0$. The quantity $\mu_w(d,x,z)$ encodes the conditional distribution $\mathbb{P}(w|d,x,z)$ from the integral equation that defines the confounding bridge $h_0$. The quantities $\mu,\nu,\mu(d),\mu(v)$ embed various reweighting distributions: $\mathbb{P}(x,w)$, $\tilde{\mathbb{P}}(x,w)$, $\mathbb{P}(x,w|d)$, and $\mathbb{P}(x,w|v)$, respectively. In general, the quantity $\int[\phi(x)\otimes \phi(w)]\mathrm{d}\mathbb{Q}$ encodes the distribution $\mathbb{Q}$ as a function in $\mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{W}}$. The characteristic property for $k_{\mathcal{X}}$ and $k_{\mathcal{W}}$ ensures that the mapping $\mathbb{Q}\mapsto \int [\phi(x)\otimes \phi(w)]\mathrm{d}\mathbb{Q}$ is injective, so that the RKHS representation of the reweighting distribution $\mathbb{Q}$ is unique.
These representations abstractly suggest estimators. For example, for $\theta_0^{ATE}(d)$ the estimator should be of the form $\hat{\theta}^{ATE}(d)=\langle \hat{h}, \phi(d)\otimes \hat{\mu}\rangle_{\mathcal{H}}$, where $\hat{h}$ is an estimator of the confounding bridge $h_0$ and $\hat{\mu}$ is an estimator of the mean embedding $\mu$. I propose a regularized kernel estimator of the confounding bridge function in the spirit of two stage least squares (2SLS): first project $\phi(W)$ onto $\phi(D)\otimes \phi(X)\otimes \phi(Z)$ to obtain $\hat{\mu}_w(D,X,Z)$, then project $Y$ onto $\phi(D)\otimes \phi(X)\otimes \hat{\mu}_w(D,X,Z)$ to obtain $\hat{h}$. I estimate unconditional mean embeddings $\hat{\mu},\hat{\nu}$ with simple averages, and I estimate conditional mean embeddings $\hat{\mu}(d),\hat{\mu}(v)$ with projections similar to $\hat{\mu}_w(d,x,z)$.\footnote{When $V$ is discrete, a simple average suffices; see the discussion in Appendix~\ref{section:application_details}.}
\subsection{Generalized regression loss}
It is not obvious that $\hat{\theta}^{ATE}(d)=\langle \hat{h}, \phi(d)\otimes \hat{\mu}\rangle_{\mathcal{H}}$ has a closed form expression in terms of kernel matrices. In this section, I state the generalized regression losses that define the estimator. These losses generalize the loss in~\eqref{eq:loss}. Along the way, I discuss the connection to 2SLS and provide intuition for these techniques. In the next section, I formally prove that a closed form expression exists and then solve for it, generalizing the expression in~\eqref{eq:sol}.
Similar to 2SLS, I estimate the confounding bridge $\hat{h}$ in two stages. In the first stage, I estimate the conditional mean embedding $\hat{\mu}_w(d,x,z)$. Let $n$ be the number of observations of $(d_i,x_i,w_i,z_i)$ used to estimate the conditional mean embedding $\hat{\mu}_w(d,x,z)$ with regularization parameter $\lambda$. The generalized regression loss for the regression of $\phi(W)$ on $\phi(D,X,Z)$ is
$$
\hat{E}=\operatorname*{\arg\!\min}_{E \in \mathcal{L}_2(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}} \otimes \mathcal{H}_{\mathcal{Z}})} \frac{1}{n}\sum_{i=1}^n \|\phi(w_i)-E^* \phi(d_i,x_i,z_i) \|^2_{\mathcal{H}_{\mathcal{W}}} + \lambda \|E\|^2_{\mathcal{L}_2(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}} \otimes \mathcal{H}_{\mathcal{Z}})}
$$
so that $\hat{\mu}_w(d,x,z)=\hat{E}^* \phi(d,x,z)$. In this notation, $E^*$ is the adjoint operator of $E$, and $\mathcal{L}_2(\mathcal{H}_{\mathcal{W}},\mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}} \otimes \mathcal{H}_{\mathcal{Z}})$ is an RKHS whose elements are operators of the form $E:\mathcal{H}_{\mathcal{W}} \rightarrow \mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}} \otimes \mathcal{H}_{\mathcal{Z}}$.
In the second stage, I estimate the confounding bridge $\hat{h}$. Let $m$ be the number of observations of $(\dot{y}_i,\dot{d}_i,\dot{x}_i,\dot{z}_i)$ used to estimate the confounding bridge $\hat{h}$ with regularization parameter $\xi$. This notation allows the analyst to use different quantities of observations $(n,m)$ to estimate $\hat{\mu}_w(d,x,z)$ and $\hat{h}$, or to reuse the same observations. To estimate $\hat{h}$, I regress $Y$ on $\hat{\mu}(D,X,Z):=\phi(D)\otimes \phi(X)\otimes \hat{\mu}_w(D,X,Z)$. The generalized regression loss for the regression of $Y$ on $\hat{\mu}(D,X,Z)$ is
$$
\hat{h}=\operatorname*{\arg\!\min}_{h\in\mathcal{H}}\frac{1}{m}\sum_{i=1}^{m}\left\{\dot{y}_i-\langle h, \hat{\mu}(\dot{d}_i,\dot{x}_i,\dot{z}_i)\rangle_{\mathcal{H}} \right\}^2+\xi\|h\|^2_{\mathcal{H}}.
$$
\textcolor{black}{\cite[eqs. 5, 6]{mastouri2021proximal} propose the same generalized regression losses for the confounding bridge. The original draft of this paper \cite{singh2020kernel_original} misquoted the regression loss for $\hat{E}$ from \cite[Section 4.1]{singh2019kernel}.\footnote{By misplacing the parentheses as $\mathbb{E}[\phi(D)\otimes \phi(X) \otimes \phi(W)|D,X,Z]$ instead of $\phi(D)\otimes \phi(X) \otimes \mathbb{E}[\phi(W)|D,X,Z]$, $\hat{E}$ was undefined.} This correction was pointed out by \cite{mastouri2021proximal} and an anonymous referee prior to the current draft. However, the closed form \cite[Proposition 2]{mastouri2021proximal} differs as detailed below.}
Finally, when estimating $\hat{\theta}^{ATT}$ and $\hat{\theta}^{CATE}$, one must also estimate the conditional mean embeddings $\hat{\mu}(d)$ and $\hat{\mu}(v)$. The losses for these conditional mean embeddings mirror the loss for the conditional mean embedding $\hat{\mu}_w(d,x,z)$.
