# Orthogonal Policy Learning Under Ambiguity111I am grateful to my advisors
Martin Weidner and Toru Kitagawa for their help and support throughout my
graduate studies. I would also like to thank Timothy Christensen, Sukjin Han,
Whitney Newey, Tomasz Olma, Joris Pinkse, Andrei Zeleneev, as well as
participants to the NeurIPS EconML Workshop, IAAE 2022 Conference, Bristol
Econometric Study Group for helpful comments and discussions.
†Department of Economics, University College London. E-mail<EMAIL_ADDRESS>
Riccardo D’Adamo†

###### Abstract

This paper studies the problem of estimating individualized treatment rules
when treatment effects are partially identified, as it is often the case with
observational data. By drawing connections between the treatment assignment
problem and classical decision theory, we characterize several notions of
optimal treatment policies in the presence of partial identification. The
proposed framework allows to incorporate user-defined constraints on the
policies, such as restrictions for transparency or interpretability, while
also ensuring computational feasibility. We show that partial identification
leads to a novel statistical learning problem with risk directionally – but
not fully – differentiable with respect to an infinite-dimensional nuisance
component. We propose an estimation procedure that ensures Neyman-
orthogonality with respect to the nuisance component and provide statistical
guarantees that depend on the amount of concentration around the points of
non-differentiability in the data-generating process. The proposed method is
illustrated using data from the Job Partnership Training Act study.

Job Market Paper

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##  1 Introduction

The problem of choosing an optimal treatment assignment based on data is
ubiquitous in economics and other fields, including medicine and marketing.
Individuals often display heterogeneous responses to the same treatment.
Decision-makers in policy and industry are therefore interested in leveraging
the growing availability of rich granular data to tailor treatment assignment
to individuals based on their characteristics. As a result, a fast-growing
literature has emerged focused on developing procedures for estimation of
individualized treatment rules. While a variety of approaches have been
recently established, these typically assume that the available data allow to
provide credible point estimates for the effect of the treatment, that is
treatment effects are point identified. While of important stylized value,
this assumption is often hard to justify in many empirical settings. For
example, economists have long been aware that popular quasi-experimental and
observational research designs, such as instrumental variables (IV), allow to
point identify treatment effects only for specific sub-populations (Imbens and
Angrist, 1994). Even in randomized control trials, point identification of the
treatment effects is often precluded due to non-random attrition, e.g. when
participants dropout from a program or the researcher is denied information on
the outcome variable (Lee, 2009). In such settings, the data may only provide
partial knowledge about the treatment response in the form of credible bounds,
i.e. the treatment effects are partially identified. As a result, the
decision-maker may have ambiguous evidence on whether a candidate policy
should be preferred to another, so that only a partial ordering of policies
can be deduced in general. While informative from a scientific perspective, a
partial ordering of policies is unsatisfying when the ultimate goal of the
analysis is to select a single policy to be implemented in the real world. In
this scenario, a decision-maker has to confront two sources of ambiguity. The
first source concerns ambiguous knowledge of the treatment response $\tau$
conditional on knowledge of distribution of the data $P$, due to partial
identification. The second source is the lack of knowledge of the distribution
$P$, which must be estimated from the data.

In this paper, we develop methods to handle both sources of ambiguity within
the framework of “empirical welfare maximization” (Kitagawa and Tetenov,
2018), also referred to as “policy learning” (Athey and Wager, 2021). This
approach considers treatment policies that are exogenously constrained to have
low complexity in terms of Vapnik-Chervonenkis (VC) dimension. This
encompasses many practical settings of interest, as policies often have to
satisfy requirements imposed for institutional or practical reasons, such as
fairness, budget or interpretability. The empirical welfare maximization (EWM)
method selects the optimal policy as the maximizer of the empirical analogue
of the population welfare, formulated as the average of the individual
outcomes in the target population. The EWM estimation procedure has the
convenient structure of an empirical risk minimization problem, which is
exploited by Kitagawa and Tetenov (2018) and Athey and Wager (2021) to study
its statistical properties.

We extend the EWM framework to settings with partial identification by making
several contributions. First, we study the problem of assigning treatment
under partial identification at the population level (i.e. where the
distribution of the data $P$ is known) from a general perspective. In
particular, we show how classic optimality criteria for decision under
ambiguity, such as minimax risk and minimax regret, can be applied in the
context of welfare maximization. Our unified framework accommodates different
attitudes towards ambiguity and a wide range of popular identification
assumptions, including Manski (1990) and Manski and Pepper (2000) bounds. Our
analysis delivers several notions of optimal treatment policies, which we
refer to as ambiguity-robust: they are “robust” in the sense that each of them
delivers a notion of single optimal policy in the presence of partial
identification, while they all reduce to the same optimal treatment assignment
in the special case of point-identification. As part of this analysis, we
establish general conditions on the identification sets under which the
treatment assignment problem can be expressed in a simplified form, leading to
computationally tractable sample analogues. In particular, we show that all
ambiguity-robust policies can be represented as maximizers of a “surrogate”
welfare, in which identification bounds are combined to form a proxy for the
partially identified CATE. The surrogate welfare depends on several nuisance
components, and its specific form is determined by the identification
assumptions and attitude towards ambiguity held by the decision-maker.

We then propose an algorithm for computing the estimated ambiguity-robust
policy and provide statistical guarantees on its performance in terms of the
regret convergence of the surrogate welfare. Similarly to Athey and Wager
(2021) and Foster and Syrgkanis (2019), our procedure leverages insights from
the literature on double/de-biased machine learning (Chernozhukov et al.,
2022) by making use of Neyman-orthogonalized estimates of the surrogate
welfare. This, coupled with sample-splitting, allows us to guarantee fast
rates of convergence for the estimated ambiguity-robust optimal policy while
imposing minimal requirements on the estimation of the nuisance components.
One unique feature of the partially identified setting studied in this paper
is the restricted degree of smoothness enjoyed by the welfare criterion. In
particular, we show that popular choices of identification assumptions and
optimality criteria for choice under ambiguity lead to surrogate welfare
criteria that are only directionally differentiable with respect to the data-
generating process. We highlight the importance of this feature for the
problem at hand and develop new theoretical results showing how the extent of
non-differentiability in the data-generating process affects the statistical
properties of the learning procedure. To the best of our knowledge, we are the
first to investigate the role of non-differentiabilities in the context of
semiparametric statistical learning problems. Our results are therefore of
independent interest and may be relevant beyond the treatment assignment
problem of this paper.

Finally, we apply the proposed method to experimental data from the Job
Training Partnership Act study, a dataset that has been extensively used to
study the effect of subsidized job training on labor market outcomes. We study
the optimal participation of workers into the job training programme based on
their education and previous earnings, and show that allowing for partial
identification delivers substantially different programme participation
policies compared to existing methods that assume point-identification.

###  1.1 Related literature

The results of this paper contribute to the recent literature on EWM methods,
e.g. Kitagawa and Tetenov (2018), Athey and Wager (2021), Mbakop and Tabord-
Meehan (2021), Viviano (2019), Sun (2021), and more broadly to the literature
studying statistical treatment choice, including Manski (2004), Dehejia
(2005), Hirano and Porter (2009), Stoye (2009), Chamberlain (2011),
Christensen et al. (2022), Kitagawa et al. (2022).222See also Hirano and
Porter (2020) and referenes therein.

Kitagawa and Tetenov (2018) introduced the EWM method and provided theoretical
results showing its optimality when implemented with experimental data. Athey
and Wager (2021) leverage insights from the recent literature on orthogonal
machine learning (Chernozhukov et al., 2022) and propose doubly-robust
estimation of the treatment effect which leads to optimal learning rates even
with observational data. We build on their work by adopting Neyman-orthogonal
estimates while we relax the fundamental assumption that treatment effects are
point identified. Cui and Tchetgen (2021) also develop procedures for learning
optimal treatments rules with instrumental variables but consider
unconstrained policy classes. Similarly to Athey and Wager (2021), they ensure
point-identification of treatment response by restricting their analysis to
the effect on compliers.

Kasy (2016), Han (2019) and Byambadalai (2022) provide methods for comparing
policies in the presence of covariates and partial identification of treatment
effects. The focus of their work is on characterizing the partial ordering of
policies in terms of their associated welfare rather than resolving the
ambiguity and estimating an optimal treatment rule.

In a series of papers, Manski (2009, 2010, 2011) studies the problem of a
social planner who must choose treatment for a population under partial
knowledge of the treatment response in the absence of covariates. He shows
that when the sign of the treatment effect is ambiguous, the minimax regret
criterion leads to policies that randomize treatment in the population. While
our study of the population problem is inspired by Manski’s work in this area,
the focus of our paper is on deterministic rules assigning individualized
treatment, i.e. based on (potentially continuous) covariates. Stoye (2012),
Ishihara and Kitagawa (2021) and Yata (2021) consider treatment assignment
under partial identification from a finite-sample minimax perspective, while
Christensen et al. (2022) adopt a local-asymptotic approach. However, these
works do not consider individualization of the treatment assignment.

More closely related to our work is Kallus and Zhou (2018), who extend the EWM
framework to learn an optimal policy in the presence of partially identified
treatment effects under violations of unconfoundedness. In particular, they
target welfare improvement with respect to a baseline pre-existing policy and
consider partial-identification of the welfare criterion within Rosenbaum’s
sensitivity model (Rosenbaum, 1987). Adjaho and Christensen (2022) and Kido
(2022) examine policies with maximin welfare guarantees when the target
population lies in a Wasserstein neighborhood of the experimental population.
The identification assumptions (and associated estimation procedures)
considered in these papers are distinct and do not nest those covered by our
framework. As a result, our contributions are complementary to these works.

Russell (2020) considers estimation of the optimal policy under partial
identification within a “probably approximately correct” learning framework
(Valiant, 1984). His proposed procedure has the advantage of side-stepping
direct estimation of the identified set, and can be applied in the context of
incomplete models for which the identification bounds cannot typically be
obtained in closed form. However, in the context of the identification
assumptions considered in this paper (e.g. Manski bounds), the theoretical
results in Russell (2020) require that the covariates have discrete support.
On the other hand, our proposed procedure requires computation of the
identification bounds in closed form but accommodates continuous covariates.

In independent work, Pu and Zhang (2021) study policy learning under ambiguity
from a classification perspective and derive an optimal policy which coincides
with one notion of ambiguity-robust policy studied in this paper. However, our
estimation procedure crucially differs from theirs for the use of Neyman-
orthogonalization which, combined with a refined proof-strategy that accounts
for the lack of full-differentiability in the welfare criterion, allows us to
guarantee considerably faster rates of convergence. In this sense, our results
extend and improve on those in Pu and Zhang (2021).

Finally, we contribute to a body of literature dealing with estimation and
inference for directionally-differentiable functionals. Hirano and Porter
(2012) show that if a target estimand is not differentiable in the parameters
of the data distributions, then no asymptotically unbiased or regular
estimator exists. Ponomarev (2022) studies efficient estimation of
directionally differentiable functionals from a local minimax perspective.
Fang and Santos (2018) and Kitagawa et al. (2020) provide inference results
for directionally differentiable functions from a frequentist and Bayesian
perspective, respectively. Also motivated by partial identification,
Christensen et al. (2022) consider estimation of treatment rules when the
welfare criterion is only directionally differentiable with respect to a
finite-dimensional nuisance component. Our framework instead involves
infinite-dimensional nuisance components and therefore our analysis must
account for the lack of differentiability with novel theoretical results that
substantially differ from Christensen et al. (2022).

The rest of this paper is organized as follows. Section 2 introduces the
setup. Section 3 presents several notions of ambiguity-robust optimal
policies. Section 4 presents the proposed estimation procedure for the
ambiguity-robust optimal policy. Section 5 provides statistical guarantees for
the estimated optimal policy. Section 6 presents an empirical illustration
based on the Job Training Partnership Act Study. Section 7 concludes the
paper. Proofs and extensions are given in the Appendix.

Notation. Throughout the paper, for $d\in\mathbb{N}$, let $\mathbb{R}^{d}$
denote the Euclidean space, with $\|\cdot\|_{p}$ and
$\langle\cdot,\cdot\rangle$ being the usual $\ell_{p}$-norm and inner product,
respectively. For two vectors $x\in\mathbb{R}^{p}$ and $y\in\mathbb{R}^{q}$,
$x\subset y$ means that $x$ is a sub-vector of $y$. For a symmetric matrix
$A$, $\lambda_{{\rm max}}(A)$ denotes its largest eigenvalue. Unless otherwise
stated, the expectation $\mathbb{E}[\cdot]$, probability $\mathbb{P}(\cdot)$,
and variance ${\rm Var}(\cdot)$ operators will be taken with respect to the
underlying distribution of observables $P$. Given a random variable
$Z\in\mathcal{Z}$ with $\mathcal{Z}\subseteq\mathbb{R}^{d}$, the associated
probability measure $P_{Z}$, and a function $f\,:\,\mathcal{Z}\to\mathcal{W}$
with $\mathcal{W}\subseteq\mathbb{R}^{q}$, we define
$\norm{f}_{Lp(P_{Z})}=\left(\mathbb{E}_{P_{Z}}\left[\|f(Z)\|^{p}_{p}\right]\right)^{1/p}$
for $p\in(0\,\infty)$. We extend this definition to $p=\infty$ in the natural
way. For a sequence of real numbers $x_{n}$ and $y_{n}$, $x_{n}=o(y_{n})$ and
$x_{n}=O(y_{n})$ mean, respectively, that $x_{n}/y_{n}\to 0$ and $x_{n}\leq
Cy_{n}$ for some constant $C$ as $n\to\infty$. For real numbers $a,b$,
$a\lesssim b$ means that there exists a constant $C$ such that $a\leq Cb$. For
a positive real number $a$, $\lfloor a\rfloor$ denotes its nearest smallest
integer. The notation $\rightarrow_{p}$ denotes convergence in probability.

##  2 Setup

Let $Y_{i}\in\mathbb{R}$ be an outcome measuring utility, $D_{i}\in\\{0,1\\}$
a binary treatment, $X_{i}\in\mathcal{X}\subseteq\mathbb{R}^{k_{x}}$ a set of
pre-treatment covariates for an individual $i$ from an i.i.d. population of
interest. We use standard notation to define the potential outcomes
$Y_{i}(0),Y_{i}(1)$. The conditional average treatment effect (CATE)
$\tau:\mathcal{X}\to\mathbb{R}$ is then defined as

$\displaystyle\tau(x)=y_{1}(x)-y_{0}(x),\quad
y_{d}(x)=\mathbb{E}[Y_{i}(d)|X_{i}=x],\,d=0,1,$

where the expectation is taken with respect to the distribution of the
population, and we will henceforth suppress the $i$-subscript for convenience.
The decision-maker (DM) is interested in choosing a deterministic treatment
assignment rule (or policy) $\pi:\mathcal{X}\to\\{0,1\\}$, which maps from the
support of individual pre-treatment covariates to the binary decision “treat”
($\pi(x)=1$) or “do not treat” ($\pi(x)=0$). Following Manski (2004), we
define the utilitarian social welfare associated with a policy $\pi$ and a
given configuration of the expected potential outcomes
$y_{0}(\cdot),y_{1}(\cdot)$ as

$\displaystyle W_{y_{0},y_{1}}(\pi)$
$\displaystyle=\mathbb{E}_{P_{X}}\left[y_{1}(X)\cdot\pi(X)+y_{0}(X)\cdot(1-\pi(X))\right]$
(1)
$\displaystyle=\underbrace{\mathbb{E}_{P_{X}}\left[\pi(X)\cdot\tau(X)\right]}_{=:I_{\tau}(\pi)}+\mathbb{E}_{P_{X}}\left[y_{0}(X)\right],$

where $I_{\tau}(\pi)$ represents the average impact of policy $\pi$. The
optimal policy for a given configuration of the CATE function is the one that
maximizes the associated welfare:

$\displaystyle\pi^{*}=\operatorname*{argmax}_{\pi\in\Pi}W_{y_{0},y_{1}}(\pi)=\operatorname*{argmax}_{\pi\in\Pi}I_{\tau}(\pi),$
(2)

where $\Pi$ is a family of candidate policies.333Throughout the paper, we will
assume that the maximization problem in (2) has at least one solution. If
multiple solutions exist, the DM is assumed to arbitrarily pick $\pi^{*}$ from
the set of maximizers. The DM has knowledge of the CATE through the
distribution $P\in\mathcal{P}$ of observable random variables $W$, where
$(Y,D,X)\subseteq W$. In particular, we denote $\mathcal{T}(P)$ the set of
plausible CATE functions associated with a certain distribution of
observables. When the DM has perfect knowledge of $P$ and $\mathcal{T}(P)$ is
a singleton, i.e. $\tau$ is point-identified, she can obtain $\pi^{*}$ by
solving (2).

