Original: Toward Fair and Robust Optimal Treatment Regimes
Revised: Toward Fair and Robust Optimal Treatment Regimes

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: We propose a new framework for robust nonparametric estimation of optimaltreatment regimes under flexible fairness constraints.
Revised: We propose a new framework for robust nonparametric estimation of optimal treatment regimes under flexible fairness constraints.

Original: Under standard regularityconditions we show that the resulting estimators possess the double robustnessproperty.
Revised: Under standard regularity conditions we show that the resulting estimators possess the double robustness property.

Original: We use this framework to characterize the trade-off between fairness andthe maximum welfare that is achievable by the optimal treatment policy.
Revised: We use this framework to characterize the trade-off between fairness and the maximum utility that is achievable by the optimal treatment policy.

Original:  Introduction
Revised:  Introduction

Original: In today’s world, an increasing number of decisions that affect people’s lives are automatically madeby machine learning models.
Revised: In today’s world, an increasing number of decisions that affect people’s lives are automatically made by machine learning models.

Original: Such decision-making systems are implemented in various settingsranging from financial investment to healthcare policy.
Revised: Such decision-making systems are implemented in various settings ranging from financial investment to healthcare policy.

Original: Considering the importance of such decisionsat an individual and societal level, it is crucial to ensure that the underlying models are not onlyaccurate but fair .
Revised: Considering the importance of such decisions at an individual and societal level, it is crucial to ensure that the underlying models are not only accurate but fair.

Original: In this work, by fairness we mean that the models are not biased so that they donot systematically benefit or harm a specific group of people, such as a minority ethnic group.
Revised: In this work, by fairness we mean that the models are not biased so that they do not systematically benefit or harm a specific group of people, such as a minority ethnic group.

Original: Theneed to address such algorithmic biases has given rise to an explosion of works studying algorithmicfairness (e.g., see [3] for a review).
Revised: The need to address such algorithmic biases has given rise to an explosion of works studying algorithmic fairness (e.g., see [3] for a review).

Original: However, despite the considerable amount of studies in thisarea, comparatively little attention has been given to fairness in causal inference.
Revised: However, despite the considerable amount of studies in this area, comparatively little attention has been given to fairness in causal inference.

Original: In this work, wepropose a novel framework for estimating optimal treatment assignments or regimes in a fair androbust manner, leveraging recent developments in counterfactual optimization [20, 21].
Revised: In this work, we propose a novel framework for estimating optimal treatment assignments or regimes in a fair and robust manner, leveraging recent developments in counterfactual optimization [20, 21].

Original: 1.1 Related Work
Revised: 1.1 Related Work

Original: Much of the earlier work on estimating optimal treatment regimes involves postulating a parametricmodel for the outcome regression function [e.g., 4, 10, 30, 37].
Revised: Much of the earlier work on estimating optimal treatment regimes involves postulating a parametric model for the outcome regression function [e.g., 4, 10, 30, 37].

Original: More robust approaches based on theidea of doubly robust estimation have also been proposed, for example, in [45, 46].
Revised: More robust approaches based on the idea of doubly robust estimation have also been proposed, for example, in [46, 47].

Original: In recent studies[1, 13, 22], flexible nonparametric approaches are discussed where an optimal policy is deployedfrom a pre-specified class that can encode problem-specific constraints.
Revised: In recent studies [1, 13, 22], flexible nonparametric approaches are discussed where an optimal policy is deployed from a pre-specified class that can encode problem-specific constraints.

Original: However, they do not providemeans to incorporate general fairness constraints.
Revised: However, they do not provide means to incorporate general fairness constraints.

Original: In order to mitigate algorithmic biases where model performance varies over sensitive features, awide array of fairness criteria have been developed typically by placing restrictions on the jointdistribution of model outcomes and sensitive features.
Revised: In order to mitigate algorithmic biases where model performance varies over sensitive features, a wide array of fairness criteria have been developed typically by placing restrictions on the joint distribution of model outcomes and sensitive features.