\subsection{Closed form}
I present a closed form solution for the confounding bridge estimator, generalizing kernel instrumental variable regression \cite[Algorithm 1]{singh2019kernel} to my extended RKHS construction.
\begin{algorithm}[Estimation of confounding bridge]\label{algorithm:bridge}
Let $\odot$ mean elementwise product. Then
\begin{align*}
\mathbf{A}&=\mathbf{K}_{DD}\odot \mathbf{K}_{XX}\odot \mathbf{K}_{ZZ}\in\mathbb{R}^{n\times n},\quad
\dot{\mathbf{A}}=\mathbf{K}_{D\dot{D}}\odot \mathbf{K}_{X\dot{X}}\odot \mathbf{K}_{Z\dot{Z}}\in\mathbb{R}^{n\times m}, \\
\mathbf{B}&=(\mathbf{A}+n\lambda \mathbf{I})^{-1} \dot{\mathbf{A}}\in\mathbb{R}^{n\times m},\quad \quad \quad
\mathbf{M}=\mathbf{K}_{\dot{D}\dot{D}}\odot \mathbf{K}_{\dot{X}\dot{X}}\odot \{\mathbf{B}^{\top}\mathbf{K}_{WW} \mathbf{B}\} \in\mathbb{R}^{m\times m}, \\
\boldsymbol{\hat{\alpha}}&=(\mathbf{M}\mathbf{M}^{\top}+m\xi \mathbf{M})^{-1}\mathbf{M}\dot{\mathbf{Y}} \in \mathbb{R}^{m} ,\quad
\hat{h}(d,x,w)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot \mathbf{K}_{\dot{X}x}\odot \{\mathbf{B}^{\top} \mathbf{K}_{Ww}\}]\in\mathbb{R}
\end{align*}
where $(\lambda,\xi)$ are ridge penalty hyperparameters.
\end{algorithm}
See Appendix~\ref{section:alg_deriv} for the derivation, which begins with an original proof that such an $\boldsymbol{\hat{\alpha}} \in\mathbb{R}^m$ even exists. \textcolor{black}{\cite[Proposition 2]{mastouri2021proximal} show that a matrix representation $\boldsymbol{\hat{\alpha}}\in \mathbb{R}^{n\times m}$ exists, rather than the vector representation $\boldsymbol{\hat{\alpha}} \in\mathbb{R}^m$ in Algorithm~\ref{algorithm:bridge}. The vector of representation of Algorithm~\ref{algorithm:bridge} is similar to kernel ridge regression.
} The elementwise products arise because tensor product RKHSs correspond to product kernels. For example, the kernel of $\mathcal{H}_{\mathcal{D}}\otimes \mathcal{H}_{\mathcal{X}}\otimes \mathcal{H}_{\mathcal{Z}}$ is $k(d,x,z;d',x',z')=k_{\mathcal{D}}(d,d')k_{\mathcal{X}}(x,x')k_{\mathcal{Z}}(z,z')$ so its kernel matrix is $\mathbf{K}_{DD}\odot \mathbf{K}_{XX}\odot \mathbf{K}_{ZZ}$. $\mathbf{M}$ is essentially the kernel matrix for the conditional mean embedding $\hat{\mu}(d,x,z)=\phi(d)\otimes \phi(x)\otimes \hat{\mu}_w(d,x,z)$. Interpreting these expressions, $\boldsymbol{\hat{\alpha}}$ is clearly the regularized empirical projection of $Y$ onto $\phi(D)\otimes \phi(X)\otimes \hat{\mu}_w(D,X,Z)$ in the spirit of 2SLS.
Next, I present closed form solutions for treatment effects, e.g. $\hat{\theta}^{ATE}(d)=\langle \hat{h}, \phi(d)\otimes \hat{\mu}\rangle_{\mathcal{H}}$, building on $\hat{h}$ from Algorithm~\ref{algorithm:bridge}. Whereas \cite[Algorithm 3.1]{singh2020kernel} estimate treatment effects assuming selection on observables, Algorithm~\ref{algorithm:treatment} estimates treatment effects assuming access to negative controls. For $\theta_0^{DS}$, let $\tilde{n}$ be the number of observations of $(\tilde{x}_i,\tilde{w}_i)$ drawn from population $\tilde{\mathbb{P}}$.
\begin{algorithm}[Estimation of treatment effects]\label{algorithm:treatment}
Treatment effect estimators have the closed form solutions
\begin{enumerate}
\item $\hat{\theta}^{ATE}(d)=n^{-1}\sum_{i=1}^n \boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot \mathbf{K}_{\dot{X}x_i}\odot \{\mathbf{B}^{\top} \mathbf{K}_{Ww_i}\}]$
\item $\hat{\theta}^{DS}(d,\tilde{\mathbb{P}})=\tilde{n}^{-1}\sum_{i=1}^{\tilde{n}} \boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot \mathbf{K}_{\dot{X}\tilde{x}_i}\odot \{\mathbf{B}^{\top} \mathbf{K}_{W\tilde{w}_i}\}]$
\item $\hat{\theta}^{ATT}(d,d')=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d'}\odot \{[\mathbf{K}_{\dot{X}X}\odot \{\mathbf{B}^{\top}\mathbf{K}_{WW}\}](\mathbf{K}_{DD}+n\lambda_1\mathbf{I})^{-1}\mathbf{K}_{Dd}\}]$
\item $\hat{\theta}^{CATE}(d,v)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{\dot{D}d}\odot \mathbf{K}_{\dot{V}v}\odot \{[\mathbf{K}_{\dot{X}X}\odot \{\mathbf{B}^{\top}\mathbf{K}_{WW}\}](\mathbf{K}_{VV}+n\lambda_2\mathbf{I})^{-1}\mathbf{K}_{Vv}\}] $
\end{enumerate}
where $(\lambda_1,\lambda_2)$ are ridge regression penalty hyperparameters. In $\hat{\theta}^{CATE}(d,v)$, $\boldsymbol{\hat{\alpha}}$ is the coefficient for the confounding bridge that includes $V$.
\end{algorithm}
See Appendix~\ref{section:alg_deriv} for the derivation. I give theoretical values for the regularization parameters that balance bias and variance in Section~\ref{sec:consistency} below. In particular, I specify $(\lambda,\xi)$ in Theorem~\ref{theorem:consistency_bridge} and $(\lambda_1,\lambda_2)$ in Theorem~\ref{theorem:consistency_treatment}. In Appendix~\ref{section:tuning}, I propose a practical tuning procedure based on the closed form solution of leave-one-out cross validation (LOOCV) to empirically balance bias and variance, and I discuss the time complexity.
\subsection{Summary}
To fix ideas, I summarize the end-to-end procedure for $\hat{\theta}^{ATE}(d)$. For simplicity, I suppose that the analyst re-uses observations in the two stages of confounding bridge estimation. I provide additional discussion in Appendix~\ref{section:alg_deriv} for researchers who are new to kernel methods.
\begin{algorithm}[End-to-end details]\label{algorithm:concrete_main}
Given $n$ observations of outcome $Y$, treatment $D$, covariates $X$, negative control outcome $W$, and negative control treatment $Z$,
\begin{enumerate}
\item Specify the kernels $k_{\mathcal{D}},k_{\mathcal{X}},k_{\mathcal{W}},k_{\mathcal{Z}}$.
\begin{enumerate}
\item For multivariate objects, e.g. $X=(X_1,...,X_p)$, use the product of scalar kernels
$$
k_{\mathcal{X}}(x,x')=\prod_{j=1}^p k_{\mathcal{X}_1}(x_1,x_1')\cdot ...\cdot k_{\mathcal{X}_p}(x_p,x_p').
$$
\item Tune the scalar kernel hyperparameters. For example, if the treatment kernel $k_{D}$ is chosen as the Gaussian kernel, a standard heuristic is to use the median interpoint distance among observed treatment values.
\item Compute the kernel matrices, e.g. $\mathbf{K}_{DD}\in\mathbb{R}^{n\times n}$ with $(i,j)$-th entry $k_{\mathcal{D}}(d_i,d_j)$.
\end{enumerate}
\item Specify the regularization hyperparameters $(\lambda,\xi)$.
\begin{enumerate}
\item For $\lambda$, I derive the closed form solution of LOOCV in Appendix~\ref{section:tuning}.
\item The same procedure applies to $\xi$, plugging in the chosen value of $\lambda$.
\end{enumerate}
\item Estimate the confounding bridge $\hat{h}$ in two stages, using $(\lambda,\xi)$.
\begin{enumerate}
\item Estimate the distribution in the integral equation $\hat{\mathbb{P}}(w|d,x,z)$ via its mean embedding $\hat{\mu}_w(d,x,z)$ with regularization $\lambda$ as
$$
[\hat{\mu}_w(d,x,z)](w)=\mathbf{K}_{wW} (\mathbf{K}_{DD}\odot \mathbf{K}_{XX}\odot \mathbf{K}_{ZZ}+n\lambda \mathbf{I})^{-1}[\mathbf{K}_{Dd}\odot \mathbf{K}_{Xx} \odot \mathbf{K}_{Zz}].
$$
\item Regress $Y$ onto $\phi(D)\otimes \phi(X)\otimes \hat{\mu}_w(D,X,Z)$ with regularization $\xi$. It turns out that $\hat{h}(d,x,w)=\boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{Dd}\odot \mathbf{K}_{Xx}\odot \{\mathbf{B}^{\top} \mathbf{K}_{Ww}\}]\in\mathbb{R}$ where
\begin{align*}
\mathbf{A}&=\mathbf{K}_{DD}\odot \mathbf{K}_{XX}\odot \mathbf{K}_{ZZ}\in\mathbb{R}^{n\times n},\quad
\mathbf{B}=(\mathbf{A}+n\lambda \mathbf{I})^{-1} \mathbf{A}\in\mathbb{R}^{n\times n}, \\
\mathbf{M}&=\mathbf{K}_{DD}\odot \mathbf{K}_{XX}\odot \{\mathbf{B}^{\top}\mathbf{K}_{WW} \mathbf{B}\} \in\mathbb{R}^{n\times n},\quad
\boldsymbol{\hat{\alpha}}=(\mathbf{M}\mathbf{M}^{\top}+n\xi \mathbf{M})^{-1}\mathbf{M}\mathbf{Y} \in \mathbb{R}^{n}.
\end{align*}
\end{enumerate}
\item Estimate the counterfactual distribution $\hat{\mathbb{P}}(x,w)$ via its mean embedding $\hat{\mu}$ as
$$
[\hat{\mu}](x,w)=\frac{1}{n}\sum_{i=1}^n k_{\mathcal{X}}(x_i,x)k_{\mathcal{W}}(w_i,w)\in\mathbb{R}.
$$
\item Estimate the dose response $\hat{\theta}^{ATE}(d)$ by combining $\hat{h}$ and $\hat{\mu}$ according to $
\hat{\theta}^{ATE}(d)=\langle \hat{h}, \phi(d)\otimes \hat{\mu} \rangle_{\mathcal{H}}
$. To do so, match the common arguments $(x,w)$ of $\hat{h}$ and $\hat{\mu}$. In summary,
$$
\hat{\theta}^{ATE}(d)=\frac{1}{n}\sum_{i=1}^n \boldsymbol{\hat{\alpha}}^{\top}[\mathbf{K}_{Dd}\odot \mathbf{K}_{Xx_i}\odot \{\mathbf{B}^{\top} \mathbf{K}_{Ww_i}\}]\in\mathbb{R}.
$$
\end{enumerate}
\end{algorithm}
\section{Consistency}\label{sec:consistency}
To define the learning problem in Section~\ref{sec:problem}, I placed three assumptions: availability of negative controls (Assumption~\ref{assumption:negative}); existence and completeness of the confounding bridge (Assumption~\ref{assumption:solution}); and invariance of the confounding bridge for transfer learning (Assumption~\ref{assumption:covariate}). To construct an algorithm in Section~\ref{section:algorithm}, I assumed RKHS regularity (Assumption~\ref{assumption:RKHS}). To guarantee uniform consistency in this section, I place three final assumptions: original space regularity (Assumption~\ref{assumption:original}); smoothness and effective dimension of conditional expectation operators (Assumption~\ref{assumption:smooth_op}); and smoothness and effective dimension of the confounding bridge (Assumption~\ref{assumption:smooth_bridge}). I first prove uniform consistency of the confounding bridge, then uniform consistency of treatment effects. As before, for $\theta_0^{CATE}$ I extend the stated assumptions from $X$ to $(V,X)$.
\textcolor{black}{\cite[Theorem 2]{mastouri2021proximal} analyze excess risk of a surrogate loss for a kernel two stage regression estimator of the confounding bridge, with finite sample rates, under smoothness and effective dimension assumptions. Excess risk of a surrogate loss corresponds to projected mean square error for the confounding bridge.
}
\subsection{Confounding bridge}
I require weak regularity conditions on the original spaces of the outcome $Y$, treatment $D$, covariates $(V,X)$, and negative controls $(W,Z)$.
\begin{assumption}[Original space regularity conditions]\label{assumption:original}
Assume
\begin{enumerate}
\item $Y \in\mathcal{Y}\subset \mathbb{R}$ is bounded, i.e. there exists $C<\infty$ such that $|Y|\leq C$ almost surely.
\item $\mathcal{D}$, $\mathcal{X}$, $\mathcal{W}$, and $\mathcal{Z}$ are Polish spaces.
\end{enumerate}
\end{assumption}
To simplify notation and analysis, I require that the outcome $Y\in\mathbb{R}$ is a bounded scalar. More generally, $\mathcal{Y}$ could be a separable Hilbert space. I preserve generality for treatment, covariates, and negative controls. A Polish space is a separable and completely metrizable topological space. Random variables with support in a Polish space may be discrete or continuous and low, high, or infinite dimensional. As such, I allow for treatment, covariates, and negative controls that could even be texts, graphs, or images.
Next, I place a smoothness and effective dimension conditions in the sense of~\eqref{eq:prior} and~\eqref{eq:prior2} for the conditional mean embedding $\mu_{w}(d,x,z)$. In anticipation of later analysis, I articulate this assumption abstractly. Consider the abstract conditional mean embedding $\mu_{a}(b):=\int \phi(a)\mathrm{d}\mathbb{P}(a|b)$ where $a\in\mathcal{A}_{\ell}$ and $b\in\mathcal{B}_{\ell}$. I will ultimately consider the three different conditional mean embeddings $\mu_w(d,x,z)$, $\mu(d)$, and $\mu(v)$ indexed by $\ell \in\{0,1,2\}$. As previewed in Section~\ref{sec:problem}, the conditional expectation operator $E_{\ell}:\mathcal{H}_{\mathcal{A}_{\ell}}\rightarrow\mathcal{H}_{\mathcal{B}_{\ell}}$, $f(\cdot)\mapsto \mathbb{E}[f(A_{\ell})|B_{\ell}=\cdot]$ encodes the same information as $\mu_{a}(b)$. In particular,
$$
\mu_{a}(b)=\int \phi(a)\mathrm{d}\mathbb{P}(a|b) =[E_{\ell}\phi(\cdot)](b) =[E_{\ell}^* \phi(b)](\cdot),\quad a\in\mathcal{A}_{\ell},\quad b\in\mathcal{B}_{\ell}
$$
where $E_{\ell}^*:\mathcal{H}_{\mathcal{B}_{\ell}} \rightarrow \mathcal{H}_{\mathcal{A}_{\ell}}$, $g(\cdot)\mapsto \mathbb{E}[g(B_{\ell})|A_{\ell}=\cdot]$ is the adjoint of $E_{\ell}$. I denote the space of Hilbert-Schmidt operators between $\mathcal{H}_{\mathcal{A}_{\ell}}$ and $\mathcal{H}_{\mathcal{B}_{\ell}}$ by $\mathcal{L}_2(\mathcal{H}_{\mathcal{A}_{\ell}},\mathcal{H}_{\mathcal{B}_{\ell}})$, which is an RKHS in its own right.
\begin{assumption}[Smoothness and spectral decay for conditional expectation operator]\label{assumption:smooth_op}
Assume $E_{\ell}\in [\mathcal{L}_2(\mathcal{H}_{\mathcal{A}_{\ell}},\mathcal{H}_{\mathcal{B}_{\ell}})]^{^{c_{\ell}}}$ and $\eta_j(\mathcal{H}_{\mathcal{B}_{\ell}})\leq C j^{-b_{\ell}}$.
\end{assumption}
To specialize the assumption, all one has to do is specify $\mathcal{A}_{\ell}$ and $\mathcal{B}_{\ell}$. For example, for $\mu_{w}(d,x,z)$, $\mathcal{A}_0=\mathcal{W}$ and $\mathcal{B}_0=\mathcal{D}\times \mathcal{X}\times \mathcal{Z}$. By assuming smoothness and effective dimension of $E_0$, I assume smoothness and effective dimension of $\mu_w(d,x,z)$. I explicitly specialize the assumption in Appendices~\ref{section:consistency_proof1} and~\ref{section:consistency_proof2}.
The final assumption that I place in order to prove uniform consistency of the confounding bridge estimator $\hat{h}$ is that the confounding bridge $h_0$ is smooth in the sense of~\eqref{eq:prior} with low effective dimension in the sense of~\eqref{eq:prior2}. Recall that the features for the RKHS $\mathcal{H}$ are $\phi(d,x,w)=\phi(d)\otimes \phi(x)\otimes \phi(w)$. Recall from Theorem~\ref{theorem:representation_treatment} that we must solve the integral equation $\gamma_0(d,x,z)=\langle h_0, \mu(d,x,z) \rangle_{\mathcal{H}}$ where $\mu(d,x,z)=\phi(d)\otimes \phi(x)\otimes \mu_w(d,x,z)$ is a mean embedding. By construction,
$$
\langle \mu(d,x,z), \mu(d',x',z') \rangle_{\mathcal{H}}=k_{\mathcal{D}}(d,d')k_{\mathcal{X}}(x,x')\int k_{\mathcal{W}}(w,w')\mathrm{d}\mathbb{P}(w|d,x,z)\mathrm{d}\mathbb{P}(w'|d',x',z'),
$$
which we may interpret as the inner product of a space $\mathcal{H}_{\mu}\subset \mathcal{H}$. In words, the kernel of $\mathcal{H}_{\mu}$ is simply the kernel of $\mathcal{H}$ after integrating out $(w,w')$ according to the conditional distribution $\mathbb{P}(w|d,x,z)$ from the integral equation. Formally, $\mathcal{H}_{\mu}$ can be viewed as an RKHS of functions evaluated on mean embeddings instead of features \cite{szabo2016learning}.
\begin{assumption}[Smoothness and spectral decay for confounding bridge]\label{assumption:smooth_bridge}
Assume $h_0\in \mathcal{H}_{\mu}^c$ and $\eta_j(\mathcal{H}_{\mu})\leq C j^{-b}$.
\end{assumption}
As a technical aside, an analyst may introduce additional nonlinearity by enriching the model and enriching this assumption. A richer model would instead allow $\gamma_0(d,x,z)=H_0\mu(d,x,z)$ where $H_0:\mathcal{H}\rightarrow \mathbb{R}$ is a nonlinear mapping in a richer RKHS. In such case, the smoothness and effective dimension assumptions would be placed on $H_0$ rather than $h_0$ and a notion of H\"older continuity is required \cite[Table 1]{szabo2016learning}. For clarity, I omit this complexity.
Under these conditions, I arrive at the first main result: uniform consistency of the confounding bridge. This result appears to be the first finite sample uniform analysis of nonparametric instrumental variable regression in the RKHS. By allowing $n\neq m$, I allow
the possibility of asymmetric sample splitting and the use of observations from different
data sets. The finite sample rates are expressed in terms of $(n,m)$. The parameter $a>0$ characterizes the ratio between the sample sizes.
\begin{theorem}[Consistency of confounding bridge]\label{theorem:consistency_bridge}
Suppose Assumptions~\ref{assumption:solution}, \ref{assumption:RKHS}, \ref{assumption:original}, \ref{assumption:smooth_op} with $\mathcal{A}_0=\mathcal{W}$ and $\mathcal{B}_0=\mathcal{D}\times \mathcal{X}\times \mathcal{Z}$, and~\ref{assumption:smooth_bridge} hold. Set $\lambda=n^{-\frac{1}{c_0+1/b_0}}$ and $n=m^{\frac{a(c_0+1/b_0)}{c_0-1}}$ where $a>0$.
\begin{enumerate}
\item If $a\leq (c+3)/(c+1/b)$ then
$\|\hat{h}-h_0\|_{\infty}=O_p(m^{-\frac{1}{2}\frac{a(c-1)}{c+3}})$ with $\xi=m^{-\frac{a}{c+3}}$.
\item If $a\geq (c+3)/(c+1/b)$ then $\|\hat{h}-h_0\|_{\infty}=O_p(m^{-\frac{1}{2}\frac{c-1}{c+1/b}})$ with $\xi=m^{-\frac{1}{c+1/b}}$.
\end{enumerate}
\end{theorem}
See Appendix~\ref{section:consistency_proof1} for exact finite sample rates and intermediate results in RKHS norm. At $a=(c+3)/(c+1/b)$, the convergence rate $m^{-\frac{1}{2}\frac{c-1}{c+1/b}}$ attains the rate of single stage kernel ridge regression with respect to $m$ \cite{fischer2017sobolev}. This rate is calibrated by $c$, the smoothness of confounding bridge and $b$, the effective dimension of its RKHS.
The rate of Theorem~\ref{theorem:consistency_bridge} requires the ratio between sample sizes to be $n=m^{\frac{c+3}{c+1/b}\cdot \frac{(c_0+1/b_0)}{c_0-1}}$, implying $n\gg m$. In practice, the analyst often uses the same observations to estimate $\hat{\mu}_w(d,x,z)$ and $\hat{h}$.
\begin{corollary}[Reusing samples]\label{corollary:reuse}
If samples are reused to estimate $\hat{\mu}_w(d,x,z)$ and $\hat{h}$, then $n=m$, $a=(c_0-1)/(c_0+1/b_0)$, $\xi=n^{-\frac{c_0-1}{(c_0+1/b_0)(c+3)}}$, and $\|\hat{h}-h_0\|_{\infty}=O_p(n^{-\frac{1}{2}\frac{c_0-1}{c_0+1/b_0}\frac{c-1}{c+3}})$.
\end{corollary}
This rate adapts to the smoothness $c_0$ of the conditional distribution as well as the smoothness $c$ of the confounding bridge. The slow rate reflects the challenge of a uniform norm guarantee in an ill posed inverse problem. A faster rate could be possible under further assumptions, which I leave to future work.
\subsection{Treatment effects}
Recall from Theorem~\ref{theorem:representation_treatment} that $\theta_0^{ATT}$ and $\theta_0^{CATE}$ contain conditional mean embeddings $\mu(d)$ and $\mu(v)$, respectively. I estimate these conditional mean embeddings by regularized projections in Algorithm~\ref{algorithm:treatment}. To control bias and variance, I place smoothness and effective dimension conditions in the sense of~\eqref{eq:prior} and~\eqref{eq:prior2} for $\mu(d)$ and $\mu(v)$ as well. As previewed in the discussion about $\mu_w(d,x,z)$ and $E_0$, a conditional mean embedding corresponds to a conditional expectation operator. As before, all one has to do is specify $\mathcal{A}_{\ell}$ and $\mathcal{B}_{\ell}$ to specialize the assumption. For $\mu(d)$, $\mathcal{A}_1=\mathcal{X}\times \mathcal{W}$ and $\mathcal{B}_1=\mathcal{D}$; for $\mu(v)$, $\mathcal{A}_2=\mathcal{X}\times \mathcal{W}$ and $\mathcal{B}_2=\mathcal{V}$.
Under these conditions, I arrive at the second main result: uniform consistency of the treatment effect estimators. For simplicity, I specialize to the scenario with the fastest rates. Recall $\tilde{n}$ is the number of observations drawn from the alternative population $\tilde{\mathbb{P}}$.
\begin{theorem}[Consistency of treatment effects]\label{theorem:consistency_treatment}
Suppose Assumption~\ref{assumption:negative} holds, as well as the conditions of Theorem~\ref{theorem:consistency_bridge}. Set $(\lambda,\lambda_1,\lambda_2)=(n^{-\frac{1}{c_0+1/b_0}},n^{-\frac{1}{c_1+1/b_1}},n^{-\frac{1}{c_2+1/b_2}})$, $\xi=m^{-\frac{1}{c+1/b}}$, and $n=m^{\frac{c+3}{c+1/b}\cdot \frac{(c_0+1/b_0)}{c_0-1}}$.
\begin{enumerate}
\item Then
$$
\|\hat{\theta}^{ATE}-\theta_0^{ATE}\|_{\infty}=O_p\left(m^{-\frac{1}{2}\frac{c-1}{c+1/b}}+n^{-\frac{1}{2}}\right).
$$
\item If in addition Assumption~\ref{assumption:covariate} holds, then
$$
\|\hat{\theta}^{DS}(\cdot,\tilde{\mathbb{P}})-\theta_0^{DS}(\cdot,\tilde{\mathbb{P}})\|_{\infty}=O_p\left( m^{-\frac{1}{2}\frac{c-1}{c+1/b}}+\tilde{n}^{-\frac{1}{2}}\right).
$$
\item If in addition Assumption~\ref{assumption:smooth_op} holds with $\mathcal{A}_1=\mathcal{X}\times \mathcal{W}$ and $\mathcal{B}_1=\mathcal{D}$, then
$$
\|\hat{\theta}^{ATT}-\theta_0^{ATT}\|_{\infty}=O_p\left(m^{-\frac{1}{2}\frac{c-1}{c+1/b}}+n^{-\frac{1}{2}\frac{c_1-1}{c_1+1/b_1}}\right).
$$
\item If in addition Assumption~\ref{assumption:smooth_op} holds with $\mathcal{A}_2=\mathcal{X}\times \mathcal{W}$ and $\mathcal{B}_2=\mathcal{V}$, then
$$
\|\hat{\theta}^{CATE}-\theta_0^{CATE}\|_{\infty}=O_p\left(m^{-\frac{1}{2}\frac{c-1}{c+1/b}}+n^{-\frac{1}{2}\frac{c_2-1}{c_2+1/b_2}}\right).
$$
\end{enumerate}
\end{theorem}
See Appendix~\ref{section:consistency_proof2} for exact finite sample rates. Inspecting the rates, we see that each one is a sum of the rate for the confounding bridge from Theorem~\ref{theorem:consistency_bridge} with the rate for the appropriate mean embedding estimation procedure. The rates adapt to the smoothness parameters $(c,c_0,c_1,c_2)$ and effective dimension parameters $(b,b_0,b_1,b_2)$ of the confounding bridge $h_0$ and the conditional expectation operators $E_0$, $E_1$, and $E_2$. Equivalently, the rates adapt to the smoothness and effective dimension of the confounding bridge $h_0$ and the conditional distributions $\mathbb{P}(w|d,x,z)$, $\mathbb{P}(x,w|d)$, and $\mathbb{P}(x,w|v)$.
The goal of this project is to propose dose response and heterogeneous treatment effect estimators to ultimately inform policy and medical decisions. For this reason, I prove a uniform guarantee that strictly controls error for \textit{any} level of treatment, rather than a mean square guarantee that controls error for the \textit{average} level of treatment. Uniform guarantees come at the cost of slower rates. In negative control treatment effect estimation, the ill posedness of the confounding bridge learning problem compounds this phenomenon. Theorem~\ref{theorem:consistency_treatment} appears to be the first finite sample analysis for nonparametric negative control treatment effects, and it holds under weak assumptions. Obtaining faster rates, perhaps via further assumptions, is an important direction for future work.
\section{Simulation and application}\label{section:experiments}
\subsection{Simulations}
I evaluate the empirical performance of the new estimators. I focus on dose response with negative controls, and consider various designs with varying sample sizes. Specifically, I compare the new algorithm that uses negative controls (\verb|N.C.|) with an existing RKHS algorithm for nonparametric treatment effects (\verb|T.E.|) \cite{singh2020kernel} that ignores unobserved confounding and instead classifies negative controls as additional covariates. Whereas the new algorithm involves reweighting a confounding bridge, the previous algorithm involves reweighting a regression. For each design, sample size, and algorithm, I implement 100 simulations and calculate mean square error (MSE) with respect to the true counterfactual function.
\begin{figure}[ht]
\begin{centering}
\begin{subfigure}[b]{0.30\textwidth}
\centering
\includegraphics[width=\textwidth]{img/results_n1.eps}
\vspace{-15pt}
\caption{Quadratic}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.30\textwidth}
\centering
\includegraphics[width=\textwidth]{img/results_n2.eps}
\vspace{-15pt}
\caption{Sigmoid}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.30\textwidth}
\centering
\includegraphics[width=\textwidth]{img/results_n3.eps}
\vspace{-15pt}
\caption{Peaked}
\end{subfigure}
\par
\vspace{-5pt}
\caption{\label{fig:sim_design}
Simulation results for various designs}
\end{centering}
\end{figure}
Specifically, I adapt the continuous treatment effect design proposed by \cite{colangelo2020double}. Whereas the original setting studied by \cite{colangelo2020double} has no unobserved confounding, my modification does have unobserved confounding. The goal is to learn the counterfactual function $\theta_0^{ATE}(d)$, which may be a quadratic, sigmoid, or peaked function. A single observation consists of the tuple $(Y,W,D,Z,X)$ for outcome, negative control outcome, treatment, negative control treatment, and covariates. $(Y,D)$ are continuous scalars. In the baseline experiment, $X\in\mathbb{R}^5$ and $(Z,W)\in\mathbb{R}$.
To explore the role of sample size, I consider $n\in\{100,500,1000,5000,10000\}$. To explore the role of dimension, I focus on the quadratic design, fix sample size at $n=1000$, and then vary $dim(X)\in\{1,5,10,50,100\}$, $dim(Z)\in\{1,5,10\}$, or $dim(W)\in\{1,5,10\}$. This range of sample sizes and dimensions is common in epidemiology research. Figures~\ref{fig:sim_design} and~\ref{fig:sim_dimension} visualize results. Across designs, sample sizes, and dimensions, the use of negative controls to adjust for unobserved confounding improves performance. The improvement is generally increasing in $n$ and $dim(Z)$ but decreasing in $dim(X)$ and $dim(W)$. Intuitively, $(X,W)$ are the variables used in the reweighting step, which is common across the two estimators \verb|N.C.| and \verb|T.E.|; as this step becomes relatively more important, the estimators become more similar. See Appendix~\ref{section:simulation_details} for implementation details as well as additional simulations that confirm: (i) robustness to tuning; (ii) improvement when treatment is discrete; and (iii) inefficiency in the absence of unobserved confounding.
\begin{figure}[ht]
\begin{centering}
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{img/figure_p.eps}
\vspace{-15pt}
\caption{Covariate}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{img/figure_z.eps}
\vspace{-15pt}
\caption{N.C. treatment}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{img/figure_w.eps}
\vspace{-15pt}
\caption{N.C. outcome}
\end{subfigure}
\par
\vspace{-5pt}
\caption{\label{fig:sim_dimension}
Simulation results for various dimensions}
\end{centering}
\end{figure}
\subsection{Dose response of cigarette smoking}
Estimating the effect of cigarette smoking on infant birth weight is challenging for several reasons. First, pregnant women are classified as a vulnerable population, so they are typically excluded from clinical trials of any kind. When the treatment of interest causes harm, ethical considerations preclude randomization. Therefore \textit{observational} data are the only option. Second, pregnancy induces many physiological changes, so medical knowledge predicts different dose response curves for women who are pregnant compared to women who are not pregnant. For example, plasma volume increases 35\%, cardiac output increases 40\%, and glomerular filtration rate (a measure of kidney function) increases 50\% during pregnancy \cite{cunningham2014williams}. Therefore the shape of the dose response curve for pregnant women is an unknown, \textit{nonparametric} quantity. Third, medical records exclude an \textit{unobserved confounder} known to be crucial for maternal-fetal health: household income \cite{joseph2007socioeconomic}.
In this section, I argue that medical records include variables that satisfy the properties of \textit{negative controls} for unobserved income. I provide preliminary results and outline directions for future work on this topic. Finally, I discuss what issues may arise if there are additional unobserved confounders. The purpose of this case study is to illustrate how the proposed estimators may be useful in epidemiology research, though the findings are not conclusive.
I estimate the dose response curve of cigarette smoking on infant birth weight using a data set of singleton births in the state of Pennsylvania between 1989 and 1991 assembled by \cite{almond2005costs} and subsequently analyzed by \cite{cattaneo2010efficient}. I focus on Pennsylvania because smoking data are available for over 95\% of mothers. I focus on singleton births because multiple gestations reflect a variety of factors and result in different fetal growth trajectories. 21\% of women report smoking during pregnancy, and I subset to this sample. I consider the subpopulations of (a) nonhispanic white women who smoke ($n=73,834$), (b) nonhispanic black women who smoke ($n=17,625$), and (c) hispanic women who smoke ($n=2,152$). Formally, I estimate $\theta_0^{CATE}(d,v)$ where $D\in\mathbb{R}$ is the number of cigarettes smoked per day, and $V$ concatenates mother's race $V_1 \in\{\text{white, black, hispanic}\}$ and mother's smoking status $V_2 \in\{0,1\}$. See Appendix~\ref{section:application_details} for further discussion.
The classification of variables extensively relies on domain knowledge, so I sought the expertise of physicians from the Department of Obstetrics, Gynecology \& Reproductive Biology at Harvard Medical School. Together, we arrived at the classification given in Appendix~\ref{section:application_details}, based on a canonical textbook \cite{cunningham2014williams}. Figure~\ref{dag:nc_smoking} illustrates the model. Demographics, alcohol consumption, prenatal care, existing medical conditions, county, and year serve as covariates $X$ since they may be associated with both smoking $D$ and birth weight $Y$.
\begin{wrapfigure}{H}{0.45\textwidth}
\vspace{-30pt}
\begin{center}
\begin{adjustbox}{width=.45\textwidth}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
semithick]
\tikzstyle{every state}=[draw=black,text=black]
\node[state] (x) [fill=gray] {$X$};
\node[state] (w) [right of=x, fill=gray, label={[align=center]above: birth order,\\[-10pt]sex,\\[-10pt]Rh}] {$W$};
\node[state] (z) [left of=x, fill=gray, label={[align=center]above:edu-\\[-10pt]cation}] {$Z$};
\node[state] (d) [below left of=x, fill=gray, label={below:smoking}] {$D$};
\node[state] (y) [below right of=x, fill=gray, label={below:birth weight}] {$Y$};
\node[state] (u) [above of=x, label={income}] {$U$};
\path (u) edge node {$ $} (z)
edge node {$ $} (x)
edge node {$ $} (w)
edge node {$ $} (d)
edge node {$ $} (y)
(x) edge node {$ $} (w)
edge node {$ $} (z)
edge node {$ $} (d)
edge node {$ $} (y)
(z) edge node {$ $} (d)
(w) edge node {$ $} (y)
(d) edge node {$ $} (y);;
\end{tikzpicture}
\end{adjustbox}
\vspace{-25pt}
\caption{Smoking DAG}
\label{dag:nc_smoking}
\end{center}
\vspace{-20pt}
\end{wrapfigure}
Education serves as a negative control treatment $Z$ because it reflects unobserved confounding due to household income $U$ but has no direct medical effect on birth weight $Y$. Formally, we require $Z\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} Y | D,U,X$: education is independent of birth weight after conditioning on smoking, income, and observed covariates. Prenatal care and weight gain are the observed covariates that, along with smoking and income, justify the conditional independence between education and birth weight.
Infant birth order and sex serve as a negative control outcomes $W$ because family size reflects household income $U$ but is not directly caused by smoking $D$ or education $Z$. Formally, we require $W\raisebox{0.05em}{\rotatebox[origin=c]{90}{$\models$}} D,Z | U,X$: family size is independent of smoking and education after conditioning on income and observed covariates. Age and marriage status are the observed covariates that, along with income, justify the conditional independence between education and family size. We also include Rh sensitization as a negative control outcome because it is one of the few medical conditions not affected by smoking (it is caused by blood type).
\begin{figure}[ht]
\begin{centering}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{img/white.eps}
\vspace{-25pt}
\caption{White smoking mothers}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{img/black.eps}
\vspace{-25pt}
\caption{Black smoking mothers}
\end{subfigure}
\par
\vspace{-10pt}
\caption{\label{fig:smoke}
Effect of cigarette smoking on birth weight for different subpopulations; $\theta_0^{CATE}(d,v)$ where $D\in\mathbb{R}$ is the number of cigarettes smoked per day, and $V$ concatenates mother's race $V_1 \in\{\text{white, black, hispanic}\}$ and mother's smoking status $V_2 \in\{0,1\}$.
}
\end{centering}
\end{figure}
\begin{wrapfigure}{R}{0.45\textwidth}
\vspace{-10pt}
\begin{center}
\begin{subfigure}[b]{0.45\textwidth}
\setcounter{subfigure}{2}
\centering
\includegraphics[width=\textwidth]{img/hispanic.eps}
\vspace{-25pt}
\caption{Hispanic smoking mothers}
\end{subfigure}
\vspace{-25pt}
\caption{Effect of cigarette smoking on birth weight for different subpopulations; $\theta_0^{CATE}(d,v)$ where $D\in\mathbb{R}$ is the number of cigarettes smoked per day, and $V$ concatenates mother's race $V_1 \in\{\text{white, black, hispanic}\}$ and mother's smoking status $V_2 \in\{0,1\}$.}
\label{fig:smoke2}
\end{center}
\vspace{-20pt}
\end{wrapfigure}
I implement both the new algorithm (\verb|N.C.|) and an existing RKHS algorithm for continuous treatment effect (\verb|T.E|) \cite{singh2020kernel} that ignores unobserved confounding. For the method that ignores unobserved confounding, I classify negative controls as additional covariates. Figures~\ref{fig:smoke} and~\ref{fig:smoke2} visualize results for white, black, and hispanic smoking mothers. The effect of cigarettes smoked per day $D$ on birth weight in grams $Y$ is generally negative with similar shapes across subpopulations. The counterfactual birth weights for black and hispanic mothers are lower than for white mothers when the number of cigarettes is high. The main finding is that using negative controls leads to higher dose response curves. Under the stated causal assumptions, the gap between \verb|N.C.| and \verb|T.E.| is the magnitude of unobserved confounding due to income. These preliminary results support the clinical hypothesis that poverty is an unmeasured confounder that affects infant birth weight. Unobserved poverty may substantially mislead observational studies that fail to account for it. See Appendix~\ref{section:application_details} for implementation details.
An unanticipated result is that the dose response curves appear nonmonotonic; estimated counterfactual birth weight increases before it decreases. This phenomenon prevails across subpopulations, and it can be seen in not only \verb|N.C.| but also \verb|T.E.| and the raw data. We propose two conjectures based on the data and domain knowledge. Both of these conjectures are ways in which the data generating process may violate the causal assumptions in Assumption~\ref{assumption:negative}.
First, it may be that measurement error contaminates observations. In the raw data, it appears that when the number of cigarettes was between one and 10 it may have been rounded up to 10. Indeed, Figures~\ref{fig:smoke} and~\ref{fig:smoke2} document substantial point masses at multiples of 10. This phenomenon would violate our causal model, since it would mean that when the true treatment value $d$ was less than 10, we observe $D=10$, $Z=z$, yet $Y=Y^{(d,z)}$. In such case, estimates of the dose response for $d<10$ may be unreliable. How to account for measurement error in negative control estimation remains an open question.
Second, it could be that another unobserved confounder exists, is not detected by the negative controls, and disproportionately affects women who reported smoking less than 10 cigarettes. Previous studies suggest that rural-urban classification, poverty, and psychosocial stress are possible confounders \cite{hobel2008psychosocial}. In our analysis, we account for rural-urban classification as an observed covariate and we account for poverty via negative controls, but we did not find plausible negative controls for stress in this data set; see Appendix~\ref{section:application_details} for further discussion. Indeed, psychosocial stress is notoriously difficult to measure, and it may cause both smoking and low birth weight.
We pose for future work a further analysis that adjusts for unobserved confounding due to both income and stress.
\section{Conclusion}\label{sec:conclusion}
I propose a new family of nonparametric algorithms for learning treatment effects with negative controls. The estimators are easily implemented and uniformly consistent. As a contribution to the negative control literature, I propose methods to estimate dose response curves and heterogeneous treatment effects under the assumption that treatment effects are smooth. As a contribution to the kernel methods literature, I show how the RKHS is well suited to causal inference in the presence of unobserved confounding. As a contribution to maternal-fetal medicine, I propose a toolkit for estimating dose response curves for pregnant women from medical records despite unobserved confounding. The results suggest that RKHS methods may be an effective bridge between epidemiology and machine learning.
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