Suppose now that $\mathcal{T}(P)$ is a non-singleton set, i.e. $\tau$ is
partially identified. In that case, even under perfect knowledge of $P$, there
exists a set of plausible values for the impact $I_{\tau}(\pi)$ of a candidate
policy $\pi$. Notice that partial identification of the CATE does not
necessarily imply that the DM cannot obtain the optimal policy $\pi^{*}$. In
particular, it is easy to see that under point-identification of the CATE one
has $\pi^{*}(x)=\mathbbm{1}\\{\tau(x)\geq 0\\}$ when the class of candidate
policies $\Pi$ is unrestricted, so that identification of the sign of the CATE
is sufficient to obtain the optimal policy.444Cui and Tchetgen (2021) study a
case in which sole point-identification of the sign of the CATE via an
instrumental variable allows to obtain the optimal policy. However, the
unrestricted policy class has limited relevance in many practical settings.
For example, the policy space $\Pi$ may be exogenously constrained for
institutional reasons, e.g. as policies may be required to satisfy specific
requirements for budget, fairness or interpretability. While the DM may still
hope that his specification of $\Pi$ contains the first-best policy
$\mathbbm{1}\\{\tau(X)\geq 0\\}$, it is useful to interpret $\pi^{*}$ as the
“best-in-class” policy for the chosen class $\Pi$, when this does not contain
the first-best. When $\Pi$ is constrained, the DM is not able to obtain
$\pi^{*}$ in general without full knowledge of the CATE, although a partial
ordering of policies can still be deduced (see Kasy, 2016; Han, 2019;
Byambadalai, 2022).

Under partial identification, the DM therefore faces two sources of ambiguity.
First, she does not know the distribution $P$. However, we assume that she has
access to a random sample $(W_{i})_{i=1,\dots,n}$ from which she can learn
about $P$. Second, she does not have knowledge about $\tau$ within the
identified-set $\mathcal{T}(P)$, even under perfect knowledge of $P$. The
broad objective of this paper is to provide a framework that allows the DM to
handle both sources of ambiguity. We will approach the problem in two steps.
First, we will study the decision problem faced by the DM under perfect
knowledge of $P$. In particular, we will handle the ambiguity arising from
partial identification of $\tau$ using well-known optimality criteria for
decision under ambiguity. Each of the optimality criteria we consider will
deliver a corresponding notion of optimal policy, which we call “ambiguity-
robust”. The ambiguity-robust optimal policy is a unique treatment assignment
rule that is preferred to all other policies in $\Pi$ according to preferences
of the DM, and that coincides with the usual notion of optimal policy
$\pi^{*}$ in (2) in the special case of point identification of the CATE. In
the next section, we study several notions of ambiguity-robust optimal policy.

In the second part of our analysis, we study how to handle the ambiguity in
$P$ by showing how the random sample $(W_{i})_{i=1,\dots,n}$ can be used to
obtain an estimate $\widehat{\pi}_{n}$ for the ambiguity-robust optimal
policy. The estimation procedure and the associated statistical guarantees are
presented in Section 4 and 5, respectively.

######  Remark 1.

Unrestricted policy classes may also be precluded for practical reasons
related to the estimation of the optimal policy. For example, the researcher
may need to condition on a large number of covariates $X$ for identification
of the treatment effects, but only be interested in assigning treatment based
on a restricted set of the covariates $\widetilde{X}\subset X$ (e.g. because
she may not observe the full set of covariates when assigning treatment to new
individuals from the population). In that case, a practical way to side-step
computation of an estimate for the lower-dimensional CATE,
$\mathbb{E}[Y_{i}(1)|\widetilde{X}_{i}=\widetilde{x}]-\mathbb{E}[Y_{i}(0)|\widetilde{X}_{i}=\widetilde{x}]$,
is to impose restrictions directly on the policy class $\Pi$ and estimate the
optimal policy based on the estimated higher-dimensional CATE via the sample
analogue of (2).

##  3 Ambiguity-robust optimal policies

The study of decision under ambiguity has a long tradition in decision theory
and has received considerable attention in the context of treatment assignment
problems (see Manski, 2011, for a review). In this section we review some
classical optimality criteria for decision under ambiguity and study how they
can be applied in the context of the treatment assignment problem at hand,
leading to several notions of ambiguity-robust optimal policy.

A well-known optimality criterion for decision under ambiguity is minimax risk
(see, e.g. Wald, 1950). In the context of our treatment assignment problem we
can interpret welfare as negative risk, and this criterion leads to the
optimal maximin welfare policy

$\displaystyle\pi^{*}_{\texttt{MMW}}=\operatorname*{argmax}_{\pi\in\Pi}\min_{\left(y_{0},y_{1}\right)\in\mathcal{Y}(P)}W_{y_{0},y_{1}}(\pi),$
(3)

where $\mathcal{Y}(P)$ is the ambiguity set for
$\left(y_{0}(\cdot),y_{1}(\cdot)\right)$ identified from the distribution $P$
of observables random variables. The optimal maximin welfare policy maximizes
the lowest possible welfare under any configuration of the expected potential
outcome functions in the identified set $\mathcal{Y}(P)$. An alternative
application of minimax risk optimality in the context of treatment assignment
is maximin impact, leading to the optimal policy

$\displaystyle\pi^{*}_{\texttt{MMI}}=\operatorname*{argmax}_{\pi\in\Pi}\min_{\tau\in\mathcal{T}(P)}I_{\tau}(\pi),$
(4)

where $\mathcal{T}(P)$ denotes the ambiguity set for the CATE function. The
optimal maximin impact policy maximizes the lowest possible impact under any
configuration of the CATE in the identified set $\mathcal{T}(P)$. Notice that
the minimax welfare criterion reflects an extreme degree of pessimism with
regards to outcomes associated with both treatment and non-treatment
scenarios; on the other hand, the minimax impact criterion reflects an extreme
degree of pessimism with regards to the impact of the policy, thus directly
raising the threshold for treatment.555In the empirical application of Section
6, both minimax welfare and minimax impact criteria result in $\pi(x)=0$ for
the entire population. Despite its intuitive appeal, minimax optimality has
been criticised for being too conservative and often delivering decisions that
are especially sensitive to changes in the ambiguity set.666In his classic
textbook, Berger goes as far as saying that “In actually making decisions, the
use of the minimax principle is definitely suspect.” (Berger, 1985).

An alternative criterion that alleviates some of these concerns is minimax
regret, with corresponding optimal policy

$\displaystyle\pi^{*}_{\texttt{MMR}}$
$\displaystyle=\operatorname*{argmin}_{\pi\in\Pi}\max_{\left(y_{0},y_{1}\right)\in\mathcal{Y}(P)}\left[\left(\max_{\pi\,:\,\mathcal{X}\to\\{0,1\\}}W_{y_{0},y_{1}}(\pi)\right)-W_{y_{0},y_{1}}(\pi)\right]$
(5)
$\displaystyle=\operatorname*{argmin}_{\pi\in\Pi}\max_{\tau\in\mathcal{T}(P)}\left[\left(\max_{\pi\,:\,\mathcal{X}\to\\{0,1\\}}I_{\tau}(\pi)\right)-I_{\tau}(\pi)\right],$

The minimax regret criterion delivers a policy that minimizes the largest
possible distance between attained welfare and the highest level of welfare
attainable by the “oracle” treatment rule
$\pi^{*}=\mathbbm{I}\left\\{\tau(x)\geq 0\right\\}$ that has knowledge of the
true $\tau$. Minimax regret optimality has been advocated by Manski (2004) for
its balanced consideration of the possible states of nature and for delivering
more “reasonable” decisions rules in practice, compared to minimax risk
approaches.

######  Remark 2.

An alternative version of the minimax regret criterion is minimax regret with
respect to the welfare attained by the best-in-class policy in $\Pi$,
resulting in the objective

$\displaystyle\pi^{*}_{\texttt{MMR2}}=\operatorname*{argmin}_{\pi\in\Pi}\max_{\tau\in\mathcal{T}(P)}\left[\left(\max_{\pi\in\Pi}I_{\tau}(\pi)\right)-I_{\tau}(\pi)\right].$
(6)

While these two versions of the minimax regret criterion can be expected to
enjoy similar properties, the first version we have considered is considerably
more tractable. In fact, the innermost maximization in (5) has the closed-form
solution
$\max_{\pi\,:\,\mathcal{X}\to\\{0,1\\}}W_{\tau}(\pi)=\mathbb{E}_{P_{X}}\left[\max\left\\{\tau(X),0\right\\}\right]$.
As we show in Proposition 2 below, this allows to more explicitly characterize
the properties of the optimization problem and the resulting optimal policy,
as well as reduce the computational burden in solving the empirical analogue
of the problem. For this reason we will focus on the version in (5) of the
criterion. We also note that whenever the class $\Pi$ is “well-specified”, in
the sense that $\mathbbm{I}\left\\{\tau(x)\geq 0\right\\}\in\Pi$ for all
$\tau\in\mathcal{T}(P)$, the two optimality criteria are equivalent.

One critical drawback in the application of the optimality criteria just
presented to the treatment assignment problem of this paper is that the
optimal policies cannot be obtained in closed form. This is due to the form of
(3), (4) and (5) involving several nested optimizations whose solutions cannot
be easily characterized at the current level of generality when $X$ includes
continuously distributed covariates and $\Pi$ may be arbitrarily restricted,
which are both primary cases of interest of this paper. To make progress, we
impose the following restrictions on the ambiguity sets for the expected
potential outcomes and CATE.

######  Assumption 3.1 (Rectangular identified set for $(y_{0},y_{1})$).

The identified set for $(y_{0},y_{1})$ is rectangular, that is, $\mathcal{Y}$
is of the form

$\mathcal{Y}=\\{\left(y_{0}(\cdot),y_{1}(\cdot)\right):\left(y_{0}(x),y_{1}(x)\right)\in\mathcal{Y}(x)\\},$

where $\mathcal{Y}(x)$ is a compact subset of $\mathbb{R}^{2}$.

######  Assumption 3.2 (Rectangular identified set for $\tau$).

The identified set for $\tau$ is rectangular, that is, $\mathcal{T}$ is of the
form

$\displaystyle\mathcal{T}=\\{\tau(\cdot):\tau(x)\in[\underline{\tau}(x),\overline{\tau}(x)]\\},$

where $|\overline{\tau}(x)|<\infty$, $|\underline{\tau}(x)|<\infty$ for all
$x\in\mathcal{X}$.

Assumptions 3.1 and 3.2 impose separation of the identified sets for the
expected potential outcomes and CATE across the support of the covariates
$\mathcal{X}$.777Notice that Assumption 3.1 implies Assumption 3.2, but not
viceversa. They are typically satisfied by identification schemes that do not
impose shape restrictions on counterfactual outcomes with respect to the
covariates $X_{i}$. These assumptions are widely adopted in the partial
identification literature, and we refer the reader to Appendix B in Kasy
(2016) for an extensive review of identification schemes that result in
rectangular identified sets. Below we present three examples of identification
schemes for the CATE that satisfy this assumption.

######  Example 3.1 (Manski bounds).

Suppose there exists a binary instrument $Z_{i}\in\\{0,1\\}$ that satisfies
the well know exogeneity and exclusion restrictions
$Y_{i}(0),Y_{i}(1),D_{i}(0),D_{i}(1)\perp Z_{i}|X_{i}$, where $Y_{i}(d)$ and
$D_{i}(z)$ denote the counterfactual outcome and treatment functions,
respectively. If the instrument $Z_{i}$ also satisfies the overlap condition

$\displaystyle\eta\leq\mathbb{P}(Z_{i}=1|X_{i})\leq 1-\eta,\quad\eta>0,$

and the monotonicity condition (also known as no-defiers condition):

$\displaystyle\mathbb{P}\big{(}D_{i}(1)\leq D_{i}(0)|X_{i}\big{)}=1\quad{\rm
or}\quad\mathbb{P}\big{(}D_{i}(1)\geq D_{i}(0)|X_{i}\big{)}=1,$

then seminal work by Imbens and Angrist (1994) shows point-identification of
the conditional local average treatment effect (LATE):

$\displaystyle\mathbb{E}[Y_{i}(1)-Y_{i}(0)\,|\,D_{i}(1)\neq
D_{i}(0),\,X_{i}=x].$

Let us now assume that $Y\in[Y_{L},Y_{U}]$, i.e. the outcome is bounded, and
define

$\displaystyle h(z,x)=\mathbb{E}[Y_{i}|Z_{i}=z,X_{i}=x],$ $\displaystyle
m(d,z,x)=\mathbb{E}[Y_{i}|D_{i}=d,Z_{i}=z,X_{i}=x],$ $\displaystyle
p(z,x)=\mathbb{P}(D_{i}=1|Z_{i}=z,X_{i}=x),$ $\displaystyle
z(x)=\mathbb{P}(Z_{i}=1|X_{i}=x).$

The identified sets for the expected potential outcomes $y_{0}(x)$ and
$y_{1}(x)$ are contained within the bounds

$\displaystyle\overline{y}_{0}(x)$
$\displaystyle=\min_{z\in\\{0,1\\}}\big{\\{}m(0,z,x)\cdot(1-p(z,x))+Y_{U}\cdot
p(z,x)\big{\\}},$ $\displaystyle\underline{y}_{0}(x)$
$\displaystyle=\max_{z\in\\{0,1\\}}\big{\\{}m(0,z,x)\cdot(1-p(z,x))+Y_{L}\cdot
p(z,x)\big{\\}},$

and

$\displaystyle\overline{y}_{1}(x)$
$\displaystyle=\min_{z\in\\{0,1\\}}\big{\\{}m(1,z,x)\cdot
p(z,x)+Y_{U}\cdot(1-p(z,x))\big{\\}},$ $\displaystyle\underline{y}_{1}(x)$
$\displaystyle=\max_{z\in\\{0,1\\}}\big{\\{}m(1,z,x)\cdot
p(z,x)+Y_{L}\cdot(1-p(z,x))\big{\\}}.$

The identified set for the CATE is then contained within the bounds

$\displaystyle\overline{\tau}(x)$
$\displaystyle=\overline{y}_{1}(x)-\underline{y}_{0}(x),$
$\displaystyle\underline{\tau}(x)$
$\displaystyle=\underline{y}_{1}(x)-\overline{y}_{0}(x).$

If no further functional form assumption on the distribution of potential
outcomes is made, these bounds are sharp (Heckman and Vytlacil, 2001) and the
sharp identified sets for the average potential outcomes and CATE respectively
satisfy Assumption 3.1 and Assumption 3.2.

######  Example 3.2 (Balke-Pearl).

Suppose that the same assumptions as in Example 3.1 hold, and additionally the
monoticity assumption is strengthened to

$\displaystyle\mathbb{P}\big{(}D_{i}(1)\geq D_{i}(0)|X_{i}\big{)}=1,$

that is, the direction of the monotonicity is known and positive. The bounds
for the potential outcomes simplify to

$\displaystyle\overline{y}_{0}(x)$
$\displaystyle=m(0,0,x)\cdot(1-p(0,x))+Y_{U}\cdot p(0,x),$
$\displaystyle\underline{y}_{0}(x)$
$\displaystyle=m(0,0,x)\cdot(1-p(0,x))+Y_{L}\cdot p(0,x),$
$\displaystyle\overline{y}_{1}(x)$ $\displaystyle=m(1,1,x)\cdot
p(1,x)+Y_{U}\cdot(1-p(1,x),$ $\displaystyle\underline{y}_{1}(x)$
$\displaystyle=m(1,1,x)\cdot p(1,x)+Y_{L}\cdot(1-p(1,x)),$

and the CATE is contained within the bounds

$\displaystyle\overline{\tau}(x)$
$\displaystyle=h(1,x)-h(0,x)+p(0,x)\cdot\big{(}m(1,0,x)-Y_{L}\big{)}+(1-p(1,x))\cdot\big{(}Y_{U}-m(0,1,x)\big{)},$
$\displaystyle\underline{\tau}(x)$
$\displaystyle=h(1,x)-h(0,x)+p(0,x)\cdot\big{(}m(1,0,x)-Y_{U}\big{)}+(1-p(1,x))\cdot\big{(}Y_{L}-m(0,1,x)\big{)}.$

where $p(0,x)$ and $1-p(1,x)$ identify the proportions of always-takers and
never-takers at $X_{i}=x$, respectively. If no further functional form
assumption on the distribution of outcomes for non-compliant populations is
made, these bounds are sharp (Balke and Pearl, 1997) and the sharp identified
sets for the average potential outcomes and CATE respectively satisfy
Assumption 3.1 and Assumption 3.2.

######  Example 3.3 (Manski-Pepper bounds).

Suppose that instead of full exogeneity, the instrumental variable $Z_{i}$
satisfies the weaker “monotone IV” condition

$\displaystyle\mathbb{E}[Y_{i}(d)|Z_{i}=0,X_{i}]\leq\mathbb{E}[Y_{i}(d)|Z_{i}=1,X_{i}],\quad
d=0,1.$ (7)

Manski and Pepper (2000) show that when the outcome is bounded one has

$\displaystyle\sum_{z=0,1}\mathbb{P}\left(Z_{i}=z|X_{i}\right)\cdot\max_{z_{1}\leq
z}$
$\displaystyle\left\\{m(d,z_{1},X_{i})\cdot\mathbb{P}(D_{i}=d|Z_{i}=z_{1},X_{i})+Y_{L}\cdot\mathbb{P}(D_{i}=1-d|Z_{i}=z_{1},X_{i})\right\\}$
$\displaystyle\qquad\qquad\qquad\leq\,\mathbb{E}[Y_{i}(d)|X_{i}]\,\leq$
$\displaystyle\sum_{z=0,1}\mathbb{P}\left(Z_{i}=z|X_{i}\right)\cdot\min_{z_{2}\geq
z}$
$\displaystyle\left\\{m(d,z_{2},X_{i})\cdot\mathbb{P}(D_{i}=d|Z_{i}=z_{2},X_{i})+Y_{L}\cdot\mathbb{P}(D_{i}=1-d|Z_{i}=z_{2},X_{i})\right\\}.$

Upper (lower) bounds for the CATE are obtained by combining upper (lower)
bounds for $\mathbb{E}[Y_{i}(1)|X_{i}=x]$ with the lower (upper) bound for
$\mathbb{E}[Y_{i}(0)|X_{i}=x]$:

$\displaystyle\overline{\tau}(x)$
$\displaystyle=z(x)\cdot\psi_{1,1}(x;Y_{U})+(1-z(x))\cdot\min\left\\{\psi_{0,1}(x;Y_{U}),\psi_{1,1}(x;Y_{U})\right\\}$
$\displaystyle-z(x)\cdot\max\left\\{\psi_{0,0}(x;Y_{L}),\psi_{1,0}(x;Y_{L})\right\\}-(1-z(x))\cdot\psi_{0,0}(x;Y_{L}),$
$\displaystyle\underline{\tau}(x)$
$\displaystyle=z(x)\cdot\max\left\\{\psi_{0,1}(x;Y_{L}),\psi_{1,1}(x;Y_{L})\right\\}+(1-z(x))\cdot\psi_{0,1}(x;Y_{L})$
$\displaystyle-z(x)\cdot\psi_{1,1}(x;Y_{U})-(1-z(x))\cdot\min\left\\{\psi_{0,0}(x;Y_{U}),\psi_{1,0}(x;Y_{U})\right\\},.$

where

$\displaystyle\psi_{z,d}\left(x;Y_{(\cdot)}\right)=m(d,z,x)\cdot(d\cdot
p(z,x)+(1-d)\cdot(1-p(z,x))+Y_{(\cdot)}\cdot(d\cdot(1-p(z,x))+(1-d)\cdot
p(z,x)).$

Under no further assumption on the distribution of potential outcomes, these
bounds are sharp (Manski and Pepper, 2000) and satisfy Assumptions 3.1 and
3.2.

Having restricted the identified sets $\mathcal{Y}$ and $\mathcal{T}$ as in
Assumptions 3.1-3.2, we are now able to provide a simpler characterization of
the maximin welfare and maximin impact policies.

######  Proposition 1.

Define $\underline{y}_{d}(x)=\min_{y_{d}(x)\in\mathcal{Y}(x)}y_{d}(x)$ and
$\overline{y}_{d}(x)=\max_{y_{d}(x)\in\mathcal{Y}(x)}y_{d}(x)$. Under
Assumption 3.1 the optimal maximin welfare policy is

$\displaystyle\pi^{*}_{\emph{{MMW}}}=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot(\underline{y}_{1}(X)-\underline{y}_{0}(X))\Bigg{]}.$
(8)

Furthermore, under Assumption 3.2 the optimal maximin impact policy is

$\displaystyle\pi^{*}_{\emph{{MMI}}}=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot\underline{\tau}(X)\Bigg{]}.$
(9)

Proposition 1 shows that the optimal maximin welfare and maximin impact
policies maximize surrogate versions of the welfare substituting the
unidentified CATE with the difference in the lower bounds of potential
outcomes $\underline{y}_{1}(x)-\underline{y}_{0}(x)$ and the lower bound for
CATE $\underline{\tau}(x)$ at every point in the covariate space,
respectively. Notice that when $\mathcal{Y}(x)$ is rectangular with respect to
the two potential outcomes, i.e.
$y_{0}(x)\in\mathcal{Y}_{0}(x),y_{1}(x)\in\mathcal{Y}_{1}(x)$ and
$\mathcal{Y}(x)=\mathcal{Y}_{0}(x)\times\mathcal{Y}_{1}(x)$, we have
$\underline{\tau}(x)=\underline{y}_{1}(x)-\overline{y}_{0}(x)$, thus
highlighting the “pessimistic” nature of the maximin impact criterion.

######  Remark 3.

The maximin welfare and maximin impact optimal policies coincide when
$y_{0}(\cdot)$ is point-identified. This case is relevant when $y_{0}(x)$
represents the (conditional) average outcome under the status-quo in the
entire population and is typically point-identified from observational data.

The simplification of these two maximin problems into single maximisation
problems has important advantages for the study of the optimal policies and
their estimation from the data. In fact, the sample analogues of optimizations
(8) and (9) are amenable to standard computation procedures for a variety of
policy classes $\Pi$. Furthermore, their solution can be studied using tools
for empirical risk minimisation problems, as discussed in Section 4.

Despite the involvement of an additional maximization problem compared to
maximin welfare and maximin impact, Assumption 3.2 allows to provide a simpler
characterization also for the minimax regret optimal policy.

######  Proposition 2.

Under Assumption 3.2 the optimal minimax welfare regret policy is

$\displaystyle\pi^{*}_{\emph{{MMR}}}=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot\widetilde{\tau}(X)\Bigg{]}$
(10)

where

$\displaystyle\widetilde{\tau}(x)=\overline{\tau}(x)\cdot\mathbbm{1}\big{\\{}\overline{\tau}(x)\geq
0\big{\\}}+\underline{\tau}(x)\cdot\mathbbm{1}\big{\\{}\underline{\tau}(x)\leq
0\big{\\}}$ (11)

This simpler characterization of the minimax regret problem as a single
maximization sheds light on the properties of its associated optimal policy.
In particular, we see that the objective function symmetrically treats
individuals whose expected treatment effect sign is identified by assigning as
surrogate for the CATE their outer bound, i.e. the CATE upper (lower) bound
for individuals with identified positive (negative) sign for CATE. Individuals
for which the sign of the treatment effect is ambiguous are assigned an
intermediate point within their respective CATE bounds. The location of this
intermediate point depends on the extent to which the identified set lies in
the positive or negative region. Intuitively, the criterion prioritizes
correct treatment allocation to individuals who unambiguously benefit from (or
are harmed by) the treatment and down-weights the importance of individuals
for which the sign of the treatment response is ambiguous within the treatment
allocation problem. As an extreme case, individuals with CATE bounds exactly
symmetric around 0 (i.e. $\overline{\tau}(x)=-\underline{\tau}(x)$) are given
no consideration in the solution of the treatment allocation problem. This
intuition can be further supported by noticing that the original welfare
maximization under point-identification in (2) can be re-casted as the
weighted classification problem

$\displaystyle\pi^{*}=\operatorname*{argmin}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\bigg{[}\mathbbm{1}\big{\\{}(2\pi(X)-1)\neq\text{sign}(\tau(X))\big{\\}}\cdot|\tau(X)|\bigg{]},$

of which the minimax welfare regret optimal policy in (11) turns out to solve
the minimax version under Assumption 3.2:

$\displaystyle\pi^{*}_{\texttt{MMR}}=\operatorname*{argmin}_{\pi\in\Pi}\max_{\tau\in\mathcal{T}}\mathbb{E}_{P_{X}}\bigg{[}\mathbbm{1}\big{\\{}(2\pi(X)-1)\neq\text{sign}(\tau(X))\big{\\}}\cdot|\tau(X)|\bigg{]}.$

It is from this minimax classification risk perspective that Pu and Zhang
(2021) obtain and study the minimax regret policy, which they call the “IV-
optimal policy”.

An alternative version of minimax regret optimality which has been used in the
context of treatment choice is minimax regret with respect to a baseline
policy. Kallus and Zhou (2018) assume the existence of a fixed policy
$\pi_{\texttt{B}}$ from which the DM does not want to unnecessarily deviate.
They define the optimal policy as minimizing regret with respect to this
baseline policy:

$\displaystyle\pi^{*}_{\texttt{MMRB}}$
$\displaystyle=\operatorname*{argmin}_{\pi\in\Pi}\max_{\tau\in\mathcal{T}}\left(I_{\tau}(\pi_{\texttt{B}})-I_{\tau}(\pi)\right)$
$\displaystyle=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot\left(\overline{\tau}(X)\cdot\mathbbm{1}\left\\{\pi_{\texttt{B}}(X)\geq
0\right\\}+\underline{\tau}(X)\cdot\mathbbm{1}\left\\{\pi_{\texttt{B}}(X)<0\right\\}\right)\Bigg{]},$

where the second equality uses Assumption 3.2. While potentially appealing in
certain settings, e.g. when $\pi_{\texttt{B}}$ represents the existing
standard of care in a medical setting, this optimality criterion suffers the
potential drawback of requiring the DM to specify (and motivate) the baseline
policy for it to be operational. Adopting the never-treat baseline policy,
i.e. $\pi_{\texttt{B}}(x)=0,\,\forall x\in\mathcal{X}$, could be seen as an
appealing “agnostic” choice, which however makes this criterion default to
maximin impact and thus inherit its potentially undesirable properties.

The last notion of ambiguity-robust optimal policy that we present in this
section is based on the Hurwicz criterion (Hurwicz, 1951), arguably one of the
most widely used in decision-making under ambiguity. In the context of the
treatment assignment problem at hand, the Hurwicz criterion leads to the
ambiguity-robust policy

$\displaystyle\pi^{*}_{\texttt{HurW},\delta_{0},\delta_{1}}=$
$\displaystyle\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot\big{(}\\{\delta_{1}\cdot\overline{y}_{1}(X)+(1-\delta_{1})\cdot\underline{y}_{1}(X)\\}-\\{\delta_{0}\cdot\overline{y}_{0}(X)+(1-\delta_{0})\cdot\underline{y}_{0}(X)\\}\big{)}\Bigg{]},$

where $\delta_{1}\in[0,1]$ and $\delta_{0}\in[0,1]$ are user-defined weights
reflecting the degree of optimism with respect to the outcomes under treatment
and non-treatment, respectively. It is easy to see that the maximin welfare
criterion in (3) corresponds to the choice $\delta_{1}=0,\delta_{0}=0$. An
analogous notion of optimality focused on impact rather than welfare, leads to
the optimal policy

$\displaystyle\pi^{*}_{\texttt{HurI},\delta}=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\Bigg{[}\left(2\pi(X)-1\right)\cdot\big{(}\delta\cdot\overline{\tau}(X)+(1-\delta)\cdot\underline{\tau}(X)\big{)}\Bigg{]},$

where $\delta\in[0,1]$ controls the degree of optimism with respect to the
effect of treatment, with the maximin impact optimal policy corresponding to
the choice $\delta=0$. Under Assumption 3.1 and
$\mathcal{Y}(x)=\mathcal{Y}_{0}(x)\times\mathcal{Y}_{1}(x)$, the Hurwicz
Impact criterion is nested into the Hurwicz Welfare for the choice of
parameters $\delta=\delta_{0}=1-\delta_{1}$; unlike Hurwicz Welfare, however,
the Hurwicz Impact criterion is still well-defined under the weaker Assumption
3.2. Interestingly, when $\delta=1/2$ and $\Pi$ is well-specified, in the
sense that it contains the first-best assignment
$\mathbbm{1}\\{\overline{\tau}(x)+\underline{\tau}(x)\geq 0\\}$, we have

$\displaystyle\pi^{*}_{\texttt{MMR}}(x)=\mathbbm{1}\\{\widetilde{\tau}(x)\geq
0\\}=\mathbbm{1}\\{\overline{\tau}(x)+\underline{\tau}(x)\geq
0\\}=\pi^{*}_{\texttt{HurI},\frac{1}{2}}(x).$

Therefore the minimax regret and Hurwicz impact optimal policies coincide
under correct specification of $\Pi$, as they assign treatment based on the
middle point between the upper and lower CATE bounds. When $\Pi$ is not well-
specified, however, minimax regret optimality is not nested into any of the
Hurwicz-type criteria just presented, thus highlighting the radically
different attitude towards ambiguity implied by minimax regret compared to
maximin welfare/impact. In particular, minimax regret is the only criterion of
those presented (along with Hurwicz impact under $\delta=1/2$) that treats
symmetrically individuals with identified sets symmetric around 0, in the
sense that $\widetilde{\tau}_{1}(x)=-\widetilde{\tau}_{2}(x)$ whenever
$\overline{\tau}_{1}(x)=-{\underline{\tau}}_{2}(x)$ and
$\underline{\tau}_{1}(x)=-{\overline{\tau}}_{2}(x)$. For this reason, minimax
regret does not reflect an optimistic/pessimistic attitude towards ambiguity
but rather an “opportunistic” one, in light of its prioritization of correct
treatment assignment to individuals whose CATE sign is unambiguously
identified.

MinimaxRegretHurwiczWelfareHurwiczImpactMaximinWelfareMaximin
Impact$\delta_{0}=\delta_{1}=0$$\Pi$ well-specified,
$\delta=\frac{1}{2}$$\delta=0$$y_{0}(x)$ point-
identified$\delta=\delta_{1}=1-\delta_{0}$ Figure 1: Relationship between
ambiguity-robust optimal policies

###  3.1 A common framework

While accommodating a wide range of attitudes towards ambiguity, the notions
of optimality presented in Section 5.2 share a common structure. In fact, by
virtue of Assumptions 3.1 and 3.2, the corresponding optimal policies can all
be written as

$\displaystyle\pi^{*}(P)=\operatorname*{argmax}_{\pi\in\Pi}Q(P;\pi),\qquad
Q(P;\pi):=\mathbb{E}_{P_{X}}\Big{[}\left(2\pi(X)-1\right)\cdot\Gamma(P;X)\Big{]},$
(12)

for a specific score function888The term ‘score function’ is borrowed from
Athey and Wager (2021). $\Gamma(P;\,\cdot\,)$, where we have highlighted the
dependence of the score on the distribution $P$. The specific dependence on
$P$ is determined by the optimality criterion (as summarized in Table 1) as
well as the identification assumptions (e.g. Balke-Pearl, Manski-Pepper etc.).
This common structure also nests the point identified setting as the special
case $\Gamma(P;X)=\tau(X)$ and thus suggests that existing estimation
procedures for this special case can be extended to the partially identified
setting.

However, one peculiar feature of the partially identified setting is the
restricted degree of smoothness enjoyed by the objective function, in
particular the differentiability of the scores with respect to $P$. Under
point-identification of the CATE via standard unconfoundedness assumptions,
one has $\Gamma(P;x)=\mathbb{E}[Y|D=1,X=x]-\mathbb{E}[Y|D=1,X=x]$ and the full
differentiability of the score with respect to the expectation
$\mathbb{E}[Y|D,X]$ is immediately apparent. However, for the minimax regret
criterion we notice that the score is directionally differentiable999 Let
$P\in\mathcal{P}$ be a probability distribution on which the function
$f:\mathcal{P}\to\mathbb{R}$ depends. We say that $f$ is directionally
differentiable at $P_{0}$ if the limit  $\displaystyle\lim_{t\downarrow
0}\frac{f(P_{0}+t(h-P_{0}))-f(P_{0})}{t}=\dot{f}_{P_{0}}[h]$ exists for every
$h\in\mathcal{P}$, in which case $\dot{f}_{P_{0}}[\cdot]$ denotes the
directional derivative of $f$ at $P_{0}$. If it exists, the directional
derivative $\dot{f}_{P_{0}}[\cdot]$ is positively homogeneous of degree one
but not necessarily linear. If $\dot{f}_{P_{0}}[\cdot]$ is linear then f is
fully differentiable at $P_{0}$. with respect to $P$ at $\overline{\tau}(x)=0$
or $\underline{\tau}(x)=0$. Even when $\Gamma(P;x)$ depends smoothly on
expected outcomes/CATE bounds, lack of full differentiability of the scores
can arise through a lack of differentiability of the expected outcomes and
CATE bounds themselves. In fact, many popular identification assumptions,
including the Manski and Manski-Pepper bounds from Examples 3.1 and 3.3,
deliver bounds that are only directionally differentiable with respect to
identified parameters due to the presence of $\min/\max$ operators (see
Chernozhukov et al., 2013, and examples therein). Whether a consequence of the
optimality criterion or the identification assumptions, lack of full
differentiability of the scores is a unique and pervasive feature of the
treatment assignment problem under partial identification, one that has not
been explicitly acknowledged in the most recent contributions in this
area.101010The only exception is Christensen et al. (2022), who deal with
estimation of optimal treatment decisions in the absence of individualization.
A major contribution of this paper is to account for the role played by the
lack of full differentiability when we establish procedures for estimating
ambiguity-robust optimal policies in Section 4.

Table 1: Optimality criteria and associated scores Optimality criterion | $\Gamma(P;x)$
---|---
Maximin Welfare | $\underline{y}_{1}(x)-\underline{y}_{0}(x)$
Maximin Impact | $\underline{\tau}(x)$
Minimax Regret (oracle) | $\overline{\tau}(x)\cdot\mathbbm{1}\big{\\{}\overline{\tau}(x)\geq 0\big{\\}}+\underline{\tau}(x)\cdot\mathbbm{1}\big{\\{}\underline{\tau}(x)\leq 0\big{\\}}$
Minimax Regret (baseline) | $\overline{\tau}(x)\cdot\mathbbm{1}\big{\\{}\pi_{\texttt{B}}(x)=1\big{\\}}+\underline{\tau}(x)\cdot\big{\\{}\pi_{\texttt{B}}(x)=0\big{\\}}$
Hurwicz (welfare) |  $\big{(}\\{\delta_{1}\cdot\overline{y}_{1}(x)+(1-\delta_{1})\cdot\underline{y}_{1}(x)\\}-\\{\delta_{0}\cdot\overline{y}_{0}(x)+(1-\delta_{0})\cdot\underline{y}_{0}(x)\\}\big{)}$ $\delta_{1},\delta_{0}\in[0,1]$
Hurwicz (impact) | $\delta\cdot\overline{\tau}(x)+(1-\delta)\cdot\underline{\tau}(x),\,\delta\in[0,1]$

##  4 Estimation

In this section we present the statistical framework underlying the problem of
estimation of optimal treatment rules under partial identification. We will
discuss heuristics underlying several features of the estimation problem, and
then present our proposed estimation procedure.

We work in a learning setting where the estimand $\pi^{*}(P)$ is as in (12),
and we observe an i.i.d. sample $(W_{i})_{i=1,\dots,n}$ of size $n$ from the
unknown distribution $P$ of the observed random variables $W\in\mathcal{W}$,
$X\subset W$. To retain generality of the framework, we do not specify the
exact dependence of the functional $\Gamma(P;x)$ on $P$, which will depend on
the choice of optimality criterion for the resolution of ambiguity (maximin
welfare, minimax regret etc.) and identification assumptions determining the
identification sets $\mathcal{Y}(P),\mathcal{T}(P)$. However, we will assume
that the scores depend on $P$ only through a vector of nuisance functions
$g:\mathcal{V}\to\mathbb{R}^{J}$ specified by the moment equations

$\displaystyle\mathbb{E}[U-g(V)\mid V]=0,$ (13)

where $U$ and $V$ are random vectors with $U\subseteq W$ and $X\subseteq
V\subset W$. Furthermore, we will stipulate that the dependence of
$\Gamma(g;x)$ on the nuisance functions $g$ from the possibly infinite-
dimensional space $\mathcal{G}$ can be reduced as

$\displaystyle\Gamma(g;x)=\Gamma(\theta(x),x),$

where, for a fixed $x$, the parameter
$\theta(x)\in\Theta_{x}\subseteq\mathbb{R}^{M}$ is a finite-dimensional vector
of conditional moments of $U$ deduced from $g$. This latter restriction rules
out scores $\Gamma(g;X)$ that at a single point in the covariate space depend
on exhaustive evaluations of the nuisance functions $g$ over continuous
supports. This is the case, for example, in versions of the CATE bounds from
Examples 3.1-3.3 featuring instruments with continuous support $\mathcal{Z}$.
In those settings, the CATE bounds depend on objects such as
$\sup_{z\in\mathcal{Z}}\mathbb{E}[Y\mid Z=z,\,X=x]$, and are therefore not
covered by the results of this paper. Finally, we will assume that that
$\Gamma(\theta(x);x)$ can be expressed as

$\displaystyle\Gamma(\theta(x);x)=\varphi_{0}(\theta(x);x)+\sum_{\ell=1}^{L}a_{\ell}\cdot\varphi_{\ell}(\theta(x);x)\cdot\mathbbm{1}\left\\{\varphi_{\ell}(\theta(x);x)\geq
0\right\\},\quad a_{\ell}\in\\{-1,1\\},$ (14)

where the functions
$\varphi_{\ell}(\theta(x);x):\Theta_{x}\times\mathcal{X}\to\mathbb{R}$ are
fully differentiable with respect to $\theta(x)$ for all $x\in\mathcal{X}$.
While seemingly ad-hoc, this restriction is sufficiently general to
accommodate a wide range of popular partial identification assumptions for the
CATE as well as optimality criteria for the resolution of ambiguity. In
particular, formulation (14) accommodates linear combinations of $\min$/$\max$
operators, which typically feature in many identification bounds for the CATE
with discrete instruments. In fact, our framework can be shown to be
applicable to any combination of the optimality criteria discussed in Section
3 and the identification schemes contained in the recent survey paper by
Swanson et al. (2018).111111Albeit not directly accommodated by formulation
(14), our framework and theoretical results also apply to scores that feature
a finite number of nested linear combinations of $\min/\max$ operators. We
discuss this extension in Appendix A.

######  Example 4.1 (continues=balkePearl).

Under the identification assumptions of the Balke-Pearl bounds and resolution
of ambiguity via Minimax Regret, we have

$\displaystyle g$ $\displaystyle=(h,m,p),$ $\displaystyle\theta(x)$
$\displaystyle=(h(1,x),h(0,x),m(1,0,x),m(0,1,x),p(1,x),p(0,x)),$

and

$\displaystyle\Gamma(g;x)=\varphi_{1}(\theta(x);x)\cdot\mathbbm{1}\left\\{\varphi_{1}(\theta(x);x)\geq
0\right\\}-\varphi_{2}(\theta(x);x)\cdot\mathbbm{1}\left\\{\varphi_{2}(\theta(x);x)\geq
0\right\\},$

where $\varphi_{1}(\theta(x);x)=\overline{\tau}(\theta(x);x)$,
$\varphi_{2}(\theta(x);x)=-\underline{\tau}(\theta(x);x)$ are differentiable
with respect to $\theta(x)$.

In this framework, a natural approach for estimation is via the so-called
“empirical risk minimisation” (ERM) principle (Vapnik, 1998), in which the
estimate for the optimal policy is obtained as the maximiser of a sample
analogue of the population objective $Q$:

$\displaystyle\widehat{\pi}_{n}=\operatorname*{argmax}\Big{\\{}\widehat{Q}_{n}(\pi)\,:\,\pi\in\Pi\Big{\\}},\qquad\widehat{Q}_{n}(\pi)=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\widehat{\Gamma}_{i}$
(15)

where $\widehat{\Gamma}_{i}$ is some suitable estimate for $\Gamma(g;X_{i})$.
The ERM approach is a cornerstone of statistical learning theory and is at the
foundation of many traditional and modern estimation methods in statistics,
econometrics and machine learning. The ERM principle has also guided much of
the recent literature on individualized treatment rules, where different
variations have been applied under the names of “outcome-weighted learning”
(Zhao et al., 2012) and “empirical welfare maximization” (Kitagawa and
Tetenov, 2018). A major challenge in the implementation of (15) comes from the
presence of the nuisance functions $g$, which are typically unknown and thus
need to be estimated. Assuming that we have access to appropriate
algorithms/nonparametric procedures for estimation the nuisance functions, one
simple approach would be to use the sample $(W_{i})_{i=1,\dots,n}$ to obtain
the estimates $\widehat{g}$ and then form plug-in estimates for the score as
$\widehat{\Gamma}_{i}=\Gamma(\widehat{g};X_{i})$. While seemingly natural,
this “naive plug-in” approach has undesirable properties. In particular,
policies estimated via the naive plug-in approach can typically only be shown
to converge at sub-optimal rates to their population counterparts, unless very
restrictive assumptions are imposed on first-stage estimators for the nuisance
functions $g$ (see, e.g., Foster and Syrgkanis, 2019).

One crucial reason underlying the undesirable statistical properties of the
naive plug-in approach is that the resulting objective function estimate
$\widehat{Q}_{n}$ is overtly sensitive to error in estimating the nuisance
functions $g$. In order to gain intuition, it is useful to consider the
following expansion of the population objective function
$Q(g;\pi)=\mathbb{E}_{P_{X}}\left[(2\pi(X)-1)\cdot\Gamma(g;X)\right]$,

$\displaystyle Q(\widetilde{g};\pi)-Q(g;\pi)=\frac{\partial
Q(g+t(\widetilde{g}-g);\pi)}{\partial
t}\biggr{\rvert}_{t=0}+\Delta(\widetilde{g},g;\pi)+O\left(\|\widetilde{g}-g\|_{L_{2}(P)}^{2}\right)$
(16)

where

$\displaystyle\Delta(\widetilde{g},g;\pi)=$
$\displaystyle\mathbb{E}_{P_{X}}\left[(2\pi(X)-1)\cdot\left(\sum_{\ell=1}^{L}a_{\ell}\cdot\varphi_{\ell}(g;X)\cdot\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widetilde{g};X)\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X)\geq
0\right\\}\right)\right)\right].$

Von Mises expansions like the above are at the heart of the theory of
orthogonal machine learning (Chernozhukov et al., 2022). In our setting, it
allows to describe the impact of a small deviation from $g$ in the direction
$\widetilde{g}-g$ as consisting of three terms. The first term is the so-
called “pathwise derivative” of $Q(g;\pi)$ and typically scales with
$\|\widetilde{g}-g\|_{L_{1}(P)}$. The second term
$\Delta(\widetilde{g},g;\pi)$ is due the presence of the type of non-
differentiabilities arising under partial identification, and is unique to the
framework of this paper. This term accounts for the bias that arises from
misclassifying whether the component functions $\varphi_{\ell}$ are above or
below 0, as we move away from $g$ in the direction $\widetilde{g}-g$. The
third term is a second-order remainder scaling with the mean-square distance
between $\widetilde{g}$ and $g$. A central feature of our proposed estimation
procedure is the construction of a new objective function, called Neyman-
orthogonal, with reduced sensitivity to local perturbations away from $g$. For
this purpose, we will assume that there exists functionals
$\alpha_{\ell}(\\{g,f\\};V)$ such that for every $\widetilde{g}\in\mathcal{G}$

$\displaystyle\varphi_{\ell}(\widetilde{g};x)=\mathbb{E}[\langle\alpha_{\ell}(\\{\widetilde{g},f\\};V),\widetilde{g}(V)\rangle\mid
X=x],\qquad\ell=1,\dots,L,$

where $f\in\mathcal{F}$ is a vector of additional nuisance functions defined
analogously to $g$.121212We will also assume that for the $j$-th entry of the
Riesz-representer we have
$\alpha^{(j)}_{\ell}(\\{(\widetilde{g}_{-j},\widetilde{g}_{j}),\widetilde{f}\\},x)=\alpha^{(j)}_{\ell}(\\{\widetilde{g}_{-j},\widetilde{f}\\},x)$,
where $\widetilde{g}_{-j}$ denotes the exclusion of the $j$-th entry
$\widetilde{g}_{j}$ from the vector of nuisance functions $\widetilde{g}$.
This restriction is sufficiently general to accommodate component functions
$\varphi_{\ell}(\theta(x);x)$ that feature linear combinations of products of
the parameters $\theta(x)$, thus encompassing all the discussed identification
schemes, including Examples 3.1-3.3. We then construct Neyman-orthogonal
formulations for the component functions as

$\displaystyle\varphi_{\ell}^{\texttt{NO}}(\\{g,f\\};w)=\varphi_{\ell}(\theta;x)+\phi_{\ell}(\\{g,f\\};w),\quad\phi_{\ell}(\\{g,f\\};w)=\langle\alpha_{\ell}(\\{g,f\\},v),u-g(v)\rangle.$

The functionals $\alpha_{\ell}$ are the Riesz-representers of
$\varphi_{\ell}$, while the functionals $\phi_{\ell}$ are their so-called
influence function adjustments. We refer the reader to Ichimura and Newey
(2022) for their properties and general methods for their calculation, while
we provide below their specific form for the Balke-Pearl CATE bounds of
Example 3.2.131313See also Kennedy (2022) for a user-friendly discussion of
methods for computation influence function adjustments.

######  Example 4.2 (continues=balkePearl).

Following Ichimura and Newey (2022), we compute the influence function
adjustment $\phi_{U}(\\{g,f\\},W_{i})$ for the CATE upper bound by taking the
Gateaux derivative of $\overline{\tau}(g;X)$, which yields

$\displaystyle\phi_{U}(\\{g,f\\};W_{i})=$
$\displaystyle\underbrace{\left[\frac{Z_{i}}{z(X_{i})}-\frac{1-Z_{i}}{1-z(X_{i})}\right]}_{\alpha_{U}^{(1)}(\\{g,f\\},V_{i})}\cdot\big{(}Y_{i}-h(Z_{i},X_{i})\big{)}$
$\displaystyle+\underbrace{\left[\frac{D_{i}(1-Z_{i})}{1-z(X_{i})}+\frac{(1-D_{i})Z_{i}}{z(X_{i})}\right]}_{\alpha_{U}^{(2)}(\\{g,f\\},V_{i})}\cdot\big{(}Y_{i}-m(D_{i},Z_{i},X_{i})\big{)}$
$\displaystyle+\underbrace{\left[(m(1,0,X_{i})-Y_{L})\cdot\frac{1-Z_{i}}{1-z(X_{i})}-(Y_{U}-m(0,1,X_{i}))\cdot\frac{Z_{i}}{z(X_{i})}\right]}_{\alpha_{U}^{(3)}(\\{g,f\\},V_{i})}\cdot\big{(}D_{i}-p(Z_{i},X_{i})\big{)},$

where the associated Riesz-representer is
$\alpha_{U}(\\{g,f\\},V_{i})=(\alpha_{U}^{(1)},\alpha_{U}^{(2)},\alpha_{U}^{(3)})^{\prime}$
with $f=z(x)$ and $V_{i}=(D_{i},Z_{i},X_{i})^{\prime}$. The influence function
and Riesz-representer for the CATE lower bound are obtained by interchanging
$Y_{U}$ and $Y_{L}$ in the expressions above.

Finally we construct Neyman-orthogonal formulations for the scores as

$\displaystyle\Gamma^{\texttt{NO}}(\\{g,f\\};w)=\varphi_{0}^{\texttt{NO}}(\\{g,f\\};w)+\sum_{\ell=1}^{L}a_{\ell}\cdot\varphi^{\texttt{NO}}_{\ell}(\\{g,f\\};w)\cdot\mathbbm{1}\left\\{\varphi_{\ell}(g;x)\geq
0\right\\},$

which are then used to form the Neyman-orthogonal objective function

$\displaystyle
Q^{\texttt{NO}}(\\{\cdot,\cdot\\};\pi)=\mathbb{E}_{P_{W}}\Big{[}\left(2\pi(X)-1\right)\cdot\Gamma^{\texttt{NO}}(\\{\cdot,\cdot\\};W)\Big{]}.$

Our construction of Neyman-orthogonal scores features the addition of the
influence function adjustments $\phi_{\ell}$ to the component functions
$\varphi_{\ell}$ outside the indicators, but crucially not inside.
Heuristically, the influence function adjustments serve the purpose of
reducing the bias induced by the evaluation of the component functions
$\varphi_{\ell}(\cdot;x)$ away from $g$. Since the indicators vary
discontinuously with $g$ it is not possible to linearly approximate the
dependence of the indicators on the nuisance functions at the point of
discontinuity $\varphi_{\ell}(g;x)=0$. As a result, it is not possible to
reduce the bias induced by the presence of the indicators (represented by the
term $\Delta(\widetilde{g},g,\pi)$ in (16)) by means of influence function
adjustments, whose de-biasing properties implicitly rely on the validity of
such linear approximation.141414On the contrary, naively adding the influence
function adjustments inside the indicators would lead to a bias increase,
rather than a reduction. Notice that $Q^{\texttt{NO}}(\\{g,f\\};\pi)=Q(g;\pi)$
by the mean-zero property of the influence function adjustments, so that
orthogonalization of the objective does not change the notion of optimal
policy $\pi^{*}(P)$. Nonetheless, for the orthogonalized objective we have
that

$\displaystyle
Q^{\texttt{NO}}(\\{\widetilde{g},\widetilde{f}\\};\pi)-Q^{\texttt{NO}}(\\{g,f\\};\pi)=\Delta(\widetilde{g},g;\pi)+O\left(\|\widetilde{g}-g\|_{L_{2}(P)}^{2}+\|\widetilde{f}-f\|_{L_{2}(P)}^{2}\right).$
(17)

Comparing the above with (16), we see that the von Mises expansion for the
orthogonalized objective does not feature the pathwise derivative term,
implying that $Q^{\texttt{NO}}(\,\cdot\,;\pi)$ is less sensitive to deviations
away from $g$ compared to the original objective $Q(\,\cdot\,;\pi)$. As shown
in Section 5, this property will generally translate in improved statistical
guarantees for the estimated policy when the nuisance functions have to be
learned from the data. It should however be noticed that the term
$\Delta(\widetilde{g},g;\pi)$ still appears in the relevant expansion after
orthogonalization. The contribution of this term is quantified in Section 5,
where it is shown to be of first-order importance for the statistical
properties of the estimation procedures.

The second key component of our approach is the use of sample-splitting, which
is a commonly employed method in semiparametric inference (Chernozhukov et
al., 2022) and statistical learning (Foster and Syrgkanis, 2019). The main
purpose of sample-splitting is to reduce the risk of overfitting that
generally arises from using the same data to estimate the nuisance functions
as well as the optimal policy, as in the naive plug-in approach. Similarly to
Athey and Wager (2021), we employ a particular form of sample-splitting known
as K-fold cross-fitting (described below). This procedure ensures that in the
estimate for $\Gamma^{\texttt{NO}}(\\{g,f\\};W_{i})$, the estimates for the
nuisance functions $\\{g,f\\}$ are independent from the data-point $W_{i}$ for
that same unit. This independence property is crucial for the theoretical
guarantees of our proposed method.

Our proposed estimation procedure is therefore as follows. We first randomly
split the data into $K$ evenly-sized folds and for each fold $k=1,\dots,K$ we
obtain estimates $\\{\widehat{g}^{(-k)},\widehat{f}^{(-k)}\\}$ using data from
the remaining $K-1$ folds. These estimates are then used to form cross-fitted
Neyman-orthogonal estimates for the scores

$\displaystyle\widehat{\Gamma}_{i}=\widehat{\Gamma}^{\texttt{NO}}\left(\\{\hat{g}^{-k(i)},\hat{f}^{-k(i)}\\};W_{i}\right),\qquad
i=1,\dots,n,$ (18)

where $k(i)\in\\{1,\dots,K\\}$ denotes the fold containing the $i$-th
observation. Finally, the estimated optimal policy rule $\widehat{\pi}_{n}$ is
obtained via the optimization problem (15).

##  5 Statistical guarantees for the estimated policy

Let $\widehat{\pi}_{n}$ be the estimated treatment policy defined in (15),
with estimated scores as in (18). Following Manski (2004), we assess the
performance of the estimated policy in terms of (statistical) regret with
respect to population optimal policy. Let the population ambiguity-robust
optimal policy be
$\pi^{*}_{n}(P)\in\operatorname*{argmax}_{\pi\in\Pi_{n}}Q(P;\pi)$, where we
have included the $n$-subscript to the policy class $\Pi_{n}$ to allow this to
depend on the sample size for generality. The statistical regret of an
estimated policy $\widehat{\pi}_{n}$ is defined as

$\displaystyle
R_{n}(P;\widehat{\pi}_{n})=\mathbb{E}_{P_{n}}\big{[}Q(P;{\pi}^{*}_{n})-Q(P;\widehat{\pi}_{n})\big{]}\geq
0,$ (19)

where $\mathbb{E}_{P_{n}}$ is the expectation with respect to the i.i.d.
sample of observable random variables $(W_{i})_{i=1,\dots,n}$ used to estimate
$\widehat{\pi}_{n}$. The next few subsections build up to a final result
providing asymptotic convergence guarantees for $\widehat{\pi}_{n}$ to
$\pi^{*}_{n}$ in terms of statistical regret.

###  5.1 Assumptions

We make the following assumptions.

######  Assumption 5.1 (VC-class).

There exists constants $0\leq\nu<1/2$ and $N\geq 1$ such that
$\emph{VC}(\Pi_{n})\lesssim n^{\nu}$ for all $n\geq N$.

Assumption 5.1 restricts the policy class to have finite VC-dimension, which
is a standard requirement for controlling the complexity of a policy class in
the classification literature. The VC-dimension of the policy-class $\Pi$ is
defined as the largest interger $m$ such that there exist points
$x_{1},\dots,x_{m}$ that are shattered by $\Pi$, i.e. where the policy values
$\pi(x_{1}),\dots,\pi(x_{m})$ can take on all $2^{m}$ possible combinations in
$\\{0,1\\}^{m}$ (for more on the VC-dimension, see Wainwright, 2019). Several
practically relevant classes of treatment rules satisfy this requirement,
including the linear-index and quadrant rules used in the empirical
application of Section 6. Our assumption allows the VC-dimension of the policy
class to grow moderately with the sample size, thus allowing the treatment
rule to depend on high-dimensional covariates.

######  Assumption 5.2 (Regularity conditions for data-generating process).

  * (i)

There exist constants $\mathcal{C}_{1,\varphi},\mathcal{C}_{1,\alpha}$ such
that for all
$\\{\widetilde{g},\widetilde{f}\\}\in\mathcal{G}\times\mathcal{F}$

$\displaystyle\norm{\varphi_{\ell}(\widetilde{g};X)-\varphi_{\ell}(g;X)}_{L_{\infty}(P_{X})}$
$\displaystyle\leq\mathcal{C}_{1,\varphi}\cdot\norm{\widetilde{g}-g}_{L_{\infty}(P_{V})},$
$\displaystyle\norm{\alpha_{\ell}(\\{\widetilde{g},\widetilde{f}\\};V)-\alpha_{\ell}(\\{g,f\\};V)}_{L_{\infty}(P_{V})}$
$\displaystyle\leq\mathcal{C}_{1,\alpha}\cdot\left(\norm{\widetilde{g}-g}_{L_{\infty}(P_{V})}+\norm{\widetilde{f}-f}_{L_{\infty}(P_{V})}\right),$

for $\ell=0,\dots,L$.

  * (ii)

There exist constants $\mathcal{C}_{2,\varphi},\mathcal{C}_{2,\alpha}$ such
that for all
$\\{\widetilde{g},\widetilde{f}\\}\in\mathcal{G}\times\mathcal{F}$

$\displaystyle\norm{\varphi_{\ell}(\widetilde{g};X)-\varphi_{\ell}(g;X)}_{L_{2}(P_{V})}$
$\displaystyle\leq\mathcal{C}_{2,\varphi}\cdot\norm{\widetilde{g}-g}_{L_{2}(P_{V})},$
$\displaystyle\norm{\alpha_{\ell}(\\{\widetilde{g},\widetilde{f}\\};V)-\alpha_{\ell}(\\{g,f\\};V)}_{L_{2}(P_{V})}$
$\displaystyle\leq\mathcal{C}_{2,\alpha}\cdot\left(\norm{\widetilde{g}-g}_{L_{2}(P_{V})}+\norm{\widetilde{f}-f}_{L_{2}(P_{V})}\right),$

for $\ell=0,\dots,L$.

  * (iii)

There exist constants $\mathcal{C}_{3,\varphi},\mathcal{C}_{3,\alpha}$ such
that for all
$\\{\widetilde{g},\widetilde{f}\\}\in\mathcal{G}\times\mathcal{F}$

$\displaystyle\norm{\varphi_{\ell}(\widetilde{g};X)}_{L_{\infty}(P_{X})}$
$\displaystyle\leq\mathcal{C}_{3,\varphi},$
$\displaystyle\norm{\alpha_{\ell}(\\{\widetilde{f},\widetilde{g}\\};V)}_{L_{\infty}(P_{V})}$
$\displaystyle\leq\mathcal{C}_{3,\alpha},$

for $\ell=0,\dots,L$.

  * (iv)

The irreducible noise $\varepsilon_{i}\coloneqq U_{i}-g(V_{i})$ is a sub-
Gaussian vector conditional on $V_{i}$, with conditional variance ${\rm
Var}(\varepsilon_{i}\mid V_{i})=\Sigma(V_{i})$ satisfying $\norm{\lambda_{\rm
max}(\Sigma(V))}_{L_{\infty}(P_{V})}\leq\overline{\lambda}<\infty$.

Assumptions 5(i) and 5(ii) impose Lipschitz continuity of the component
functions and Riesz-representers with respect to the nuisance component in the
$L_{\infty}$ and $L_{2}$-norm, respectively. These requirements are typically
met under mild conditions within the framework of this paper. For the Balke-
Pearl bounds of Example 3.2, these assumptions hold under the overlap
condition whenever $\mathcal{G}$ and $\mathcal{F}$ are subsets of the space of
bounded functions151515That is, there exists a constant $B>0$ such that
$\|\\{\widetilde{g},\widetilde{f}\\}\|_{L_{\infty}(P_{V})}\leq
B,\,\forall\\{\widetilde{g},\widetilde{f}\\}\in\mathcal{G}\times\mathcal{F}$,
which is automatically satisfied since $U_{i}=(Y_{i},D_{i},Z_{i})^{\prime}$ is
a vector of random variables with bounded support. Assumption 5.2(iii) is a
uniform bound on the component functions and Riesz-representers, the former
implying uniform boundedness of the scores $\Gamma(g;\cdot)$. Assumption
5.2(iv) is a standard requirement in statistical learning theory restricting
the tail behaviour of the statistical noise $\varepsilon_{i}$. It is
automatically satisfied when $U_{i}$ has bounded support, as in the Balke-
Pearl bounds, but also allows for outcomes with unbounded support whose
conditional distributions have sufficiently thin tails. Together with
Assumption 5.2(iii), this assumption implies sub-gaussianity of
$\Gamma^{\texttt{NO}}(\\{g,f\\},W_{i})$.

The next two assumptions impose requirements on the estimators for the
nuisance components.

######  Assumption 5.3 (Regularity conditions for fist-step estimators).

  * (i)

The estimators of the nuisance functions
$\\{\widehat{g}_{n},\widehat{f}_{n}\\}$ belong to the function classes
$\mathcal{G}\times\mathcal{F}$ with probability 1.

  * (ii)

There exists a constant $\mathcal{C}_{4}>0$ such that

$\displaystyle\norm{\widehat{g}_{n}-g}_{L_{\infty}(P_{V})}\leq\mathcal{C}_{4},$
$\displaystyle\norm{\widehat{f}_{n}-f}_{L_{\infty}(P_{V})}\leq\mathcal{C}_{4},$

with probability approaching 1 as $n\to\infty$.

Part (i) of Assumption 5.3 is needed to ensure the validity of the Lipschitz
continuity requirements of Assumption 5.2 for the component functions and
Riesz-representers when evaluated at the first-stage estimates. In the context
of the Balke-Pearl bounds, it is satisfied when
${\widehat{g}_{n},\widehat{f}_{n}}$ are uniformly bounded and the estimated
propensity score $\widehat{z}(X_{i})$ is uniformly bounded away from 0 and 1,
with probability one. The first condition is satisfied by virtually any
estimation procedure when the outcomes $U_{i}$ have bounded support. The
second requirement can be guaranteed under appropriate trimming of the
estimated propensities. Part (ii) requires that estimation errors for the
nuisance components are uniformly bounded, which is satisfied under Assumption
5.3(i) when $\mathcal{G}\times\mathcal{F}$ is a subset of the space of bounded
functions. When $U_{i}$ has unbounded support and
$\mathcal{G}\times\mathcal{F}$ includes unbounded functions, a more primitive
condition for (ii) would be uniform consistency of the first stage estimates,
that is
$\|\\{\widehat{g}_{n},\widehat{f}_{n}\\}-\\{g,f\\}\|_{L_{\infty}(P_{V})}\rightarrow_{p}0$.161616However,
it should be noted that the uniform consistency requirement is not completely
innocuous when $\\{\widehat{g}_{n},\widehat{f}_{n}\\}$ are machine learning
estimators (Farrell et al., 2021).

######  Assumption 5.4 ($L_{2}$ convergence rates).

The estimators of the nuisance functions satisfy

$\displaystyle\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}_{n}-g}^{2}_{L_{2}(P_{V})}\right]\leq\frac{r_{n}}{n^{1/2}},$
$\displaystyle\mathbb{E}_{P_{n}}\left[\norm{\widehat{f}_{n}-f}^{2}_{L_{2}(P_{V})}\right]\leq\frac{r_{n}}{n^{1/2}},$

for some sequence $r_{n}=o(1)$.

The above requirement on the $L_{2}$-convergence rates for the learners of the
nuisance functions is a standard assumption in the semiparametric inference
literature (see, e.g., Farrell, 2015, and Chernozhukov et al., 2022). It can
be shown to provably hold for traditional nonparametric estimation methods
such as sieve methods (Chen, 2007) as well as modern black-box machine
learning algorithms including Lasso (see, e.g., Farrell, 2015), deep neural
networks (Farrell et al., 2021), boosting and others, for which stronger
guarantees such as Donsker-type properties are typically not available. The
ability to invoke a mild $L_{2}$-convergence requirement is a virtue of the
combined use of Neyman-orthogonalization and sample-splitting, a key insight
brought forward by Chernozhukov et al. (2022) for semiparametric GMM
inference, and subsequently leveraged by Athey and Wager (2021) and Foster and
Syrgkanis (2019) in the context of statistical learning problems.171717Unlike
Athey and Wager (2021), our assumptions do not allow to trade-off accuracy in
the estimation across the different nuisance functions. This is because our
framework allows for $\varphi_{\ell}(g;x)$ to be a potentially non-linear
functional of the nuisance functions $g$, as is the case in Examples 3.1-3.3,
thus precluding such double-robustness property.

Finally, we present an assumption that concerns the distribution of the
component functions $\varphi_{\ell}$ at the population level.

######  Assumption 5.5 (Margin).

There exist constants $\mathcal{C}_{m}>0$ and $\gamma\geq 0$ such that

$\displaystyle\mathbb{P}_{X}\Big{(}0<|\varphi_{\ell}(g;X)|\leq t\
\Big{)}\leq\mathcal{C}_{m}t^{\gamma},\quad\forall\,t>0.$

for $\ell=1,\dots,L$.

The above assumption restricts the extent to which the distribution of the
component functions $\varphi_{\ell}(g;X)$ can concentrate around the point of
non-differentiability 0 and it is a form of “margin assumption”, first
introduced by Mammen and Tsybakov (1999). Such an assumption has been widely
used in statistics to obtain fast learning rates in classification problems
(see, e.g., Arlot and Bartlett, 2011). Notice that the above formulation for
the margin assumption restricts the concentration of probability for the
distribution of the components functions in a neighbourhood of 0, but still
allows for arbitrary probability mass at 0.

######  Example 5.1 ($\gamma=1$).

Suppose $X$ contains an absolutely continuous covariate $\mathrm{\tilde{x}}$
and $\varphi_{\ell}(g;X)\cdot\mathbbm{1}\\{\mathrm{\tilde{x}}\neq 0\\}$ is
absolutely continuous with density bounded above by $\overline{f}$ for
$\ell=1,\dots,L$. Then Assumption (5.5) holds with $\gamma=1$ and
$\mathcal{C}_{m}=2\overline{f}$.

######  Example 5.2 ($\gamma=\infty$).

Suppose there exists a $t_{0}>0$ such that
$\mathbb{P}_{X}\left(0<|\varphi_{\ell}(X)|<t_{0}\right)=0$ for
$\ell=1,\dots,L$. Then Assumption (5.5) holds with $\gamma=\infty$ and some
$\mathcal{C}_{m}>0$.

In the context of the Balke-Pearl bounds from Example 3.2 with resolution of
ambiguity via Minimax Regret, Assumption 5.5 restricts the extent to which the
CATE bounds $\overline{\tau},\underline{\tau}$ can concentrate around 0 in the
data-generating process. Under $\gamma=\infty$ the support of each CATE bound
is required to be fully separated from 0, while $\gamma=1$ requires that each
CATE bound has bounded density in a neighborhood of 0.

In the next section we present our theoretical results based on Assumptions
5.1-5.5.

###  5.2 Regret convergence rates

In this section, we provide asymptotic rates of convergence for the regret of
the estimated policy $R_{n}(P;\widehat{\pi}_{n})$ as defined in (19). In line
with the existing literature, we study uniform regret bounds that are valid
for all distributions $P\in\mathcal{P}$ satisfying Assumptions 5.1-5.5. All
results in this section are thus intended to hold uniformly in the above
sense, and we will drop the dependence on $P$ for notational convenience.

We begin by noticing that controlling the convergence of $\widehat{\pi}_{n}$
to the best-in-class policy $\pi^{*}_{n}$ intuitively requires accounting for:
1) the estimation error in the component functions $\varphi_{\ell}$ and
influence function adjustments $\phi_{\ell}$ due to estimation of the nuisance
components $\\{g,f\\}$, 2) the difference between the population Neyman-
orthogonal score and true score181818That is, we need to account for the fact
that we have added the influence function adjustments to the component
functions., and 3) the fact that we estimate our policy using a sample from
the distribution of the covariates $X_{i}$ rather than their true
distribution. We define the following quantities:

$\displaystyle\widehat{Q}_{n}^{\texttt{NO}}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\big{(}2{\pi}(X_{i})-1\big{)}\cdot\Gamma^{\texttt{NO}}(\\{\widehat{g},\widehat{f}\\};W_{i}),$
$\displaystyle{Q}^{\texttt{NO}}_{n}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\big{(}2{\pi}(X_{i})-1\big{)}\cdot\Gamma^{\texttt{NO}}(\\{g,f\\},W_{i}),$

and formalize this intuition in the next proposition.

######  Proposition 3.

The regret of $\widehat{\pi}_{n}$ obeys the following bound:

$\displaystyle R_{n}(\widehat{\pi}_{n})\leq$ $\displaystyle
2\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}\left|\widehat{Q}^{\emph{{NO}}}_{n}(\pi)-Q^{\emph{{NO}}}_{n}(\pi)\right|\right]+\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}\left|Q^{\emph{{NO}}}_{n}(\pi)-Q(\pi)\right|\right].$
(20)

The second term in the above bound accounts for points 2) and 3).
$Q_{n}(\pi)-Q(\pi)$ is a centred (mean-zero) empirical process and therefore
its uniform expectation can be shown to be
$O\left(\sqrt{\text{VC}(\Pi_{n})/n}\right)$ using symmetrization and chaining
arguments (see, e.g., Wainwright, 2019, Ch. 5.3). Controlling the first term,
which accounts for point 1), is particularly challenging and requires tailored
arguments that deal with the particular form of the population scores in (14),
in particular their lack of full differentiability.