Original: Popular fairness criteria include independence(or statistical parity) [3] and separation (or equalized odds) 
Revised: Popular fairness criteria include independence (or statistical parity) [3] and separation (or equalized odds) [9].

Original: In some cases such as risk assessmentsettings [e.g., 7], counterfactual fairness may be of interest where fairness criteria depend on potential(or counterfactual) outcomes with respect to the sensitive feature [e.g., 23, 31] or a decision variable[e.g.
Revised: In some cases such as risk assessment settings [e.g., 7], counterfactual fairness may be of interest where fairness criteria depend on potential (or counterfactual) outcomes with respect to the sensitive feature [e.g., 23, 31] or a decision variable [e.g., 7, 28, 29].

Revised: Some proposed constraint-based frameworks to flexibly incorporate such fairness criteria in classification [e.g., 28, 45], but it is not clear how to extend these frameworks to enable the design of fair optimal treatment regimes.

Original: It is also well known that there exists a fairness-accuracy tradeoff, because in somecases the most accurate models under consideration do not satisfy a chosen fairness criterion [e.g.,27, 29, 36]. 
Revised: It is also well known that there exists a fairness-accuracy tradeoff, because in some cases the most accurate models under consideration do not satisfy a chosen fairness criterion [e.g., 27, 29, 36]. However, the tradeoff between fairness and treatment utility, if any, has never been formally explored.

Original: Interestingly, little work has been done at the intersection of these two areas.
Revised: Interestingly, little work has been done at the intersection of these two areas.

Revised: Most results in the algorithmic fairness literature are not directly applicable to optimal treatment regimes where our objective function involves a particular form of counterfactual functionals.

Original: A few importantexceptions include [32] which integrates algorithmic fairness and policy learning using tools from mediation analysis, and [44] which proposes an estimator for the Pareto optimal policy that minimizesunfairness through a mixed-integer quadratic programming.
Revised: A few important exceptions include [32] which integrates algorithmic fairness and policy learning using tools from mediation analysis, and [44] which proposes an estimator for the Pareto optimal policy that minimizes unfairness through a mixed-integer quadratic programming.

Original: 1.2 Contribution
Revised: 1.2 Contribution

Original: Our method builds on a promising literature at the intersection of algorithmic fairness, causalinference, and stochastic optimization, bridging the gap between algorithmic fairness and optimaltreatment regimes.
Revised: Our method builds on a promising literature at the intersection of algorithmic fairness, causal inference, and stochastic optimization, bridging the gap between algorithmic fairness and optimal treatment regimes.

Original: At this intersection, our contribution is twofold.
Revised: At this intersection, our contribution is twofold.

Original: First, we propose a robustestimator of optimal treatment regimes under general fairness constraints.
Revised: First, we propose a robust estimator of optimal treatment regimes under general fairness constraints.

Original: We cast our estimator asa convex quadratic program that can be readily solved with off-the-shelf solvers.
Revised: We cast our estimator as a convex quadratic program that can be readily solved with off-the-shelf solvers.

Original: We show that theresulting estimators are doubly robust under standard regularity conditions.
Revised: We show that the resulting estimators are doubly robust under standard regularity conditions.

Original: Our proposed approachcontributes to [32] in terms of robustness, and to [44] in terms of ease of implementation andinterpretability.
Revised: Our proposed approach contributes to [32] in terms of robustness, and to [44] in terms of ease of implementation and interpretability.

Original: Second, by analyzing thewelfare regret bound, we characterize the trade-off betweenthe maximum possible benefit and fairness.
Revised: Second, by analyzing the regret bound, we characterize the trade-off between the maximum possible benefit and fairness.

Original: This will be useful for understanding, for example, how adesired level of fairness requires a welfare compromise.
Revised: This will be useful for understanding, for example, how a desired level of fairness requires a utility compromise.

Original:  Setup and Framework2.1 Optimal Treatment Regimes
Revised:  Setup and Framework 2.1 Optimal Treatment Regimes

Original: Suppose that we have access to an i.i.d.
Revised: Suppose that we have access to an i.i.d.

Original: sample ( Z 1 , ..., Z n ) of n tuples Z =
Revised: sample ( