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Stable Policy Learning
\maketitle
\begin{abstract}
In evidence-based policymaking, typically one experimental sample is observed, then a learned policy recommendation is implemented at scale.
Policies learned from the experimental data can perform well in expected welfare, yet random sampling in the experiment can produce recommendations with poor welfare outcomes.
In this paper, we ask: how should policy learning algorithms balance expected welfare against sampling risk?
Our main contribution is to show that algorithmic stability plays a central role in characterizing and navigating the tradeoff.
Intuitively, if a policy learning algorithm's recommendation remains stable when one experimental unit is replaced, then that algorithm has limited sampling risk.
We propose a method for policy learning called policy-vote bagging, which learns treatment decisions on many subsamples then averages their votes into treatment probabilities.
Relative to using one subsample, averaging across subsamples preserves expected welfare and improves expected utility for a risk-averse researcher.
We derive sharp bounds linking estimation accuracy, subsample size, and welfare variation, including an exact guarantee under CARA utility.
\end{abstract}
\medskip
\noindent\textit{Keywords:} Algorithmic stability; empirical welfare maximization; statistical treatment choice.
\vfill
\clearpage
\setstretch{1.45}
\section{Introduction}\label{sec:introduction}
Many economic policies are learned from experiments and then implemented at scale.
Often, one experimental sample is observed, one resulting policy is implemented, and a poor policy recommendation may be difficult to reverse.
Standard treatment-choice frameworks evaluate an algorithm by its expected welfare across hypothetical experimental samples.
This criterion does not distinguish between two algorithms that perform equally well in expectation, even though one algorithm may produce much more variable recommendations across hypothetical experiments.
The distinction matters when researchers, institutions, or societies are averse to the inherent gamble of learning a policy from randomly sampled experimental data.
This paper views the choice of a policy learning algorithm as part of the research design. Before observing the experimental sample, a researcher chooses how the data will be converted into a policy.\footnote{Choosing an algorithm before observing the data
follows the statistical decision-theoretic framework of
\citet{wald1950}. A decision rule maps possible
samples into actions. Its performance is evaluated
across samples.} The experimental sample is then collected, the algorithm recommends a policy, and that policy determines treatment in the full population. This sequence creates two sources of uncertainty in chronological order, which may be viewed as a compound lottery. First, the learned policy varies across possible experimental samples. Second, outcomes vary across people in the full population. The main text focuses on aversion to the first source of variation. Appendix~\ref{app:joint_concavity} allows aversion to both.\footnote{This compound lottery extends the veil of ignorance \citep{vickrey_1945,harsanyi_1953,rawls_1971} to before the experiment. The researcher chooses how to learn a policy without knowing which sample will be observed. The policymaker evaluates a policy without knowing whose outcome they will receive. }
Our primary contribution is to prove that algorithmic stability provides a general way to manage experimental sampling risk. We say that a policy learning algorithm is welfare stable when replacing one experimental unit changes the welfare of its learned policy by a small amount. This one-unit stability controls both welfare variation across samples and the probability that the learned policy performs poorly. It also yields an exact, finite-sample guarantee for a researcher with constant absolute risk aversion. The result is deliberately broad: it requires only that welfare be well defined, and places no restriction on the dimension of the covariates nor the complexity of the policy class.
Our secondary contribution is to propose a method for stable policy learning called policy-vote bagging. The method learns a binary treatment decision on each of many subsamples and uses the fraction voting for treatment as the final treatment probability. Its stability comes directly from the construction. If one unit is replaced, every subsample that omits that unit casts the same vote as before; only the fraction of subsamples containing it can change. Policy-vote bagging therefore achieves stability across the full experimental sample, even when the underlying base learner is unstable. At a fixed subsample size, averaging preserves expected welfare (relative to a policy based on one subsample) and improves expected utility for any increasing, concave evaluation of welfare.
Our formal results show how subsample size trades estimation accuracy for stability. Larger subsamples give the base learner more information, but they also make each unit influential in more votes. We derive matching upper and lower bounds over a class of populations and base learners described by their mean-squared estimation error. Within this class, the bounds characterize a minimax rate-optimal subsample size. Parametric, H\"older-smooth, and Gaussian-kernel examples show how familiar estimation guarantees enter the analysis.
Subsampling can be worthwhile in small experimental samples when estimation accuracy improves slowly in the sample size and sampling risk is prioritized.
\subsection{Related work}
We contribute to the literature on statistical treatment choice; see \citet{opocher_26_review} for a recent review. Expected-regret analysis \citep{manski2004} underlies empirical welfare maximization and its extensions \citep{kitagawa_tetenov_2018,mbakop_tabord_meehan_2021,athey_wager_2021}. Related work incorporates concerns about inequality and fairness into the objectives used to evaluate treatment assignments \citep{kitagawa_tetenov_2021,viviano_bradic_2024}, yielding different criteria; see Section~\ref{sec:compound}.
Our closest connection is to work that evaluates welfare risk across experimental samples. In an innovation experiment with binary outcomes, \citet{manski_tetenov_2007} evaluate treatment rules through the expectation of a concave transformation of population welfare and characterize admissible fractional treatment rules. Their criterion inspires our formalization of sampling risk. \citet{manski_tetenov_2023} study quantile performance and stochastic-dominance comparisons of sampling distributions, while \citet{kitagawa_lee_qiu_2026} study nonlinear transformations of regret, discussed in Section~\ref{sec:compound}. We connect sampling risk to algorithmic stability and derive quantitative guarantees for policy-vote bagging with covariates.
Other approaches account for statistical uncertainty when
choosing policies or reporting their performance.
These include policy choice with budget constraints and estimated
costs \citep{sun2026}, policy choice based on confidence
bounds for welfare
\citep{chernozhukov_lee_rosen_sun_2025,andrews_chen_2025},
and empirical Bayes allocation across policies with estimated
costs and benefits \citep{moon_2026}.
\citet{swaminathan_joachims_2015} account for uncertainty
in an estimator of welfare when choosing a policy.
By contrast, our criterion concerns how the actual welfare
of the learned policy varies across experimental samples.
We show how algorithmic stability controls this variation.
Our two-stage formulation resembles models of ambiguity aversion. Smooth, variational, multiplier, and recursive preferences distinguish layers of uncertainty \citep{klibanoff_et_al_2005,maccheroni_marinacci_rustichini_2006,strzalecki_2011,strzalecki_2013,denti_pomatto_2022}. Robust mean-variance analysis provides a particularly close mathematical analogy \citep{maccheroni_marinacci_ruffino_2013}. Our interpretation is different. Both the experimental-sample lottery and full-population lottery are objective and arise under a fixed data-generating process.
Single-unit stability has separate roots in econometrics and machine learning. Building on the influence framework of \citet{hampel_1974}, \citet{andrews_1986} formalizes a stability exponent based on the rate at which an estimator's largest leave-one-out change vanishes. Modern algorithmic-stability analyses use this property to obtain finite-sample statistical guarantees \citep{bousquet_elisseeff_2002}. \citet{elisseeff_evgeniou_pontil_2005} extend this framework to randomized learners, and analyze bagging and subbagging. In recent work on semiparametric inference, \citet{chernozhukov_newey_singh_syrgkanis_2026} and \citet{chen_syrgkanis_austern_2022} develop a stability-based analysis and show that subsample bagging can provide the required stability. Our definition of welfare stability shares this single-unit logic. Unlike earlier works, we apply it to the population welfare of a learned policy, and use it to control variation across experimental samples.
Bagging can stabilize an unstable base learner. Classical work studies its effects on instability and prediction error \citep{breiman_1996,buhlmann_yu_2002,buja_stuetzle_2006}. \citet{soloff_barber_willett_2024} give assumption-free stability guarantees for bagged predictors. Stable predictions need not yield stable decisions, so \citet{soloff_barber_willett_argmax_2024} address discontinuous classification by reporting sets of candidate labels, and \citet{qian_ying_lam_yin_2025} instead select models through subsample voting. Our proposal of policy-vote bagging retains the vote shares as treatment probabilities. We study how this fractional implementation controls the sampling distribution of population welfare.
The paper follows the decision problem from criterion to method. \Cref{sec:compound} defines the two-stage lottery and risk-aware regret. \Cref{sec:design} interprets algorithm choice as a tradeoff between expected welfare and sampling risk. \Cref{sec:stability} shows how stability controls experimental-sample risk. \Cref{sec:bagging} develops policy-vote bagging. \Cref{sec:rates} gives lower bounds and optimality results. \Cref{sec:discussion} concludes. Appendix~\ref{app:joint_concavity} allows risk aversion at both stages. Appendices~\ref{app:sec2_proofs}-\ref{app:secA_proofs} collect the proofs.
\section{The training-test compound lottery}\label{sec:compound}
This section separates the policy-learning problem into its two stages. We define the experimental sample used to learn a policy, and the welfare in the full population. We place them in a chronological lottery. As shorthand, we refer to the former source of uncertainty as arising from a ``training'' set, and the latter as arising from a ``test'' set. Finally, we introduce risk-aware regret as a tractable criterion integrating both sources of uncertainty.
\subsection{Data setting}
We distinguish the population that receives the policy (the ``test'' set) from the experiment used to learn it (the ``training'' set). Let $P$ denote the joint law of covariates $X$ and potential outcomes $Y(0),Y(1)$ for a generic test unit, with $X\in\mathcal X\subseteq\mathbb R^d$. The conditional average treatment effect is $\tau(x)=\mathbb E\{Y(1)-Y(0)\mid X=x\}$.
A policy specifies a treatment probability for each covariate value. Formally, $G:\mathcal X\to[0,1]$ is measurable, and $\mathcal G$ is the class of all such rules. This unconstrained class contains the oracle and the averages of policies studied below. For an independent test unit and $U^{\mathrm{te}}\sim\operatorname{Unif}(0,1)$, set $D^G=\mathbf 1\{U^{\mathrm{te}}\leq G(X^{\mathrm{te}})\}$ and $Y^G=Y^{\mathrm{te}}(D^G)$. Thus, the test outcome reflects population heterogeneity and, when $G$ is fractional, the policy's own randomization.
The policymaker may value the distribution of test outcomes as well as their mean. Under policy $G$, realized utility is $U_{\mathrm{PM}}(G)=\phi(Y^G)$ and ex ante value is $V_\phi(G)=\mathbb E\{\phi(Y^G)\}$, where $\phi$ is increasing and concave, and $\mathbb E|\phi\{Y(d)\}|<\infty$ for $d\in\{0,1\}$. Concavity places less value on mean-preserving spreads in outcomes, which we call inequality aversion.\footnote{It also evaluates idiosyncratic outcome risk.} Conditional independence of the policy randomization gives
\[
V_\phi(G)
=
\mathbb E\bigl[G(X)\phi\{Y(1)\}+\{1-G(X)\}\phi\{Y(0)\}\bigr].
\]
The relevant treatment contrast is therefore $\tau_\phi(x)=\mathbb E[\phi\{Y(1)\}-\phi\{Y(0)\}\mid X=x]$, and the unconstrained oracle treats when this contrast is nonnegative: $G_\phi^\star(x)=\mathbf 1\{\tau_\phi(x)\geq0\}$.
The training data come from a randomized experiment, sampled from the same population. The researcher observes $S=(S_1,\ldots,S_n)\sim P_{\mathrm{tr}}^n$, where $S_i=(X_i,D_i,Y_i)$ and $Y_i=D_iY_i(1)+(1-D_i)Y_i(0)$. The law $P_{\mathrm{tr}}$ draws covariates and potential outcomes from $P$ and then randomizes treatment within covariate strata. The following assumption makes this design explicit and imposes the regularity used below.
\begin{assumption}[Training sample and test outcome]\label[assumption]{ass:main}
The following conditions hold.
\begin{enumerate}[label=(\roman*),leftmargin=2.1em]
\item Integrability: $\mathbb E|Y(0)|+\mathbb E|Y(1)|<\infty$.
\item Stratified randomization: for the training experiment,
$
D\perp\{Y(0),Y(1)\}\mid X,
$ and $
\mathbb P(D=1\mid X=x)\in[\underline e,1-\underline e]
$
for some $\underline e\in(0,1/2)$ and all $x\in\mathcal X$.
\item Margin condition: there exist $C_\tau<\infty$ and $\kappa>0$ such that, for all $t>0$,
$
\mathbb P_X\{0<|\tau(X)|\leq t\}\leq C_\tau t^\kappa.
$
\end{enumerate}
\end{assumption}
Each part of Assumption~\ref{ass:main} has a separate role. Integrability requires only first moments, rather than bounded outcomes, and ensures that welfare is well defined for every $G\in\mathcal G$. Randomization identifies the CATE from the training sample. The margin condition limits how much of the test population is nearly indifferent between treatment and no treatment.
\subsection{The researcher's decision problem}
The researcher's choice precedes both the training and test lotteries. A learning algorithm is a measurable map $A:\mathcal S^n\to\mathcal G$, and at date zero the researcher chooses $A$ from an admissible class $\mathcal A$. The sequence of events is as follows. We abbreviate the researcher by R, and the policymaker by PM.
\begin{description}[leftmargin=5.2em,labelwidth=5.2em,style=nextline]
\item[$t=0$.] R decides on an algorithm $A\in\mathcal A$ before observing the training data.
\item[Training.] The randomized training experiment produces $S\sim P_{\mathrm{tr}}^n$.
\item[$t=1$.] R recommends the data-driven policy $\widehat G=A(S)$, and the PM decrees it. At this date, R evaluates the recommendation using the PM's expected utility conditional upon the training sample.
\item[Test.] An independent test unit is assigned according to $\widehat G$, and only the selected potential outcome $Y^{\widehat G}$ realizes.
\item[$t=2$.] The PM receives utility $U_{\mathrm{PM}}(\widehat G)=\phi(Y^{\widehat G})$.
\end{description}
The two lotteries enter the researcher's objective in different places. Conditional on $S$, the policymaker values the learned policy as $\mathbb E_P[\phi\{Y^{A(S)}\}\mid S]=V_\phi\{A(S)\}$. The policymaker accepts the recommendation rather than choosing among algorithms. The researcher's date-one utility is $U_{\mathrm{R}}(A)=\psi[V_\phi\{A(S)\}]$, where the increasing, concave function $\psi$ expresses aversion to variation in policy value across training samples. At date zero, R solves
\begin{equation}\label{eq:researcher_problem}
A_\star
\in
\operatorname*{arg\,max}_{A\in\mathcal A}
\mathbb E_{P_{\mathrm{tr}}^n}
\left\{
\psi\left(
\mathbb E_P\left[
\phi\left\{Y^{A(S)}\right\}
\mid S
\right]
\right)
\right\}.
\end{equation}
The criterion in \eqref{eq:researcher_problem} makes reliability across training samples part of statistical treatment choice. R can be interpreted as a benevolent adviser who dislikes exposing the population to a sampling gamble. The same reduced-form preference can represent costly policy reversal or reputational losses from a recommendation that performs poorly because of the realized sample. Alternatively, $\psi$ can be an ex ante social criterion delegated to R. The two-agent language simply keeps the training and test lotteries conceptually separate; it does not require their objectives to conflict.
\begin{remark}[Taxonomy]\label[remark]{rmk:taxonomy}
When $\psi$ is linear, \eqref{eq:researcher_problem} reduces to maximizing expected transformed welfare, or equivalently minimizing
$
R_\phi(A)
:=
V_\phi(G_\phi^\star)
-
\mathbb E_{P_{\mathrm{tr}}^n}[V_\phi\{A(S)\}].
$
If $\phi$ is concave, this objective is consistent with equality-minded or fairness-oriented treatment choice \citep{kitagawa_tetenov_2021,viviano_bradic_2024}. If $\phi$ is also linear, it is the standard expected-regret problem \citep{kitagawa_tetenov_2018}. When $\phi$ is linear and $\psi$ is concave, the PM is utilitarian but R is sampling-risk averse, which is the case studied below. Concavity of both transformations accommodates the two concerns simultaneously without requiring them to be linked. Appendix~\ref{app:joint_concavity} states the corresponding statistical results.
\end{remark}
\begin{remark}[Sampling risk averse treatment choice]\label[remark]{rmk:manski_tetenov}
Risk aversion over sampling is directly inspired by \citet{manski_tetenov_2007}. They study an innovation experiment without covariates, with binary outcomes, and with a known status quo success probability. A statistical treatment rule maps the experimental sample into the fraction of the population receiving the innovation, and the planner evaluates the rule through the sampling expectation of a concave transformation of the population mean outcome. It has the same preference structure as \eqref{eq:researcher_problem} with $\phi(y)=y$ and concave $\psi$. Their analysis studies admissibility of fractional treatment rules in that experiment. Ours connects sampling-risk preferences to algorithmic stability and welfare guarantees for learned policies with covariates. The mean-variance criterion introduced next is motivated by a local approximation to the exact expected-utility objective.
\end{remark}
The nested objective resembles several second-order models of uncertainty. Smooth ambiguity applies an outer transformation to inner expected utility; variational and multiplier preferences penalize departures from a reference model; and recursive models emphasize when uncertainty is resolved \citep{klibanoff_et_al_2005,maccheroni_marinacci_rustichini_2006,strzalecki_2011,strzalecki_2013,denti_pomatto_2022}. The robust mean-variance analysis of \citet{maccheroni_marinacci_ruffino_2013} is especially close mathematically. The distinction between objective and subjective mixtures in \citet{ghirardato_et_al_2003} helps interpret fractional policy randomization. Unlike those works, every probability law in our setting is objective, and the data-generating process is fixed.
\begin{remark}[Outcome levels versus policy gains]\label[remark]{rmk:gains}
The criterion $V_\phi(G)=\mathbb E\{\phi(Y^G)\}$ evaluates the distribution of post-policy outcome levels. Without additional structure, concavity of $\phi$ therefore represents aversion to dispersion in outcomes, not specifically dispersion in treatment gains. A gain-based alternative is
$
V_\phi^\Delta(G)
=
\mathbb E\left(\phi\left[D^G\{Y(1)-Y(0)\}\right]\right),
$
whose transformed conditional treatment contrast is
$
\tau_\phi^\Delta(x)
=
\mathbb E\left[\phi\{Y(1)-Y(0)\}-\phi(0)\mid X=x\right].
$
All compound-lottery arguments continue to apply after replacing $V_\phi$ by $V_\phi^\Delta$. In the utilitarian case $\phi(y)=y$ studied below, outcome welfare and gain welfare differ only by the policy-invariant constant $\mathbb E\{Y(0)\}$, so they induce the same policy target and regret.
\end{remark}
\subsection{Risk-aware regret}
For simplicity, take the policymaker to be utilitarian. Setting $\phi(y)=y$, population welfare is
\begin{equation*}\label{eq:welfare}
W(G)
:=
\mathbb E(Y^G)
=
\mathbb E\{Y(0)\}+\mathbb E\{\tau(X)G(X)\}.
\end{equation*}
Integrability is enough to make this welfare comparison finite.
The unconstrained oracle is $G^\star(x)=\mathbf 1\{\tau(x)\geq0\}$, with welfare $W^\star:=W(G^\star)$. Standard expected regret is $R(A):=W^\star-\mathbb E_{P_{\mathrm{tr}}^n}[W\{A(S)\}]$; when $\psi$ is linear, minimizing this quantity is exactly \eqref{eq:researcher_problem}.
A local expected-utility calculation motivates adding sampling variation to expected regret. Write $Z_A=W\{A(S)\}$ and define R's certainty equivalent by $\psi\{\mathrm{CE}_\psi(A)\}=\mathbb E\{\psi(Z_A)\}$. The next result makes the mean-variance approximation uniform.
\begin{lemma}[Mean-variance approximation]\label[lemma]{lem:taylor}
Let $\mathcal J$ be an open interval containing every feasible welfare value. Suppose $\psi\in C^3(\mathcal J)$, $\inf_{w\in\mathcal J}\psi'(w)>0$, and
$
-\frac{\psi''(w)}{\psi'(w)}=\gamma\geq0
$ for every $w\in\mathcal J.
$
There exists a finite constant $C_\psi$, depending only on $\psi$ and the feasible welfare interval, such that every $A\in\mathcal A$ satisfies
$$
\mathrm{CE}_\psi(A)
=
\mathbb E(Z_A)-\frac{\gamma}{2}\mathbb V(Z_A)+r_\psi(A)
,\quad
|r_\psi(A)|
\leq
C_\psi
\left\{
\mathbb E|Z_A-\mathbb E(Z_A)|^3+\mathbb V(Z_A)^2
\right\}.
$$
Consequently, the expansion is uniform over any sequence $\mathcal A_n$ for which the centered third moments vanish uniformly:
$\sup_{A\in\mathcal A_n}\mathbb E|Z_A-\mathbb E(Z_A)|^3\to0.
$
\end{lemma}
The lemma expresses the certainty equivalent as average welfare minus a common price for welfare variance. Constant absolute risk aversion makes that price the same across algorithms, while the smoothness condition controls the approximation uniformly.
\begin{corollary}[Optimizer approximation]\label[corollary]{cor:optimizer}
For a candidate class $\mathcal A_n$, define the mean-variance objective
$
M_\rho(A)=\mathbb E(Z_A)-\rho\mathbb V(Z_A),
$ the approximation error $
\epsilon_n=\sup_{A\in\mathcal A_n}|r_\psi(A)|,
$ and the risk aversion $
\rho=\gamma/2.
$
Consider the optimizers $A_n^{\mathrm{CE}}\in\operatorname*{arg\,max}_{A\in\mathcal A_n}\mathrm{CE}_\psi(A)$ and $A_n^{\mathrm{MV}}\in\operatorname*{arg\,max}_{A\in\mathcal A_n}M_\rho(A)$. Then
$
0
\leq
\mathrm{CE}_\psi(A_n^{\mathrm{CE}})-\mathrm{CE}_\psi(A_n^{\mathrm{MV}})
\leq
2\epsilon_n,
$
and
$
0
\leq
M_\rho(A_n^{\mathrm{MV}})-M_\rho(A_n^{\mathrm{CE}})
\leq
2\epsilon_n.
$
In particular, if the mean-variance maximizer is unique and its objective gap compared to the next best algorithm exceeds $2\epsilon_n$, then the certainty-equivalent and mean-variance maximizers coincide.
\end{corollary}
The corollary shows when the approximation can guide algorithm choice: a uniformly small approximation error implies a small loss in certainty equivalent from choosing an algorithm by its mean-variance approximation.
These calculations motivate our definition of risk-aware regret, which adds a penalty for sampling risk to ordinary expected regret. We study this criterion in its own right. The preceding results explain when it approximates the expected-utility choice.
\begin{definition}[Risk-aware regret]\label[definition]{def:riskaware}
For $\rho\geq0$, the risk-aware regret of algorithm $A$ is
\begin{equation}\label{eq:riskaware}
R_{\mathrm{RA}}(A)
:=
W^\star
-
\mathbb E[W\{A(S)\}]
+
\rho\mathbb V[W\{A(S)\}]
=
R(A)+\rho\mathbb V(Z_A).
\end{equation}
\end{definition}
The two terms in \eqref{eq:riskaware} separate expected performance from reliability. The first is ordinary expected welfare regret. The second prices variation in the welfare of the learned policy across experimental samples. The parameter $\rho$ has units inverse to welfare, so the terms are comparable. The criterion is a second-order certainty-equivalent loss.
\begin{remark}[Nonlinear regret]\label[remark]{rmk:nonlinear_regret}
Let $\operatorname{Reg}(A,S)=W^\star-W\{A(S)\}$. The mean-square-regret criterion of \citet{kitagawa_lee_qiu_2026} satisfies
$
\mathbb E\{\operatorname{Reg}(A,S)^2\}
=
R(A)^2+\mathbb V[W\{A(S)\}],
$
whereas \eqref{eq:riskaware} is $R(A)+\rho\mathbb V[W\{A(S)\}]$. The criteria share the same variance component but differ in the role of mean regret. Our criterion preserves the canonical expected-regret benchmark and allows the price of sampling risk to vary independently.
\end{remark}
\begin{remark}[Stochastic dominance]\label[remark]{rmk:sd}
\citet{manski_tetenov_2023} take the sampling distribution of welfare as the primitive object and require comparisons to respect stochastic dominance. Risk-aware regret also depends on the sampling distribution, but only through its initial two moments. A mean-variance functional need not respect first-order stochastic dominance globally because a stochastically better distribution can have a larger variance. Our criterion is therefore not a substitute for their general dominance framework. Its role is narrower: it is a locally expected-utility-based and statistically tractable target for studying how algorithmic stability controls sampling risk.
\end{remark}
Risk-aware regret is a local mean-variance criterion. It augments standard expected regret without claiming to order arbitrary welfare distributions. Using its two components, we interpret algorithm choice as a tradeoff between expected welfare and sampling risk.
\section{A tradeoff: Expected welfare versus sampling risk}\label{sec:design}
The researcher manages sampling risk by choosing among learning algorithms before seeing the data. The choice is the same $A\in\mathcal A$ introduced in \Cref{sec:compound}, and the recommendation is $\widehat G=A(S)$. Different tuning parameters are incorporated as different elements of $\mathcal A$.\footnote{We write $\mathcal A_n$ only when sample-size dependence matters. For complete policy-vote bagging, the candidates will be $\mathcal A=\{A_m:1\leq m\leq n\}$.}
\begin{example}[Deterministic rule]\label[example]{ex:plugin}
Hard thresholding can make a deterministic policy sensitive near zero. Let $\widehat\tau_h$ be a CATE estimator whose hyperparameter $h$ may index bandwidth, tree depth, leaf size, or regularization strength. A deterministic algorithm $A\in\mathcal A$ yields the policy
$
\widehat G(x)
=
\{A(S)\}(x)
=
\mathbf 1\{\widehat\tau_h(S,x)\geq0\}.
$
Here $\widehat\tau_h$ is the score function and $t\mapsto\mathbf 1(t\geq0)$ is the choice function. The hyperparameter $h$ is absorbed into the definition of the algorithm $A$. Because the choice function is discontinuous, small changes in the training sample can flip treatment decisions near zero and thereby create sampling variability in welfare.
\end{example}
\begin{example}[Fractional rule]\label[example]{ex:fractional}
A fractional candidate smooths the treatment choice near zero:
$
\widehat G(x)
=
\{A(S)\}(x)
=
\delta_c\{\widehat\tau_h(S,x)\},
$
where $\delta_c(t)=0$ if $t\leq -c$, $\delta_c=(t+c)/(2c)$ if $|t|\leq c$, and $\delta_c(t)=1$ if $t\geq c$ for some $c>0$.
The pair $(h,c)$ is again absorbed into the definition of $A$. Increasing $c$ reduces sensitivity near the treatment threshold, potentially lowering sampling risk at the cost of expected welfare.
\end{example}
A risk-aware research design chooses the best attainable combination of expected regret and welfare variation. For the candidate class $\mathcal A$, R solves
$
A_\star
\in
\operatorname*{arg\,min}_{A\in\mathcal A}
R_{\mathrm{RA}}(A).
$
The lower-left Pareto boundary summarizes the attainable tradeoff. Each algorithm generates a pair $\bigl(R(A),\mathbb V(Z_A)\bigr)$, and no algorithm on the boundary can be improved in both expected regret and welfare variance by another candidate. As in the mean-variance frontier of portfolio theory, different policy learning algorithms offer different combinations of return and risk \citep{markowitz1952}.
Here the frontier concerns population welfare across training samples, and $\rho$ determines how much expected welfare R is willing to exchange for greater reliability.
The frontier gives a geometric interpretation of the optimal algorithm. For each $\rho>0$, R selects the attainable point on the lowest iso-risk-aware-regret line. At a smooth interior solution, that line is tangent to the lower-left boundary, and a larger $\rho$ favors a point with less sampling risk. \Cref{fig:frontier} illustrates this geometry. Appendix~\ref{app:sec3_proofs} records the slope calculation.
\begin{figure}[t]
\centering
\begin{tikzpicture}[
x=1cm, y=0.9cm,
font=\normalsize,
>=Latex,
axis/.style={->,line width=0.9pt,black!75},
frontier/.style={RoyalBlue!85!black,line width=2pt},
iso/.style={
black!65,line width=1.2pt,
dash pattern=on 4pt off 3pt
}
]
\draw[axis] (0,0) -- (6.1,0);
\draw[axis] (0,0) -- (0,3.9);
\node[below=7pt] at (3.05,0)
{Expected regret $R(A)$};
\node[rotate=90,above=7pt] at (0,1.95)
{Welfare variance $\mathbb V(Z_A)$};
\pgfmathsetmacro{\rstar}{3}
\pgfmathsetmacro{\vstar}{4.8/(\rstar+0.45)-0.1}
\pgfmathsetmacro{\tangentslope}{-4.8/(\rstar+0.45)^2}
\draw[iso,domain=0.8:5.9,samples=2,variable=\x]
plot ({\x},{\vstar+\tangentslope*(\x-\rstar)});
\draw[frontier,domain=0.9:5.85,smooth,
samples=100,variable=\x]
plot ({\x},{4.8/(\x+0.45)-0.1});
\fill (\rstar,\vstar) circle[radius=2.2pt];
\node[above right=3pt] at (\rstar,\vstar) {$A_\star$};
\end{tikzpicture}
\caption{Algorithmic design and the welfare-risk frontier}
\label{fig:frontier}
\caption*{\footnotesize \textit{Notes:}
The blue curve shows the efficient combinations of expected
regret and welfare variance attainable within $\mathcal A$.
Along the dashed line, $R(A)+\rho\mathbb V(Z_A)$ is constant.
At the illustrated optimum $A_\star$, this line is tangent
to the frontier, with slope $-1/\rho$.}
\end{figure}
Estimating the frontier of expected regret and welfare variation is a separate inferential problem. The support-function methods of \citet{liu_molinari_2025} for a fairness-accuracy frontier may provide a useful starting point, but we do not estimate the boundary here. We instead ask which aspects of an algorithm control its position on the welfare variation axis. Algorithmic stability supplies the answer.
\section{The central role of algorithmic stability}\label{sec:stability}
This section shows how a property of the policy learning algorithm controls sampling risk. We define welfare stability, then use it to bound welfare variance and downside risk. Finally, we connect those guarantees to expected utility under constant absolute risk aversion.
\subsection{Welfare stability}
Welfare stability asks how much one observation can change the value of the learned policy. Fix $A:\mathcal S^n\to\mathcal G$ and write $A(S)=\widehat G_S$. For $i\in\{1,\ldots,n\}$ and $s'\in\mathcal S$, let $S^{i\leftarrow s'}=(S_1,\ldots,S_{i-1},s',S_{i+1},\ldots,S_n)$ denote the sample obtained by replacing observation $i$.
\begin{definition}[Welfare stability]\label[definition]{def:stability}
The welfare-stability parameter of $A$ is
\begin{equation*}\label{eq:stability}
\beta_n(A)
:=
\sup_{S\in\mathcal S^n}
\sup_{1\leq i\leq n}
\sup_{s'\in\mathcal S}
\left|
W\{A(S)\}
-
W\{A(S^{i\leftarrow s'})\}
\right|.
\end{equation*}
A sequence $A_n$ is \emph{welfare stable} if $\beta_n(A_n)\to0$.
\end{definition}
Welfare stability is weaker than requiring every prediction to be stable. It controls the scalar welfare of the policy, so changes at different covariate values may offset one another. Welfare stability requires only that the influence of one observation vanishes. For the generic variance bound below to vanish, the stronger rate $\beta_n(A_n)=o(n^{-1/2})$ is needed. The next result gives simple conditions under which familiar stability guarantees imply welfare stability.
\begin{lemma}[From algorithmic stability to welfare stability]\label[lemma]{lem:policy_to_welfare}
Suppose Assumption \ref{ass:main}(i) holds and define the algorithmic stability parameter
\[
\Gamma_n(A)
=
\sup_{S,i,s'}\sup_{x\in\mathcal X}
\left|
\{A(S)\}(x)-\{A(S^{i\leftarrow s'})\}(x)
\right|.
\]
Then the welfare stability parameter satisfies
$
\beta_n(A)
\leq
\Gamma_n(A) \lVert\tau\rVert_{L_1(P_X)}.
$
In particular, if $\{A(S)\}(x)=\delta\{\zeta(S,x)\}$, the choice function $\delta$ is $L_\delta$-Lipschitz, and the score function $\zeta$ has algorithmic stability parameter at most $s_n$, then
$
\beta_n(A)
\leq
L_\delta s_n\lVert\tau\rVert_{L_1(P_X)}.
$
\end{lemma}
The lemma makes welfare stability verifiable through well-known algorithmic stability results.\footnote{Here, $\Gamma_n$ is standard, while $s_n$ follows from e.g. \citet{sun_xiang_2025}.}
The next subsection turns this welfare sensitivity into variance and risk guarantees.
\subsection{Main result}
Our central result translates welfare stability into direct control of sampling risk.
\begin{theorem}[Stable policy learning]\label[theorem]{thm:stability}
If $S_1,\ldots,S_n$ are independent and $\beta_n(A)<\infty$, then
\begin{equation*}\label{eq:stable_ra}
R_{\mathrm{RA}}(A)
\leq
R(A)
+
\rho \frac{n\beta_n(A)^2}{4}.
\end{equation*}
Moreover, for every $t>0$,
\[
\mathbb P\left[
W^\star-W\{A(S)\}
\geq
R(A)+t
\right]
\leq
\exp\left\{-\frac{2t^2}{n\beta_n(A)^2}\right\}.
\]
\end{theorem}
The theorem translates algorithmic stability into two policy learning guarantees. First, it bounds the variance term in risk-aware regret. Second, it bounds the probability that the learned policy performs poorly. The welfare stability parameter must vanish faster than $n^{-1/2}$ for this worst-case variance bound to disappear.
\subsection{Utility interpretation}
Constant absolute risk aversion gives an exact link between stability and expected utility. For $\gamma>0$, define the CARA utility function $\psi_\gamma(w)=-\exp(-\gamma w)$ and the CARA certainty equivalent $\mathrm{CE}_\gamma(A)=-(1/\gamma)\log[\mathbb E\{\exp(-\gamma Z_A)\}]$.\footnote{By definition, the certainty equivalent solves $\psi_\gamma\{\mathrm{CE}_\gamma(A)\}
=\mathbb E\{\psi_\gamma(Z_A)\}$.}
\begin{proposition}[Exact CARA guarantee from stability]\label[proposition]{prop:cara_stability}
Suppose $Z_A=W\{A(S)\}$ is integrable and $A$ has welfare-stability parameter $\beta_n(A)<\infty$. Then
$
W^\star-\mathrm{CE}_\gamma(A)
\leq
R(A)
+
\frac{\gamma n}{8}\beta_n(A)^2.
$
\end{proposition}
The CARA guarantee gives a direct utility interpretation to welfare stability. It applies to any learner with algorithmic stability and requires no approximation. The next result instead uses welfare variance itself, which can be much smaller than its worst-case bound.
\begin{proposition}[Exact CARA guarantee from welfare variance]\label[proposition]{prop:cara_variance}
Suppose Assumption~\ref{ass:main}(i) holds, $\gamma>0$, and $T=\lVert\tau\rVert_{L_1(P_X)}>0$. Then every policy-learning algorithm satisfies
\[
R(A)
\leq
W^\star-\mathrm{CE}_\gamma(A)
\leq
R(A)+
\frac{e^{\gamma T}-1-\gamma T}{\gamma T^2}\mathbb V(Z_A).
\]
If $T=0$, expected regret, welfare variance, and certainty-equivalent loss are all zero.
\end{proposition}
The variance coefficient approaches $\gamma/2$ as $\gamma T\to0$, matching the local mean-variance calculation. The inequality itself holds for every $\gamma>0$ and every sample size. Any sharper bound on welfare variance can therefore also sharpen the exact utility guarantee; policy-vote bagging will provide such a bound.
Under the smooth preference assumptions of \Cref{lem:taylor}, stability also controls the error in the local mean-variance approximation. The following corollary gives a sufficient rate for that error to be negligible uniformly over the policy learning algorithms.
\begin{corollary}[Stability validates the approximation]\label[corollary]{cor:stable_taylor}
Suppose the conditions of \Cref{lem:taylor} hold for a class $\mathcal A_n$, and $\sup_{A\in\mathcal A_n}\beta_n(A)\leq\bar\beta_n$. Then the optimizer-level approximation error in \Cref{cor:optimizer} satisfies
$
\epsilon_n
\lesssim
(n\bar\beta_n^2)^{3/2}
+
(n\bar\beta_n^2)^{2}.
$
Consequently, if $n\bar\beta_n^2\to0$, the certainty-equivalent and mean-variance objectives are uniformly asymptotically equivalent, and the remainder is of smaller order than the generic variance scale $n\bar\beta_n^2$.
\end{corollary}
Generality is also the limitation of the stability theorem. Because it uses only worst-case analysis, it can be conservative for structured algorithms. Policy-vote bagging has additional symmetry that yields a sharper variance guarantee.
\section{A concrete proposal}\label{sec:bagging}
Policy-vote bagging creates a stable final policy by averaging decisions learned on overlapping subsamples. Each base learner casts a binary treatment vote, and the fraction voting for treatment becomes the final treatment probability. The base learner may be unstable on its own $m$-unit subample. The aggregate can nevertheless be stable because replacing one unit leaves every vote from a subsample that omits it unchanged. Under complete bagging, only the $m/n$ fraction of subsamples containing that unit can respond. The procedure thus builds stability across subsamples into stability across the full experimental sample. The inclusion-frequency argument appears in the bagging stability analysis of \citet{elisseeff_evgeniou_pontil_2005} and is used by \citet{chernozhukov_newey_singh_syrgkanis_2026} and \citet{chen_syrgkanis_austern_2022} to stabilize nuisance estimators for debiased inference. Here it controls the welfare of a learned treatment policy.
\subsection{Policy-vote bagging}
Complete policy-vote bagging averages over every subsample of a fixed size. For $1\leq m\leq n$, write $\mathcal I_{n,m}=\{I\subseteq\{1,\ldots,n\}:|I|=m\}$ and $S_I=(S_i:i\in I)$. A base learner maps $S_I$ into a CATE estimate $\widehat\tau_I$. We assume the learner is symmetric: permuting the observations within the subsample leaves the fitted function $\widehat{\tau}_I$ unchanged. This estimate casts the vote $\widehat G_I(x)=\mathbf 1\{\widehat\tau_I(x)\geq0\}$. The complete policy is
\begin{equation}\label{eq:bagged_policy}
\{A_m(S)\}(x)
=:
\widehat G_m^{\mathrm{comp}}(x)
=
\binom{n}{m}^{-1}
\sum_{I\in\mathcal I_{n,m}}\widehat G_I(x)
=
\mathbb E_I\{\widehat G_I(x)\mid S\}.
\end{equation}
Retaining vote shares as treatment probabilities is essential to the method. Policy-vote bagging averages the decisions $\mathbf 1\{\widehat\tau_I(x)\geq0\}$, whereas score bagging averages numerical CATE estimates before applying a threshold. Classical bagged classification also aggregates votes, but typically takes a final majority decision \citep{breiman_1996}. We retain the fractional average, since applying a final threshold can destroy its stability \citep{soloff_barber_willett_argmax_2024}. The candidate class here is $\mathcal A=\{A_m:1\leq m\leq n\}$: a smaller $m$ makes each unit influential in fewer votes, while a larger $m$ gives each base learner more information.
Computational policy-vote bagging approximates the complete average with a bounded number of subsamples. Let $I_1,\ldots,I_B$ be conditionally i.i.d. uniform draws from $\mathcal I_{n,m}$, independent of $S$. Define the realized bag collection $\mathbf I=(I_1,\ldots,I_B)$ and
\begin{equation}\label{eq:finite_bagging}
\{A_{m,B}(S,\mathbf I)\}(x)
=:
\widehat G_{m,B}(x)
=
\frac{1}{B}\sum_{b=1}^B\widehat G_{I_b}(x).
\end{equation}
Computational bagging introduces computational randomness in addition to experimental-sampling randomness. Risk-aware regret for $A_{m,B}$ averages over both sources. The displayed definitions use a deterministic base learner. For a randomized base learner, complete bagging also averages over its seed, and computational bagging draws an independent seed with each bag; this seed variation can persist even at $m=n$. Conditional on $S$, the policy in \eqref{eq:bagged_policy} is the expectation of the policy in \eqref{eq:finite_bagging}, so complete bagging removes Monte Carlo variation from aggregation. Classical random forests are computational bagging procedures with randomized tree learners \citep{breiman_2001}.
Averaging policy votes has a direct expected-utility benefit at any fixed bag size. Because welfare is affine in treatment probabilities, averaging preserves mean welfare while reducing the variation created by selecting a particular collection of bags. The next result makes this comparison for any concave evaluation of welfare.
\begin{proposition}[Averaging votes improves concave welfare evaluations]\label[proposition]{prop:bag_concave}
Suppose Assumption~\ref{ass:main}(i) holds. Let $\psi$ be increasing and concave on an open interval containing the feasible welfare interval. For every $m$ and $B\geq1$, conditional on the experimental sample,
\begin{align*}
\psi[W\{A_m(S)\}]
&\geq
\mathbb E_{\mathbf I}\left(\psi\left[W\{A_{m,B}(S,\mathbf I)\}\right]\mid S\right)
\geq
\mathbb E_I[\psi\{W(\widehat G_I)\}\mid S].
\end{align*}
All three policies have the same conditional mean welfare:
\[
W\{A_m(S)\}
=
\mathbb E_{\mathbf I}[W\{A_{m,B}(S,\mathbf I)\}\mid S]
=
\mathbb E_I\{W(\widehat G_I)\mid S\}.
\]
The same utility ordering and mean equality hold after averaging over the experimental sample.
\end{proposition}
Complete bagging is therefore weakly preferred to computational bagging, which is weakly preferred to deploying the policy from one randomly selected bag, under any increasing concave welfare evaluation. This improvement holds at a fixed $m$ and preserves expected welfare. Choosing the bag size raises the additional question of how much information each base learner should receive, to which we turn in Section~\ref{sec:rates}.
Policy-vote bagging acts like a data-adaptive smoother around the treatment threshold. At a covariate value $x$, each bag supplies a score $\widehat\tau_{I_b}(x)$, and the final policy averages the hard-threshold decision over the empirical distribution of those scores. That distribution behaves like a smoothing kernel whose shape and width are generated by subsampling rather than chosen in advance. Smaller bags can produce more dispersed scores and a wider fractional-treatment region, while larger bags can sharpen the decision boundary.\footnote{These are possible effects of bag size, rather than a monotonicity property required by our results.}
The number of bags $B$ controls the Monte Carlo resolution. This score-space view complements the covariate-space smoothing induced by forest weights \citep{wager_athey_2018}.
\subsection{Policy-vote bagging is stable}\label{sec:bag_stability}
A key benefit of policy-vote bagging is welfare stability. The next lemmas verify stability for both complete and computational variants of policy-vote bagging.
\begin{lemma}[Welfare stability of complete policy-vote bagging]\label[lemma]{lem:bag_stability}
Under Assumption \ref{ass:main}(i), the complete policy-vote bagging algorithm in \eqref{eq:bagged_policy} satisfies
$
\beta_n(A_m)
\leq
\frac{m}{n}\lVert\tau\rVert_{L_1(P_X)}.
$
The factor $m/n$ is worst-case sharp over symmetric binary base learners and data-generating processes satisfying Assumption \ref{ass:main}.
\end{lemma}
\begin{lemma}[Welfare stability of computational policy-vote bagging]\label[lemma]{lem:finite_stability}
Under Assumption \ref{ass:main}(i), the computational policy-vote bagging algorithm in \eqref{eq:finite_bagging} satisfies,
with probability at least $1-\eta$ over the computational bags,
$
\beta_n(A_{m,B}\mid\mathbf I)
\leq
\lVert\tau\rVert_{L_1(P_X)}
\left\{
\frac{m}{n}
+
\sqrt{\frac{\log(n/\eta)}{2B}}
\right\}.
$
\end{lemma}
The stability of bagging is governed by how often each unit appears. Complete bagging gives every unit the exact inclusion frequency $m/n$, whereas computational bagging uses its realized frequency among the $B$ sampled bags. These welfare guarantees complement the assumption-free prediction-stability results of \citet{soloff_barber_willett_2024}. The symmetry of complete bagging yields a sharper variance rate than worst-case sensitivity alone.
\subsection{Upper bounds for policy-vote bagging}\label{sec:bag_upper_bounds}
Stability is useful only if it can be achieved without sacrificing too much welfare. The next theorem expresses expected regret and welfare variance in terms of the bag-level CATE error, making the accuracy-stability tradeoff explicit.
Formally, the accuracy of the base learner enters through its integrated CATE error. For an independent target covariate $X$, define
\begin{equation*}\label{eq:Qmp}
\Delta_I(X)
=
\widehat\tau_I(X)-\tau(X),
\quad
Q_{m,p}
=
\mathbb E_{S,I,X}\{|\Delta_I(X)|^p\},
\quad p>1.
\end{equation*}
Here, $Q_{m,2}$ is the integrated mean-squared CATE error of a learner trained on one size-$m$ bag. We initially require only that it be finite. \Cref{sec:learning_examples} shows how standard learning rates bound this quantity in parametric, H\"older-smooth, and Gaussian-kernel examples.
\begin{theorem}[Upper bound for complete policy-vote bagging]\label[theorem]{thm:bag_upper}
Suppose Assumption \ref{ass:main} holds, the base learner is symmetric, and $Q_{m,2}<\infty$. Then
\begin{equation*}\label{eq:main_upper}
R_{\mathrm{RA}}(A_m)
\lesssim
Q_{m,2}^{\frac{\kappa+1}{\kappa+2}}
+
\rho\frac{m}{n}Q_{m,2}.
\end{equation*}
\end{theorem}
The theorem separates accuracy from stability. The first bound converts CATE error into expected welfare regret: mistakes matter most when treatment effects are large, while the margin condition limits how often small estimation errors can reverse the correct decision. The second bound shows that complete averaging reduces welfare variance by the inclusion frequency $m/n$.
Only the regret bound uses the margin condition.
\begin{corollary}[Upper bound for computational policy-vote bagging]\label[corollary]{cor:finite_bagging}
Under the conditions of \Cref{thm:bag_upper}, for every $B\geq1$,
\[
R_{\mathrm{RA}}(A_{m,B})
\lesssim
Q_{m,2}^{\frac{\kappa+1}{\kappa+2}}
+
\rho\left(\frac{m}{n}+\frac{1-m/n}{B}\right)Q_{m,2}.
\]
\end{corollary}
The computational-bag variance bound interpolates between $Q_{m,2}$ at $B=1$ and $(m/n)Q_{m,2}$ as $B$ grows. For the deterministic base learner defined above, the Monte Carlo component vanishes as $B\to\infty$ and is zero at $m=n$, when there is only one subsample. Without a cost of computation, the bound weakly favors more bags; an interior choice of $B$ would require an explicit computation constraint.
The same variance bounds control exact CARA utility loss. Combining them with \Cref{prop:cara_variance} gives a direct preference-based guarantee for both complete and computational policy-vote bagging.
\begin{corollary}[Exact CARA guarantee for policy-vote bagging]\label[corollary]{cor:cara_bagging}
Under the conditions of \Cref{thm:bag_upper}, let $\gamma>0$ and $T=\lVert\tau\rVert_{L_1(P_X)}>0$. Then
\[
W^\star-\mathrm{CE}_\gamma(A_m)
\leq
R(A_m)+
\frac{e^{\gamma T}-1-\gamma T}{\gamma T^2}
\frac{m}{n}Q_{m,2}.
\]
For computational bagging, the same inequality holds with $A_m$ replaced by $A_{m,B}$ and $m/n$ replaced by $m/n+(1-m/n)/B$, with the certainty equivalent taken over both sampling and computational randomness. When $T=0$, both certainty-equivalent losses are zero.
\end{corollary}
For fixed $\gamma$ and $T$, exact CARA utility loss thus has the same upper-bound rate as risk-aware regret, including for the full-sample policy $m=n$.
\begin{remark}[Other CATE moments]\label[remark]{rmk:p_moment}
For every $p>1$, the regret conclusion has the corresponding moment form
$
R(A_m)
\lesssim
Q_{m,p}^{\frac{\kappa+1}{\kappa+p}}.
$
When the available primitive is an $L_2$ CATE rate, setting $p=2$ directly is sharper than first converting the MSE into a different moment. This is the route used in \Cref{thm:bag_upper}.
\end{remark}
The upper-bound analysis therefore reduces policy-vote bagging to two quantities: the base learner's CATE error and each unit's bag-inclusion frequency. The next section asks whether this decomposition can be improved and what bag size it recommends when CATE accuracy follows a standard learning rate.
\section{Lower bounds and optimality}\label{sec:rates}
Are the bounds sharp? What bag sizes do they recommend? How do the resulting rates compare with what any algorithm can achieve? We answer these questions in turn. First, we establish matching lower bounds over a high-level class of populations and base learners. Second, we characterize rate-optimal bag sizes. Third, we give a minimax benchmark over all policy-learning algorithms and use it to assess the parametric and Gaussian-kernel rates.
\subsection{Sharpness over a high-level class}\label{sec:bag_lower}
A single learning-rate parameter summarizes how the base learner improves with bag size. Suppose that, for some $\alpha \in(0,1]$, $
Q_{m,2}
\lesssim
m^{-\alpha}
$ uniformly over $1\leq m\leq n.
$
For empirical-risk minimizers, $Q_{m,2}$ is often the square of a critical radius, so localized complexity calculations can characterize $\alpha$ for particular CATE learners \citep{van_der_laan_2026}. Parametric learners have $\alpha=1$. In nonparametric problems, $\alpha<1$ reflects smoothness and dimension.
The learning rate $\alpha$ determines how bag size $m$ affects the two terms of our upper bound. Substituting the learning rate into \Cref{thm:bag_upper} gives
\begin{equation}\label{eq:rate_upper}
R_{\mathrm{RA}}(A_m)\lesssim m^{-\frac{\alpha(\kappa+1)}{\kappa+2}}+\rho\frac{m^{1-\alpha}}{n}.
\end{equation}
The first term bounds expected regret and decreases with bag size. The second term bounds the sampling-risk. Larger bags improve estimation accuracy, but each observation also enters a larger fraction of the bags. For $\rho>0$ and $\alpha<1$, accuracy improves too slowly to offset this increased influence, so the second term increases with $m$. When $\alpha=1$, the two effects exactly offset, leaving the second term constant.
Computational bagging adds variation from sampling the bags. With $B$ bags, \Cref{cor:finite_bagging} adds $\rho\{(1-m/n)/B\}m^{-\alpha}$ to the complete-bagging bound in \eqref{eq:rate_upper}. This additional term vanishes at $m=n$, where every bag is the full sample.
Our lower bound shows that neither term in \eqref{eq:rate_upper} can be improved without restricting the data-generating process or the base learner further.
\begin{theorem}[Lower bound for complete policy-vote bagging]\label[theorem]{thm:sharpness}
Fix $\kappa>0$ and $\alpha>0$. There exist a data-generating process satisfying Assumption \ref{ass:main},
and a sequence of symmetric bag-level base learners, such that for every $n\geq m\geq1$, we have
$
Q_{m,2}
\asymp
m^{-\alpha}
$ while
\begin{equation}\label{eq:sharp_lower}
R_{\mathrm{RA}}(A_m)
\gtrsim
m^{-\frac{\alpha(\kappa+1)}{\kappa+2}}
+
\rho\frac{m^{1-\alpha}}{n}.
\end{equation}
The implicit constants are independent of $n$, $m$, and $\rho$. Together, \eqref{eq:rate_upper} and \eqref{eq:sharp_lower} show that the risk-aware-regret bound is sharp up to constants over the stated high-level class.
\end{theorem}
The theorem establishes sharpness when only an integrated CATE-error rate is known. It constructs a population and a sequence of base learners for which both terms attain their upper-bound orders. Particular learners can perform better. The matching bounds nevertheless allow a worst-case comparison of bag sizes over a class with common constants.
\subsection{Optimal bag size}\label{sec:bag_choice}\label{sec:learning_examples}
The matching bounds tell us how to choose bag size when only the learning rate is known. The comparison uses the same class of populations and base learners at every bag size.
\begin{corollary}[Optimal bag size over the high-level class]\label[corollary]{cor:bag_choice}
Fix $\kappa>0$ and $\alpha>0$. Let $\mathcal H_{\alpha,\kappa}$ satisfy the uniform class conditions in Appendix~\ref{app:bag_class}. Write $R_{\mathrm{RA},H}(A_m)$ for the risk-aware regret of complete policy-vote bagging under the population and base learners specified by $H$. Uniformly over $n\geq1$ and $\rho\geq0$,
\[
\inf_{1\leq m\leq n}
\sup_{H\in\mathcal H_{\alpha,\kappa}}
R_{\mathrm{RA},H}(A_m)
\asymp
\inf_{1\leq m\leq n}
\left\{
m^{-\frac{\alpha(\kappa+1)}{\kappa+2}}
+\rho\frac{m^{1-\alpha}}{n}
\right\},
\]
where the comparison constants depend only on $\alpha,\kappa$ and the class bounds. Consequently:
\begin{enumerate}[label=(\roman*),leftmargin=2.1em]
\item If $\alpha=1$, choosing $m=n$ is minimax-rate optimal.
\item If $0<\alpha<1$ and $\rho>0$, let $m_0=
\left(\frac n\rho\right)^{
\frac{\kappa+2}{\kappa+2-\alpha}}.$ Any integer bag size with $m\asymp\min\{n,\max(1,m_0)\}$ is minimax-rate optimal.
\item For every fixed $\rho\geq0$ and every $\alpha>0$, the minimized worst-case criterion has order $n^{-\alpha(\kappa+1)/(\kappa+2)}$ as $n\to\infty$. Every sequence $m\asymp n$, including $m=n$, attains this order.
\end{enumerate}
\end{corollary}
\Cref{cor:bag_choice} clarifies when to use a bag size $m$ that is smaller than the sample size $n$.
If estimation accuracy improves quickly due to a parametric CATE $(\alpha=1)$, the researcher should use the full sample, taking $m=n$.
If estimation accuracy improves slowly due to a nonparametric CATE $(\alpha<1)$, the researcher should typically use subsamples, taking $m=\left(\frac n\rho\right)^{
\frac{\kappa+2}{\kappa+2-\alpha}}$.
Intuitively, if there is simple heterogeneity with a known functional form ($\alpha=1$), then using the full sample is fine; if there is complex heterogeneity with an unknown functional form ($\alpha<1$), then subsampling helps.
Moreover, placing a higher price $\rho$ on sampling risk favors smaller bags.
For a fixed and large price $\rho$ of sampling risk, the researcher should use $m<n$ in finite samples and $m=n$ in the large sample limit $n\rightarrow \infty$. \Cref{fig:bag_size_envelope} illustrates these comparisons.
\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{Figures/bag_size_envelope.pdf}
\caption{Bag size and the risk-aware-regret bound}
\label{fig:bag_size_envelope}
\caption*{\footnotesize \textit{Notes:}
All panels use the envelope in \eqref{eq:rate_upper}
with unit leading constants and $\kappa=1$.
Panels (a) and (b) fix $n=1000$; panel (a) uses $\rho=20$.
Dots in these panels mark the minimizing bag sizes,
treating $m$ as continuous and imposing $1\leq m\leq n$.
Panel (b) shows how greater weight on sampling risk
can favor smaller bags.
Panel (c) fixes $\rho=3$ and plots the minimizing
fraction $m^\star/n$ as the sample grows.
The dashed line marks the full-sample choice,
which eventually minimizes the envelope.}
\end{figure}
The next two examples show how the learning rate affects optimal bag size.
\begin{example}[Parametric CATE]\label[example]{ex:parametric}
Under a parametric model and correctly-specified learner, $Q_{m,2}\lesssim m^{-1}$ and
$
R_{\mathrm{RA}}(A_m)
\lesssim
m^{-(\kappa+1)/(\kappa+2)}
+
\frac{\rho}{n}.
$
The rate-optimal bag size is $m=n$.
\end{example}
Slower learning leaves more room for smaller bags to help. The next example makes this dependence explicit in terms of smoothness and dimension.
\begin{example}[H\"older-smooth CATE]\label[example]{cor:holder}
Under the H\"older model and local-polynomial learner specified in Appendix~\ref{app:holder_conditions}, let $s$ denote smoothness and $d$ the covariate dimension. Then
$
Q_{m,2}\lesssim m^{-\frac{2s}{2s+d}}
$ and $
R_{\mathrm{RA}}(A_m)\lesssim
m^{-\frac{2s(\kappa+1)}{(2s+d)(\kappa+2)}}
+\rho\frac{m^{\frac{d}{2s+d}}}{n}.
$
The rate-optimal bag size is $m=\left(\frac n\rho\right)^{
\frac{(2s+d)(\kappa+2)}
{2s(\kappa+1)+d(\kappa+2)}}$.
\end{example}
This bound need not be sharp for smooth treatment effects. More detailed guarantees for the estimator can give faster regret rates \citep{audibert_tsybakov_2007}.
\begin{remark}[Finite-sample bag choice]\label[remark]{rem}\label[remark]{rem:finite_sample_bag_choice}
A rate-optimal bag size need not minimize finite-sample risk-aware regret. At fixed $n$, the upper bound depends on the actual error curve $m\mapsto Q_{m,2}$ and the constants omitted by rate statements. Finding the best bag size requires comparing expected welfare and sampling variance. Estimating the frontier is for future work.
\end{remark}
CARA utility permits an exact comparison in simple experiments. Even with parametric estimation accuracy, sufficiently strong risk aversion can favor smaller bags.
\begin{example}[Finite-sample bag choice in a toy example]\label[example]{ex:cara_bag_choice}
Consider the experiment and base learner in Appendix~\ref{app:cara_experiment}, with $n=3$ and $Q_{m,2}=75/(64m)$. Under $\psi_\gamma(w)=-\exp(-\gamma w)$, complete policy-vote bagging satisfies $\operatorname*{arg\,max}_{m\in\{1,2,3\}}\mathrm{CE}_\gamma(A_m)=3$ if $0<\gamma<4\log 3$, and $\operatorname*{arg\,max}_{m\in\{1,2,3\}}\mathrm{CE}_\gamma(A_m)=1$ if $\gamma>4\log 3$.
\end{example}
The full sample has higher expected welfare, but strong enough risk aversion favors small bags. We conclude that the best finite-sample choice depends on more than the learning rate.
\subsection{Minimax benchmark over all algorithms}\label{sec:minimax_local}
The final question is what any policy-learning algorithm can achieve. We partially answer the question, focusing on the parametric case $(\alpha=1)$. The next theorem uses two populations that differ only in whether treatment helps a small group.
\begin{theorem}[Local minimax rate]\label[theorem]{thm:minimax}
Fix $\kappa>0$, and let $\mathcal P_n$ be the parametric experiments constructed in Appendix~\ref{app:local_experiment}. For sufficiently small $c>0$ and every $\rho\geq0$,
\begin{equation*}\label{eq:minimax}
\inf_{A:\mathcal S^n\to\mathcal G}
\sup_{P\in\mathcal P_n}
R_{\mathrm{RA},P}(A)
\asymp
n^{-(\kappa+1)/(\kappa+2)}.
\end{equation*}
The constants are independent of $n$ and $\rho$.
\end{theorem}
The lower bound comes from mistakes in treatment choice. It matches the parametric rate in \Cref{ex:parametric} for fixed $\rho$.
Gaussian kernels offer a flexible learning method with nearly the same rate. The next example uses \Cref{thm:minimax} to show when this rate is close to the best possible.
\begin{example}[Gaussian-RKHS CATE]\label[example]{cor:gaussian_rkhs}
Under the Gaussian-RKHS model and kernel ridge learner specified in Appendix~\ref{app:gaussian_conditions}, let $A_n$ be the full-sample policy and $d$ the covariate dimension. Uniformly over the stated class,
$
Q_{n,2}\lesssim\frac{(\log n)^d}{n}
$ and $
R_{\mathrm{RA}}(A_n)\lesssim
\left\{\frac{(\log n)^d}{n}\right\}^{\frac{\kappa+1}{\kappa+2}}
+\rho\frac{(\log n)^d}{n}.
$
Under the additional lower-bound conditions stated there, this rate is minimax for fixed $\rho$, up to the factor $(\log n)^{d(\kappa+1)/(\kappa+2)}$.
\end{example}
The Gaussian comparison allows the covariate distribution to vary within the class. It does not establish the same lower bound for every fixed distribution.
\section{Discussion}\label{sec:discussion}
This paper connects sampling risk in statistical treatment choice to algorithmic stability.
The compound lottery includes both variation across experimental samples and variation across the full population under the implemented policy.
The main text focuses on aversion to the first source of uncertainty, while Appendix~\ref{app:joint_concavity} allows risk aversion at both stages.
If replacing one experimental unit changes a policy recommendation only slightly, then welfare cannot vary much across experimental samples.
Policy-vote bagging builds this stability into the overall policy learning algorithm.
Future work may estimate the frontier of expected welfare and sampling risk to guide the choice of bag size. Accounting for computational costs could also guide how many bags to use. Other extensions include alternative risk measures and treatment capacity constraints.
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