######  Lemma 1.

Suppose that Assumptions 5.1-5.5 hold and define $\kappa_{n}=\lfloor
n(1-1/K)\rfloor$. Then we have

$\displaystyle\mathbb{E}_{P_{n}}\left[\sup_{\pi\in\Pi_{n}}\left|\widehat{Q}^{\emph{{NO}}}_{n}(\pi)-Q^{\emph{{NO}}}_{n}(\pi)\right|\right]=O\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}+\sqrt{\frac{\emph{VC}(\Pi_{n})}{n}}+\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}\right)^{\frac{\gamma+1}{\gamma+2}}\right).$

Lemma 1 is the central result of this paper. It provides an asymptotic rate of
convergence to zero of the empirical process
$\left|\widehat{Q}^{\texttt{NO}}_{n}(\pi)-Q^{\texttt{NO}}_{n}(\pi)\right|$
uniformly over the policy class $\Pi_{n}$, which depends on the VC-dimension
of the class and the degree of concentration of the component functions
$\varphi_{\ell}(g;X)$ around 0, as indexed by $\gamma$. In order to convey
intuition on this result we provide a brief outline of the proof, which is
based on the decomposition

$\displaystyle\widehat{Q}^{\texttt{NO}}_{n}(\pi)-Q^{\texttt{NO}}_{n}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\left[\widehat{\Gamma}^{\texttt{NO}}\left(\\{\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\\},W_{i}\right)-\Gamma^{\texttt{NO}}\left(\\{g,f\\},W_{i}\right)\right]$
(21)
$\displaystyle=A_{0}(\pi)+\sum_{\ell=1}^{L}a_{\ell}\cdot\left[A_{1,\ell}(\pi)+A_{2,\ell}(\pi)+A_{3,\ell}(\pi)\right],$

where

$\displaystyle
A_{0}(\pi)=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\left[\varphi_{0}^{\texttt{NO}}(\\{\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\\},W_{i})-\varphi_{0}^{\texttt{NO}}(\\{g,f\\},W_{i})\right],$
$\displaystyle
A_{1,\ell}(\pi)=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\Big{[}\varphi_{\ell}^{\texttt{NO}}(\\{\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\\},W_{i})-\varphi_{\ell}^{\texttt{NO}}(\\{g,f\\},W_{i})\Big{]}\cdot\mathbbm{1}\left\\{\varphi_{\ell}\left(\widehat{g}^{-k(i)};X_{i}\right)>0\right\\},$
$\displaystyle
A_{2,\ell}(\pi)=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\phi_{\ell}(\\{g,f\\};W_{i})\cdot\Big{[}\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq 0\right\\}\Big{]},$
$\displaystyle
A_{3,\ell}(\pi)=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\varphi_{\ell}(g;X_{i})\cdot\Big{[}\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq 0\right\\}\Big{]}.$

Terms $A_{0}(\pi)$ and $A_{1,\ell}(\pi)$ can be controlled using similar
arguments to Athey and Wager (2021) and are responsible for the
$O\left(r_{\kappa_{n}}/\sqrt{n}\right)$ term in the bound of Lemma 1. The de-
biasing properties of Neyman-orthogonalization combined with sample-splitting
play a crucial role in this context, as they ensure that the error in
estimating $\varphi_{\ell}(g;x)$ only has a second-order contribution. As a
result, term $A_{1,\ell}(\pi)$ scales with the mean-squared estimation error
in the nuisance functions and, under Assumption 5.4, its expectation decays
faster than $1/\sqrt{n}$ uniformly over $\Pi_{n}$.191919Notice that, by virtue
of sample-splitting, the presence of the indicator
$\mathbbm{1}\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq 0\\}$ is
immaterial when controlling the expectation of $A_{1,\ell}(\pi)$ uniformly
over $\Pi_{n}$. If plug-in (non-orthogonalized) estimates for $\varphi_{\ell}$
are instead used to form the score estimates $\widehat{\Gamma}_{i}$, the
estimation error in the nuisance functions has a first-order impact on term
$A_{1,\ell}(\pi)$. As a result, its uniform expectation would scale with the
$L_{1}$ estimation error which, under Assumption 5.4, implies the much slower
convergence $\mathbb{E}[\sup_{\pi\in\Pi_{n}}|A_{1,\ell}(\pi)|]=o(n^{1/4})$.

For term $A_{2,\ell}(\pi)$, the mean-zero property of the influence function
adjustments together with sample-splitting ensures that this term is a centred
empirical process and thus it is responsible for a
$O\left(\sqrt{\text{VC}(\Pi_{n})/n}\right)$ contribution again by
symmetrization and chaining arguments.

Finally, for term $A_{3,\ell}(\pi)$ we show that

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}A_{3,\ell}(\pi)\right]\leq\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\cdot\left(\mathbbm{1}\left\\{\varphi_{\ell}^{-k(i)}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\right|\Bigg{]},$

where the RHS can be recognized to be the classification loss of an estimator
for the sign of $\varphi_{\ell}(g;x)$ based on thresholding
$\varphi_{\ell}(\widehat{g}^{-k(i)};x)$. Rates of convergence in binary
classification problems intuitively depend on the degree of separation of the
true regression function from 0, as indexed by $\gamma$. We thus leverage
results from the literature on classification (Audibert and Tsybakov, 2007) to
quantify the contribution of $A_{3,\ell}$ in the bound of Lemma 1 in terms of
$\gamma$.

We are now ready to combine the rates of convergence for the three terms in
Proposition 3 to obtain a final regret bound for our proposed estimation
procedure.

######  Theorem 2.

Suppose Assumptions 5.2-5.5 hold. Then the regret obeys

$\displaystyle
R_{n}(\widehat{\pi}_{n})=O\left(\sqrt{\frac{\emph{VC}(\Pi_{n})}{n}}\vee\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}\right)^{\frac{\gamma+1}{\gamma+2}}\right).$

We see that regret convergence for our policy learning procedure happens at a
rate corresponding to whichever is the leading term in the asymptotic
expansion of Lemma 1, which depends on $\nu$ and $\gamma$. When the policy
class $\Pi_{n}$ has fixed VC-dimension ($\nu=0$), regret convergence happens
at rates ranging from $o(n^{1/4})$ in the least favourable case ($\gamma=0$)
to $O(\sqrt{\text{VC}(\Pi)/n})$ in the most favourable case ($\gamma=\infty$).
The latter case is in line with existing results for policy learning with
point-identified CATE, in which full-differentiability of the scores leads to
$\sqrt{\text{VC}(\Pi_{n})/n}$ learning rates (see Kitagawa and Tetenov, 2018;
Athey and Wager, 2021; Foster and Syrgkanis, 2019). For the intermediate case
$\gamma=1$ of Example 5.1 our procedure guarantees regret convergence at rate
$o(n^{1/3})$.

It is useful to compare the performance guarantees in this paper with Pu and
Zhang (2021), whose procedure involves the use of non-orthogonalized estimates
for the scores with sample-splitting. They show that the regret of a policy
estimated via the maximization $(\ref{estimationPolicy})$ based on cross-
fitted non-orthogonalized scores is upper bounded by the $L_{1}$-norm of the
estimation error in the nuisance functions. Under Assumption 5.4, this implies
$o(n^{1/4})$ convergence for the regret, which is strictly slower than our
rates for all values of $\gamma>0$. The faster speed of convergence guaranteed
by our procedure is not just due to a refined proof strategy but crucially
depends on the use of Neyman-orthogonalization, as elucidated by our
discussion of Lemma 1.

######  Remark 4.

The procedure of Pu and Zhang (2021) also differs from ours in its final
implementation, which in their case is carried out via support vector machines
(SVM) with $\Pi_{n}$ assumed to be a reproducing kernel Hilbert space. While
the use of surrogate losses (such as the hinge loss in SVM) to convexify
problem (15) can bring considerable computational benefits in terms of speed
and scalability, it comes at the cost of even slower convergence guarantees
than the $o(n^{-1/4})$ discussed above. We stress that our insights regarding
the benefits of Neyman-orthogonalization in terms of faster learning rates
apply irrespective of the final implementation. Notice also that the use of
surrogate loss functions does not guarantee convergence of the estimated
optimal policy to the best-in-class $\pi^{*}_{n}$ in general when the policy
class $\Pi_{n}$ does not contain the “first-best” policy
$\mathbbm{1}\\{\Gamma(g;x)\geq 0\\}$, as shown by the recent work of Kitagawa
et al. (2021).

##  6 Empirical application

In this section we apply the methods discussed in this paper to data from the
National Job Training Partnership Act (JTPA) Study. This study randomly
selected applicants to receive various training and services, including job-
search assistance, for a period of 18 months. The study collected background
information on applicants before random assignment and then recorded their
earnings in the 30-month period following treatment assignment. Kitagawa and
Tetenov (2018) apply their EWM method to a sample of 9,223 adult JTPA
applicants to estimate the optimal allocation of eligibility into the
programme that maximizes individual earnings across the population. In
particular, they take total individual earnings in the 30 months after
assignment as the welfare outcome measure $Y_{i}$, and consider policies that
allocate eligibility in the programme based on the individual’s observable
characteristics. Kitagawa and Tetenov’s analysis is from an intent-to-treat
perspective as they focus on the problem of deciding who should be given
eligibility to participate in the programme. Since eligibility in the JTPA
study is randomly assigned, the effect of eligibility on earnings is point
identified from the data and methods for policy learning under point-
identification can be applied in this setting. We depart from Kitagawa and
Tetenov (2018) and instead consider optimal assignment of actual participation
in the training. This analysis would be of interest to a policy-maker that
expects to achieve (close to) perfect compliance to her treatment decision,
e.g. when participation is made a condition for receipt of a generous
unemployment benefit.202020No financial incentive had been put in place to
promote compliance in the implementation of the JTPA study. Compliance in the
JTPA study is imperfect as roughly 23% of applicants’ participation status
$D_{i}=0,1$ deviates from their assigned eligibility status $Z_{i}=0,1$, as
shown in Table 2. As a result, random assignment of the eligibility instrument
$Z_{i}$ is not sufficient to point-identify the effect of participation in the
training, motivating the use of the methods proposed in this paper.

Table 2: Joint distribution of eligibility and participation, JTPA study

| Eligibility ($Z_{i}$) |
---|---|---
Participation ($D_{i}$) | 0 | 1 | Total
0 | 3047 | 2118 | 5165
1 | 43 | 4015 | 4058
Total | 3090 | 6133 | 9233

  * •

Data source: Kitagawa and Tetenov (2018) and Abadie, Angrist, and Imbens
(2002).

For partial identification of the CATE we consider the Balke-Pearl scheme of
Example 3.2, where bounds for the 30-month post-treatment earnings are
$Y_{L}=\$0$ and $Y_{U}=\$59,640$.212121The outcome upper bound corresponds to
the 97.5th percentile of the earnings distribution rather than highest
recorded value of $\$155,760$. Outcome bounds in Balke-Pearl bounds
effectively impute unidentified expected earnings for never-takers and always-
takers. Restricting expected earnings to be below such high quantile is in
effect a mild requirement which brings considerable identification power. We
compare this with point-identification of the CATE as the conditional local
average treatment effect (LATE), predicated under the assumption of no
unobserved heterogeneity. We subtract $1216 from both the CATE bounds and the
conditional LATE; this is the average cost of services per actual treatment,
estimated from Table 5 in Bloom et al. (1997). Following Kitagawa and Tetenov
(2018), we condition treatment assignment on two pre-treatment variables: the
individual’s years of education and earnings in the year prior to assignment.
Estimation of the optimal policy follows the procedure described in Section 4,
with $K=10$ evenly-sized data folds used to form cross-fitted Neyman-
orthogonal estimates for the CATE bounds and conditional LATE functions. The
nuisance functions are estimated via boosted regression trees, performed by
the MATLAB function `fitrensenmble`.222222Tuning parameters have been chosen
via cross-validation within each data-fold. For further details on the
estimation procedure we refer to the MATLAB documentation for the command.

Figure 1 demonstrates cross-fitted plug-in estimates for the CATE bounds (a)
and the LATE/minimax regret scores (b), where the size of the dots indicates
the number of individuals with different covariate values. We first notice
that the estimated CATE lower bounds are negative for the whole sample, and
thus the maximin impact optimal policy never assigns treatment in this
application. We therefore focus our analysis on minimax regret
(MMR).232323Maximin welfare also results in no treatment for the whole
population in this application. Comparison of the conditional LATE and MMR
scores highlights how partial identification leads to increased variation of
the scores across different levels of education and pre-program earnings. In
particular, the MMR scores are considerably higher (and positive) for
individuals with fewer years of education and smaller pre-programmes. The
conditional LATE estimates display less overall variation over the support of
the covariates, compared to the MMR scores, but are lower for individuals with
0 pre-programme earnings.

(a) CATE bounds

(b) Scores

Figure 2: JTPA – Plug-in cross-fitted estimates (net of $1216)

We consider three alternative choices for the candidate policy class $\Pi$.
The first is the class of quadrant treatment policies. To be assigned to
treatment according to this policy, an individual’s education and pre-program
earnings have to be above (or below) some specific threshold. Figure 3
illustrates the optimal quadrant treatment policies based on Neyman-
orthogonalized cross-fitted MMR and LATE scores, where the colored shaded
areas indicate individuals required to undertake job training by the
respective policies. The optimal MMR policy (green) assigns treatment to
individuals with education below 15 years and pre-treatment earnings below
$39,952. The optimal LATE policy (blue) selects the same threshold for
education, but selects individuals with pre-treatment earnings above $200 for
treatment. While the two policies appear similar, they substantially differ in
the proportion of population assigned to treatment (96% by the MMR policy
versus 64% by the LATE policy), as shown in Table 3. This is due to the large
concentration of individuals with pre-treatment earnings close to (or equal)
zero. As a result, 32% of individuals receive a different treatment assignment
across the two policies. Figure 3 also shows the optimal “naïve” MMR policy
based on cross-fitted but non-orthogonalized scores (yellow), which recommends
participation into the programme for the entire population.

Table 3: Treatment proportions of alternative treatment assignment policies

|  |  |  Share of Population to be treated |  Share of Population receiving same treatment as
---|---|---|---|---
|  |  |  | MMR (naïve) | MMR | LATE
Quadrant Rule |  |  |
Minimax Regret (naïve) |  |  | $1.00$ | – |  |
Minimax Regret |  |  | $0.96$ | $0.96$ | – |
LATE |  |  | $0.64$ | $0.68$ | $0.68$ | –
Linear Index Rule |  |  |
Minimax Regret (naïve) |  |  | $0.99$ | – |  |
Minimax Regret |  |  | $0.96$ | $0.96$ | – |
LATE |  |  | $0.69$ | $0.69$ | $0.70$ | –
Linear Index Rule + ${\rm\bf edu^{2}}+{\rm\bf edu^{3}}$ |  |  |
Minimax Regret (naïve) |  |  | $0.99$ | – |  |
Minimax Regret |  |  | $0.96$ | $0.97$ | – |
LATE |  |  | $0.75$ | $0.75$ | $0.75$ | –

  * •

The rows labeled “Minimax Regret (naïve)” give information on the estimated
optimal minimax regret policy based on the scores in Equation (11) with the
Balke-Pearl CATE bounds of Example 3.2, without Neyman-orthogonalization. The
rows labeled “Minimax Regret” give information on the estimated optimal
minimax regret policy with Neyman-orthogonalization. The rows labeled “LATE”
give information on optimal policy for Neyman-orthogonal scores for the
conditional LATE.

Second, we consider the class of linear treatment policies. This class
consists of policies that assign treatment to an individual according to
whether a linear index in his observable characteristics is above a certain
threshold. Figure 4 illustrates how the direction of treatment assignment as a
function of prior earnings differs between the MMR and LATE policy in a
similar fashion to the quadrant rules; contrary to the LATE policy, MMR
prioritizes treatment assignment to individuals with lower pre-program
earnings. Nonetheless, 70% of the population still receives the same treatment
under the two different policies, in light of the relatively low concentration
of individuals in the areas of the covariate space where the two policies
differ. Similarly to the quadrant policy rule, the MMR policy assigns
treatment to a larger share of the population (95%) compared to the LATE
policy (69%). The naïve MMR policy is qualitatively similar to the one using
Neyman-orthogonolization, but recommends programme participation to a larger
share of individuals.

Finally, we consider linear treatment policies that additionally include
quadratic and cubic terms for education. Figure 5 shows how the additional
flexibility in the policy class leads to rules that are less interpretable but
maintain similar qualitative features compared to the more parsimonious
classes previously considered.

Figure 3: Estimated optimal policies from the quadrant policy class
conditioning on years of education and pre-programme earnings. Figure 4:
Estimated optimal policies from the linear-index policy class Figure 5:
Estimated optimal policies from the linear-index policy class conditioning on
years of education, ${\rm(education)^{2}}$, ${\rm(education)^{3}}$, and pre-
programme earnings.

##  7 Conclusion

This paper develops a general policy learning framework for estimation of
individualized treatment rules when treatment effects are partially
identified. By drawing connections between the treatment assignment problem
and classical decision theory, we have characterized several notions of
optimal treatment policies in the presence of partial identification. We have
shown how partial identification leads to a new policy learning problem where
the risk is only directionally-differentiable with respect to a nuisance
infinite-dimensional component. We have proposed an estimation procedure that
ensures Neyman-orthogonality with respect to the nuisance components and we
have provide statistical guarantees that depend on the amount of concentration
around the points of non-differentiability in the data-generating process. Our
proposed methods are illustrated with an application to the Job Training
Partnership Act study, where we have shown that allowing for partial
identification delivers substantially different programme participation
policies compared to existing methods that assume point-identification.

There are several avenues for future research. First, it would be interesting
to extend the theory of this paper to partial identification via instrumental
variables with continuous support. Second, it would be useful to extend the
methods to more general identification sets that incorporate smoothness
restrictions on unobserved counterfactual quantities, such as those considered
in Kim et al. (2018). Finally, it would be interesting to assess the
optimality of our proposed estimation procedure by deriving minimax lower
bound rates for semiparametric statistical learning problems with
directionally-differentiable risk.

##  Appendix A Extension to nested min/max operators

In this section we describe how the proposed estimation procedure and the
theoretical results of Section 4 can be extended to scores $\Gamma(g;X)$ that
feature nested linear combinations of $\min/\max$ operators. This extension
comprises $\min/\max$ operators over multiple components, since
$\max\\{a,b,c\\}=\max\\{\max\\{a,b\\},c\\}$.

We begin by noticing that our proposed estimation described in Section 4 can
be also defined as follows. First, for each $\min\\{a(g;x),b(g;x)\\}$ (or
$\max$) operator contained in $\Gamma(g;x)$, one substitutes the operator with
$a(g;x)$ or $b(g;x)$ based on their cross-fitted plug-in (non-orthogonalized)
estimates $\widehat{a}_{i}:=a(\widehat{g}^{-k(i)};X_{i})$ and
$\widehat{b}_{i}:=b(\widehat{g}^{-k(i)};X_{i})$. Then, the selected component
is estimated ($a(g;x)$ or $b(g;x)$) is estimated by their cross-fitted Neyman-
orthogonal analogue ($\widehat{a}^{\texttt{NO}}_{i}$ or
$\widehat{b}^{\texttt{NO}}_{i}$). In the presence of nested $\min/\max$
operators, our estimation is generalized as follows. First, in succession from
the most inner to the most outer $\min/\max$, each operator is substituted
with their smallest/largest argument based on cross-fitted non-orthogonalized
estimates. Then, the selected components are estimated by their cross fitted
Neyman-orthogonal analogue. As an illustration, consider the hypothetical
score

$\displaystyle\Gamma(g;x)$
$\displaystyle=d(g;x)+\max\\{c(g;x)+\min\\{a(g;x),b(g;x)\\},0\\}$
$\displaystyle=d(g;x)+\max\\{\underbrace{c(g;x)+b(g;x)+(a(g;x)-b(g;x))\cdot\mathbbm{1}\\{a(g;x)-b(g;x)\leq
0\\}}_{\varrho(g;x)},0\\}$
$\displaystyle=d(g;x)+\varrho(g;x)\cdot\mathbbm{1}\left\\{\varrho(g;x)\geq
0\right\\}.$

Applying the above procedure to this example gives the following expression
for the estimated Neyman-orthogonal score:

$\displaystyle\Gamma^{\texttt{NO}}(\\{\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\\},W_{i})=\widehat{d}^{\texttt{NO}}_{i}+\left[\widehat{c}^{\texttt{NO}}_{i}+\widehat{b}^{\texttt{NO}}_{i}+(\widehat{a}^{\texttt{NO}}_{i}-\widehat{b}^{\texttt{NO}}_{i})\cdot\mathbbm{1}\\{\widehat{a}_{i}-\widehat{b}_{i}\leq
0\\}\right]\cdot\mathbbm{1}\left\\{\widehat{\varrho}_{i}\geq 0\right\\},$

where

$\displaystyle\widehat{\varrho}_{i}=\widehat{c}_{i}+\widehat{b}_{i}+(\widehat{a}_{i}-\widehat{b}_{i})\cdot\mathbbm{1}\\{\widehat{a}_{i}-\widehat{b}_{i}\leq
0\\}.$

We will now show how the theoretical results of Section 5 can be generalized
to this example.242424 The extension to general scores containing an arbitrary
finite number of nested $\min/\max$ operators follows immediately from our
discussion of this example. Following the arguments of Section 5.2, we have

$\displaystyle\widehat{Q}_{n}^{\texttt{NO}}(\pi)-Q_{n}^{\texttt{NO}}(\pi)$
$\displaystyle=\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot(\widehat{d}^{\texttt{NO}}_{i}-d^{\texttt{NO}}_{i})$
$\displaystyle+\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot(\widehat{c}^{\texttt{NO}}_{i}+\widehat{b}^{\texttt{NO}}_{i}-c^{\texttt{NO}}_{i}-b^{\texttt{NO}}_{i})\cdot\mathbbm{1}\\{\widehat{\varrho}_{i}\geq
0\\}$
$\displaystyle+\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot(\widehat{a}^{\texttt{NO}}_{i}-\widehat{b}^{\texttt{NO}}_{i})\cdot\mathbbm{1}\\{\widehat{a}_{i}-\widehat{b}_{i}\leq
0\\}\cdot\mathbbm{1}\\{\widehat{\varrho}_{i}\geq 0\\}$
$\displaystyle+\sum_{i=1}^{n}(2\pi(x)-1)\cdot(a^{\texttt{NO}}_{i}-b^{\texttt{NO}}_{i})\cdot\left[\mathbbm{1}\\{\widehat{a}_{i}-\widehat{b}_{i}\leq
0\\}-\mathbbm{1}\\{a_{i}-b_{i}\leq
0\\}\right]\cdot\mathbbm{1}\\{\widehat{\varrho}_{i}\geq 0\\}$
$\displaystyle+\sum_{i=1}^{n}(2\pi(x)-1)\cdot\varrho^{\texttt{NO}}_{i}\cdot\left[\mathbbm{1}\\{\widehat{\varrho}_{i}\geq
0\\}-\mathbbm{1}\\{\varrho_{i}\geq 0\\}\right].$

The first term in the expansion has the same structure as $A_{0,\ell}$ and
thus obeys the same bound. The second and third term obey the same bound as
$A_{1,\ell}$ since, by virtue of sample-splitting, the indicators
$\mathbbm{1}\\{\widehat{\varrho}_{i}\geq 0\\}$ and
$\mathbbm{1}\\{\widehat{a}_{i}-\widehat{b}_{i}\geq 0\\}$ are immaterial when
controlling the expectation of these term uniformly over $\Pi_{n}$ (see
arguments in the Proof of Lemma 1). The fourth term has the same structure as
$A_{2,\ell}+A_{3,\ell}$ except for the presence of the indicator
$\mathbbm{1}\\{\widehat{\varrho}_{i}\geq 0\\}$, which again can be shown to be
immaterial for controlling the $A_{2,\ell}$-like term by virtue of sample-
splitting. For the $A_{3,\ell}$-like term we instead have the bound

$\displaystyle\mathbb{E}$
$\displaystyle\left[\sup_{\pi\in\Pi_{n}}\frac{1}{n}\sum_{i=1}^{n}(2\pi(x)-1)\cdot(c_{i}-b_{i})\cdot\left(\mathbbm{1}\\{\widehat{c}_{i}-\widehat{b}_{i}\leq
0\\}-\mathbbm{1}\\{c_{i}-b_{i}\leq
0\\}\right)\cdot\mathbbm{1}\\{\widehat{\varrho}_{i}\geq 0\\}\right]$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|(c_{i}-b_{i})\cdot\left(\mathbbm{1}\\{\widehat{c}_{i}-\widehat{b}_{i}\leq
0\\}-\mathbbm{1}\\{c_{i}-b_{i}\leq
0\\}\right)\right|\cdot\left|\mathbbm{1}\\{\widehat{\varrho}_{i}\geq
0\\}\right|\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|(c_{i}-b_{i})\cdot\left(\mathbbm{1}\\{\widehat{c}_{i}-\widehat{b}_{i}\leq
0\\}-\mathbbm{1}\\{c_{i}-b_{i}\leq 0\\}\right)\right|\Bigg{]},$

which can be bounded in the same fashion as $A_{3,\ell}$ under a margin
assumption on $a(g;X)-b(g;X)$. Finally, the fifth term also has the same
structure as $A_{2,\ell}+A_{2,\ell}$, and can be controlled using arguments
from Section 5.2 under a margin assumption on $\rho(g;X)$.

##  Appendix B Proofs

###  B.1 Proof of Proposition 1

For the maximin welfare policy we have

$\displaystyle\min_{(y_{0},y_{1})\in\mathcal{Y}}\mathbb{E}_{P_{X}}\left[\pi(X)\cdot
y_{\pi(X)}(X)\right]$
$\displaystyle=\mathbb{E}_{P_{X}}\left[\min_{(y_{0}(x),y_{1}(x))\in\mathcal{Y}(x)}\pi(X)\cdot
y_{\pi(X)}(X)\right]$
$\displaystyle=\mathbb{E}_{P_{X}}\left[\pi(X)\cdot\underline{y}_{1}(X)+(1-\pi(X))\cdot\underline{y}_{0}(X)\right],$

where the first equality is justified by 3.1. Thus we have

$\displaystyle\operatorname*{argmax}_{\pi\in\Pi}\min_{(y_{0},y_{1})\in\mathcal{Y}}W_{\tau}(\pi)=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\left[\pi(X)\cdot\underline{y}_{1}(X)+(1-\pi(X))\cdot\underline{y}_{0}(X)\right].$

For the maximin impact policy we have

$\displaystyle\min_{\tau\in\mathcal{T}}\mathbb{E}_{P_{X}}\left[\pi(X)\cdot{\tau}(X)\right]=\mathbb{E}_{P_{X}}\left[\min_{\tau\in\mathcal{T}}\pi(X)\cdot{\tau}(X)\right]=\mathbb{E}_{P_{X}}\left[\pi(X)\cdot\underline{\tau}(X)\right].$

where the first equality is justified by Assumption 3.2. Thus we have

$\displaystyle\operatorname*{argmax}_{\pi\in\Pi}\min_{\tau\in\mathcal{T}}W_{\tau}(\pi)=\operatorname*{argmax}_{\pi\in\Pi}\mathbb{E}_{P_{X}}\left[\pi(X)\cdot\underline{\tau}(X)\right].$

The statement of the proposition again follows from the invariance of the
maximizer to positive affine transformations of the objective function.

###  B.2 Proof of Proposition 2

We notice that

$\displaystyle\max_{\tau\in\mathcal{T}}\left(\max_{\pi\,:\,\mathcal{X}\to\\{0,1\\}}W_{\tau}(\pi)-W_{\tau}(\pi)\right)$
$\displaystyle=\max_{\tau\in\mathcal{T}}\mathbb{E}_{P_{X}}\left[\left(\frac{1}{2}+\frac{1}{2}\text{sgn}(\tau(X))-\pi(X)\right)\cdot\tau(X)\right]$
$\displaystyle=\mathbb{E}_{P_{X}}\left[\max_{\tau\in\mathcal{T}}\left(\frac{1}{2}+\frac{1}{2}\text{sgn}(\tau(X))-\pi(X)\right)\cdot\tau(X)\right]$
$\displaystyle=\mathbb{E}_{P_{X}}\bigg{[}\underbrace{\max_{\tau\in\mathcal{T}}\mathbbm{1}\big{\\{}\underline{\tau}(x)\geq
0\big{\\}}\cdot\left(1-\pi(X)\right)\cdot\tau(X)}_{=(1-\pi)\cdot\mathbbm{1}\left\\{\underline{\tau}(x)\geq
0\right\\}\cdot\overline{\tau}(X)}\bigg{]}+\mathbb{E}_{P_{X}}\bigg{[}\underbrace{\max_{\tau\in\mathcal{T}}\mathbbm{1}\big{\\{}\overline{\tau}(x)\leq
0\big{\\}}\cdot-\pi(X)\cdot\tau(X)}_{=-\pi\cdot\mathbbm{1}\left\\{\underline{\tau}(x)\geq
0\right\\}\cdot\underline{\tau}(X)}\bigg{]}$
$\displaystyle\qquad+\mathbb{E}_{P_{X}}\bigg{[}\underbrace{\max_{\tau\in\mathcal{T}}\mathbbm{1}\big{\\{}\underline{\tau}(x)<0<\overline{\tau}(x)\big{\\}}\cdot\left(\frac{1}{2}+\frac{1}{2}\text{sgn}(\tau(X))-\pi(X)\right)\cdot\tau(X)}_{=-\pi\cdot\mathbbm{1}\left\\{\underline{\tau}(X)<0<\overline{\tau}(X)\right\\}(\overline{\tau}(X)-\underline{\tau}(X))+\mathbbm{1}\left\\{\underline{\tau}(X)<0<\overline{\tau}(X)\right\\}\cdot\overline{\tau}(X)}\bigg{]}$
$\displaystyle=-\mathbb{E}_{P_{X}}\left[\pi\cdot\bigg{(}\mathbbm{1}\big{\\{}\underline{\tau}(X)\geq
0\big{\\}}\cdot\overline{\tau}(X)+\mathbbm{1}\big{\\{}\overline{\tau}(X)\leq
0\big{\\}}\cdot\underline{\tau}(X)+\mathbbm{1}\big{\\{}\underline{\tau}(X)<0<\overline{\tau}(X)\big{\\}}\cdot\big{(}\overline{\tau}(X)-\underline{\tau}(X)\big{)}\bigg{)}\right]$
$\displaystyle\qquad+\mathbb{E}_{P_{X}}\left[\overline{\tau}(X)\cdot\mathbbm{1}\big{\\{}\underline{\tau}(X)\geq
0\big{\\}}+\mathbbm{1}\left\\{\underline{\tau}(X)<0<\overline{\tau}(X)\right\\}\cdot\overline{\tau}(X)\right],$

where the first equality uses the fact
$\operatorname*{argmax}_{\pi\,:\,\mathcal{X}\to\\{0,1\\}}W_{\tau}(\pi)=\frac{1}{2}+\frac{1}{2}\text{sign}(\tau(X))$,
and the second equality uses Assumption 3.2. The statement of the proposition
then follows from the invariance of the maximizer to positive affine
transformations of the objective function.

###  B.3 Proof of Proposition 3

We being by decomposing regret as follows:

$\displaystyle
Q(\pi_{n}^{*})-Q(\widehat{\pi}_{n})={\Big{[}Q(\pi_{n}^{*})-{Q}^{\texttt{NO}}_{n}(\pi_{n}^{*})\Big{]}}+\Big{[}Q^{\texttt{NO}}_{n}(\pi_{n}^{*})-\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})\Big{]}+\Big{[}\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})-Q(\widehat{\pi}_{n})\Big{]}.$
(22)

The first term is zero in expectation. The second term can be upper bounded as

$\displaystyle\Big{[}Q^{\texttt{NO}}_{n}(\pi_{n}^{*})-\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})\Big{]}\leq\Big{[}Q^{\texttt{NO}}_{n}(\pi^{*}_{n})-\widehat{Q}^{\texttt{NO}}_{n}(\pi_{n}^{*})\Big{]}+\Big{[}\widehat{Q}^{\texttt{NO}}_{n}(\pi_{n}^{*})-\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})\Big{]}\leq\sup_{\pi\in\Pi_{n}}\left|Q^{\texttt{NO}}_{n}(\pi)-\widehat{Q}^{\texttt{NO}}_{n}(\pi)\right|,$

where we have used that
$\widehat{Q}^{\texttt{NO}}_{n}(\pi^{*})-\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})\leq
0$, which follows from $\widehat{\pi}_{n}$ being the maximizer of
$\widehat{Q}^{\texttt{NO}}_{n}(\cdot)$. The third term can be further expanded
and upper bounded as follows

$\widehat{Q}^{\texttt{NO}}_{n}(\widehat{\pi}_{n})-Q(\widehat{\pi}_{n})\leq\sup_{\pi\in\Pi_{n}}\left|\widehat{Q}^{\texttt{NO}}_{n}(\pi)-Q^{\texttt{NO}}_{n}(\pi)\right|+\sup_{\pi\in\Pi_{n}}\left|Q^{\texttt{NO}}_{n}(\pi)-Q(\pi)\right|.$

Using the last two displays and taking expectations in (22) yields the desired
conclusion.

###  B.4 Proof of Lemma 1

We will establish each of the following bounds in turn:

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{0}(\pi)|\right]$
$\displaystyle=O\left(\sqrt{{\rm
VC}(\Pi_{n})\cdot\frac{r_{\kappa_{n}}}{n^{3/2}}}+\frac{r_{\kappa_{n}}}{n^{1/2}}\right),$
$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{1,\ell}(\pi)|\right]$
$\displaystyle=O\left(\sqrt{{\rm
VC}(\Pi_{n})\cdot\frac{r_{\kappa_{n}}}{n^{3/2}}}+\frac{r_{\kappa_{n}}}{n^{1/2}}\right),$
$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{2,\ell}(\pi)|\right]$
$\displaystyle=O\left(\sqrt{\frac{{\rm VC}(\Pi_{n})}{n}}\right)$
$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{3,\ell}(\pi)|\right]$
$\displaystyle=O\left(\left(\frac{r_{\kappa_{n}}}{n}\right)^{\frac{\gamma+1}{\gamma+2}}\right)$

Combining the above through decomposition (21) gives the desired final bound.

#### Bound for $A_{0}$ and $A_{1,\ell}$

We prove a bound for $\sup_{\pi_{\in}\Pi_{n}}|A_{1,\ell}(\pi)|$; it will be
immediate that $A_{0}(\pi)$ obeys the same bound. We begin with the following
decomposition

$\displaystyle A_{1,\ell}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\langle\widehat{\alpha}_{i}-\alpha_{i},U_{i}-g(V_{i})\rangle,\,$
$\displaystyle\left(=B_{1}(\pi)\right),$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\langle\widehat{\alpha}_{i}-\alpha_{i},g_{i}-\widehat{g}_{i}\rangle,\,$
$\displaystyle(=B_{2}(\pi)),$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})+\langle\alpha_{i},g_{i}-\widehat{g}_{i}\rangle\right),\,$
$\displaystyle(=B_{3}(\pi)),$

where we have used the shorthand notation
$\widehat{g}_{i}:=\widehat{g}^{-k(i)}(V_{i})$, $g_{i}=g(V_{i})$,
$\widehat{\alpha}_{i}:=(\\{\widehat{g}^{-k(i)},f^{-k(i)}\\};V_{i})$,
$\alpha_{i}:=(\\{g,f\\};V_{i})$. Starting with $B_{1}(\pi)$, the contribution
of the $k$-th fold is

$\displaystyle
B_{1}^{(k)}(\pi)=\frac{1}{n}\sum_{i:k(i)=k}(2\pi(X_{i})-1)\cdot\langle\widehat{\alpha}_{i}-\alpha_{i},\varepsilon_{i}\rangle\cdot\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}.$

The sample-splitting procedure guarantees that
$\\{\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\\}$ only depend on data from the
remaining $K-1$ folds, and thus conditioning on these estimates for the
nuisance components makes $B_{1}^{(k)}(\pi)$ a sum of independent mean-zero
terms, in light of

$\displaystyle\mathbb{E}\left[U_{i}-g(V_{i})\mid
V_{i},\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\right]=0.$

Furthermore, the terms are also sub-Gaussian since it is a linear combination
of sub-Gaussian random variables with bounded weights w.p.a 1, in light of

$\displaystyle\norm{\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}\cdot(\widehat{\alpha}_{i}-\alpha_{i})}_{L_{\infty}(P_{V})}\leq\norm{\widehat{g}^{-k}(V)-g(V)}_{L_{\infty}(P_{V})}+\norm{\widehat{f}^{-k}(V)-f(V)}_{L_{\infty}(P_{V})}\leq
2\cdot\mathcal{C}_{2}\cdot\mathcal{C}_{3},$

w.p.a 1, where the first inequality uses Assumption 5.2(i) and the second
inequality uses Assumption 5.2(iii). Having computed the variance of
$B_{1}^{(k)}(\pi)$ conditional on
$\left(\widehat{g}^{-k},\widehat{f}^{-k}\right)$

$\displaystyle
V_{n}(k)=\mathbb{E}\left[(\widehat{\alpha}^{-k}_{i}-\alpha_{i})^{\prime}\,\Sigma(V_{i})\,(\widehat{\alpha}^{-k}_{i}-\alpha_{i})\cdot\mathbbm{1}\left\\{\widehat{\varphi}_{i}\geq
0\right\\}\mid\widehat{g}^{-k},\widehat{f}^{-k}\right],$

we can apply Corollary 3 in Athey and Wager (2021) to establish the bound

$\displaystyle\frac{n}{n_{k}}\mathbb{E}\left[\sup_{\pi\in\Pi}\left|B_{1}^{(k)}(\pi)\right|\mid\hat{g}^{-k}\right]=O\left(\sqrt{V_{n}(k)\frac{\text{VC}(\Pi_{n})}{n_{k}}}\right),$
(23)

where $n_{k}$ denotes the number of observations in the $k$-th fold. Using
Assumptions 5.2(ii) and 5.4, we have

$\displaystyle\mathbb{E}\left[V_{n}(k)\right]$
$\displaystyle\leq\mathbb{E}_{P_{n}}\left[\overline{\lambda}\cdot\norm{\widehat{\alpha}_{i}-\alpha_{i}}^{2}_{L_{2}(P_{V})}\right]$
(24) $\displaystyle\leq
2\cdot\overline{\lambda}\cdot\mathcal{C}_{2,\alpha}^{2}\cdot\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}^{-k(i)}-g}_{L_{2}(P_{V})}^{2}+\norm{\widehat{f}^{-k(i)}-f}_{L_{2}(P_{V})}^{2}\right]$
$\displaystyle=O\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}\right).$

Finally, we apply (23) repeatedly for each of the $K$ data-folds and using
Jensen’s Inequality and (24) and obtain the final bound

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi}\left|B_{1}(\pi)\right|\right]=O\left(\sqrt{\text{VC}(\Pi_{n})\cdot\frac{r_{\kappa_{n}}}{n^{3/2}}}\right).$
(25)

We now turn to $B_{2}(\pi)$, for which we have

$\displaystyle B_{2}(\pi)$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\langle\widehat{\alpha}_{i}-\alpha_{i},\widehat{g}_{i}-g_{i}\rangle\cdot\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}$
$\displaystyle=\sum_{j=1}^{J}\left[\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot(\widehat{\alpha}^{(j)}_{i}-\alpha^{(j)}_{i})\cdot(\widehat{g}_{i}^{(j)}-g_{i})\cdot\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}\right]$
$\displaystyle\leq\sum_{j=1}^{J}\left[\frac{1}{n}\sum_{i=1}^{n}|\widehat{\alpha}^{(j)}_{i}-\alpha^{(j)}_{i}|\cdot|\widehat{g}_{i}^{(j)}-g_{i}|\right]$
$\displaystyle\leq\sum_{j=1}^{J}\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(\widehat{\alpha}^{(j)}_{i}-\alpha^{(j)}_{i}\right)^{2}}\times\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(g_{i}^{(j)}-g_{i}\right)^{2}},$

where the last inequality uses Cauchy-Schwarz inequality. This bound does not
depend on $\pi$ and thus holds uniformly over $\Pi_{n}$. We then apply Cauchy-
Schwarz again and use Asuumption 5.4 to verify that

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi}\left|B_{2}(\pi)\right|\right]$
$\displaystyle\leq\sum_{j=1}^{J}\mathbb{E}_{P_{n}}\left[\norm{\widehat{\alpha}^{(j)}_{i}-\alpha^{(j)}_{i}}^{2}_{L_{2}(P_{V})}\right]^{1/2}\times\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}^{(j)}_{i}-g^{(j)}_{i}}^{2}_{L_{2}(P_{V})}\right]^{1/2},$
$\displaystyle\leq
J\cdot\mathbb{E}_{P_{n}}\left[\norm{\widehat{\alpha}_{i}-\alpha_{i}}^{2}_{L_{2}(P_{V})}\right]^{1/2}\times\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}_{i}-g_{i}}^{2}_{L_{2}(P_{V})}\right]^{1/2}$
$\displaystyle\lesssim
J\cdot\mathbb{E}_{P_{n}}\left[\norm{\widehat{f}^{-k(i)}-f}^{2}_{L_{2}(P_{V})}+\norm{\widehat{g}^{-k(i)}-g}^{2}_{L_{2}(P_{V})}\right]^{1/2}\times\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}^{-k(i)}-g}^{2}_{L_{2}(P_{V})}\right]^{1/2}$
$\displaystyle=O\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}\right).$

We now turn to $B_{3}(\pi)$. We begin by considering the following telescoping

$\displaystyle\widehat{g}_{i}-g_{i}=\sum_{j=1}^{J}\left[(g^{(\bullet:j-1)}_{i},\widehat{g}^{(j:\bullet)}_{i})-(g^{(\bullet:j)}_{i},\widehat{g}^{(j+1:\bullet)}_{i})\right]=\sum_{j=1}^{J}\left(0,0,\dots,\widehat{g}^{(j)}_{i}-g^{(j)}_{i},0,\dots,0\right),$

where $g^{(\bullet:j)}$ and $g^{(j:\bullet)}$ denote, respectively, the first
and last $j$ entries of $g(V_{i})$, where we adopt the convention
$g^{(\bullet:0)}=g^{(J+1:\bullet)}=\emptyset$. We can therefore decompose
$B_{3}(\pi)$ as follows:

$\displaystyle
B_{3}(\pi)=\sum_{j=1}^{J}\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)$
$\displaystyle\times\left(\varphi((g^{(\bullet:j-1)},\widehat{g}^{(j:\bullet)}_{i});X_{i})-\varphi((g^{(\bullet:j)}_{i},\widehat{g}^{(j+1:\bullet)}_{i});X_{i})-\alpha^{(j)}(\\{(g^{(\bullet:j-1)},\widehat{g}^{(j:\bullet}),f\\})\cdot\left(\widehat{g}^{(j)}_{i}-g^{(j)}_{i}\right)\right)\cdot\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}$
$\displaystyle-\sum_{j=1}^{J}\frac{1}{n}\sum_{i=1}^{n}(2\pi(X_{i})-1)\cdot\left[\left(\alpha^{(j)}_{i}-\alpha^{(j)}(\\{(g^{(\bullet:j-1},\widehat{g}^{(j:\bullet)}),f\\})\right)\cdot\left(\widehat{g}^{(j)}_{i}-g^{(j)}_{i}\right)\right]\cdot\mathbbm{1}\\{\widehat{\varphi}_{i}\geq
0\\}.$

By the definition of the Riesz-representer and cross-fitting we have

$\displaystyle\mathbb{E}\Big{[}\varphi((g^{(\bullet:j-1)},$
$\displaystyle\widehat{g}^{(j:\bullet)}_{i});X_{i})-\varphi((g^{(\bullet:j)}_{i},\widehat{g}^{(j+1:\bullet)}_{i});X_{i})$
$\displaystyle-\alpha^{(j)}(\\{(g^{(\bullet:j-1)},\widehat{g}^{(j:\bullet}),f\\},W_{i})\cdot\left(\widehat{g}^{(j)}_{i}-g^{(j)}_{i}\right)\mid
V_{i},\widehat{g}^{-k(i)},\widehat{f}^{-k(i)}\Big{]}=0,$

where we have used the property
$\alpha^{(j)}_{\ell}(\\{(\widetilde{g}_{-j},\widetilde{g}_{j}),\widetilde{f}\\},x)=\alpha^{(j)}_{\ell}(\\{\widetilde{g}_{-j},\widetilde{f}\\},x)$.
Furthermore, the term within the expectation operator is sub-Gaussian since
uniformly bounded by Assumption 5.2(iii). Therefore the first term in the
expansion of $B_{3}(\pi)$ can be controlled uniformly using identical
arguments as for $B_{1}(\pi)$ and obeys the same bound. The second term in the
expansion of $B_{3}(\pi)$ can be bounded with identical arguments as for
$B_{2}(\pi)$ and obeys the same bound. We therefore conclude that

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi}\left|B_{3}(\pi)\right|\right]=O\left(\sqrt{\text{VC}(\Pi_{n})\cdot\frac{r_{\kappa_{n}}}{n^{3/2}}}+\frac{r_{\kappa_{n}}}{\sqrt{n}}\right).$

Combining the bounds for $B_{1}(\pi)$,$B_{2}(\pi)$ and $B_{3}(\pi)$ via the
triangle inequality finally gives the desired bound for
$\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{1,\ell}(\pi)|\right]$.

#### Bound for $A_{2,\ell}$

We first notice that

$\displaystyle\mathbb{E}\left[\phi_{\ell}(\\{g,f\\};W_{i})\cdot\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\mid V_{i},\widehat{g}^{-k(i)}\right]=0,$ (26)

by the mean-zero property of the influence function adjustments $\phi_{\ell}$
and cross-fitting. Furthermore, the term inside expectation is sub-Gaussian by
uniform boundedness of the Riesz-representer, guaranteed by Assumption
5.2(iii). Thus we can use similar arguments to those used for $B_{1}(\pi)$ to
show

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}|A_{2,\ell}(\pi)|\right]=O\left(\sqrt{\frac{{\rm
VC}(\Pi_{n})}{n}}\right).$

#### Bound for $A_{3,\ell}$

We begin by noticing that
$\varphi_{\ell}(g;X_{i})\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\leq 0$ and thus, since the “never treat” policy belongs to
any policy class $\Pi$ for which $\text{VC}(\Pi)\geq 1$, we have

$\displaystyle\sup_{\pi\in\Pi_{n}}A_{3,\ell}(\pi)=\frac{1}{2n}\sum_{i=1}^{n}\left|\varphi_{\ell}(g;X_{i})\cdot\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\right|,$

and thus we obtain the uniform bound252525For a policy class of zero VC-
dimension, (27) holds as an inequality.

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}A_{3,\ell}(\pi)\right]=\frac{1}{2}\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\cdot\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\right|\Bigg{]}.$ (27)

For the RHS in (27), we closely follow Lemma 5.2 in Audibert and Tsybakov
(2007), but we report the steps of the proof for completeness. For $\gamma>0$
and any $t>0$ we have

$\displaystyle\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\right|\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\geq\left|\varphi_{\ell}(g;X_{i})\right|\right\\}\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\geq\left|\varphi_{\ell}(g;X_{i})\right|\right\\}\cdot\mathbbm{1}\left\\{0<\left|\varphi_{\ell}(g;X_{i})\right|\leq
t\right\\}\Bigg{]}$
$\displaystyle\qquad+\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\geq\left|\varphi_{\ell}(g;X_{i})\right|\right\\}\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(g;X_{i})\right|>t\right\\}\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{0<\left|\varphi_{\ell}(g;X_{i})\right|\leq
t\right\\}\Bigg{]}$
$\displaystyle\qquad+\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|>t\right\\}\Bigg{]}$
$\displaystyle\leq\mathbb{E}\left[\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right)^{2}\right]^{1/2}\cdot\mathbb{P}\big{(}0<\left|\varphi_{\ell}(g;X_{i})\right|\leq
t\big{)}^{1/2}+\frac{\mathbb{E}\left[\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right)^{2}\right]}{t}$
$\displaystyle\leq
C_{0}^{1/2}\mathbb{E}\left[\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right)^{2}\right]^{1/2}t^{\gamma/2}+\frac{\mathbb{E}\left[\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right)^{2}\right]}{t},$

where the penultimate inequality uses Cauchy-Schwarz and Markov inequalities,
and the last inequality uses the Margin Assumption. Minimizing the last
display over $t$ gives

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}A_{3,\ell}(\pi)\right]$
$\displaystyle\leq(\gamma+2)\cdot\left(\frac{2}{\gamma}\right)^{\gamma/(\gamma+2)}\cdot\mathcal{C}_{m}^{1/(\gamma+2)}\cdot\mathbb{E}\left[\left(\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right)^{2}\right]^{\frac{\gamma+1}{\gamma+2}}$
$\displaystyle\leq(\gamma+2)\cdot\left(\frac{2}{\gamma}\right)^{\gamma/(\gamma+2)}\cdot\mathcal{C}_{m}^{1/(\gamma+2)}\cdot\mathcal{C}_{2,\varphi}^{\frac{2(\gamma+1)}{\gamma+2}}\cdot\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}^{-k}-g}_{L_{2}(P_{X})}^{2}\right]^{\frac{\gamma+1}{\gamma+2}}$

For $\gamma=0$, a similar argument gives

$\displaystyle\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(g;X_{i})\left(\mathbbm{1}\left\\{\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})\geq
0\right\\}-\mathbbm{1}\left\\{\varphi_{\ell}(g;X_{i})\geq
0\right\\}\right)\right|\Bigg{]}$
$\displaystyle\leq\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{0<\left|\varphi_{\ell}(g;X_{i})\right|\leq
t\right\\}\Bigg{]}$
$\displaystyle\qquad\qquad\qquad+\mathbb{E}\Bigg{[}\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|\cdot\mathbbm{1}\left\\{\left|\varphi_{\ell}(\widehat{g}^{-k(i)};X_{i})-\varphi_{\ell}(g;X_{i})\right|>t\right\\}\Bigg{]}$
$\displaystyle\leq
2\mathbb{E}_{P_{n}}\left[\norm{\varphi_{\ell}(\widehat{g}^{-k};X)-\varphi_{\ell}(g;X)}_{L_{2}(P_{X})}^{2}\right]^{1/2}$
$\displaystyle\leq
2\cdot\mathcal{C}_{2,\varphi}^{2}\cdot\mathbb{E}_{P_{n}}\left[\norm{\widehat{g}^{-k}-g}_{L_{2}(P_{X})}^{2}\right]^{1/2}.$

Combining the cases $\gamma>0$ and $\gamma=0$, and using the $L_{2}$-risk
bounds for $\widehat{g}^{-k}$ from Assumption 5.4 we finally get

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}A_{3,\ell}(\pi)\right]=O\left(\left(\frac{r_{\kappa_{n}}}{\sqrt{n}}\right)^{\frac{\gamma+1}{\gamma+2}}\right).$

###  B.5 Proof of Theorem 2

The Neyman-orthogonalized score $\Gamma^{\texttt{NO}}(\\{g,f\\};W_{i})$
satisfies the assumptions of Corollary 3 in Athey and Wager (2021), and thus
it can be applied verbatim to show that

$\displaystyle\mathbb{E}\left[\sup_{\pi\in\Pi_{n}}\left|Q^{\texttt{NO}}_{n}(\pi)-Q(\pi)\right|\right]=O\left(\sqrt{\frac{{\rm
VC}(\Pi_{n})}{n}}\right).$

Combining the above bound with Lemma 1 via Proposition 3 gives the statement
of the theorem.

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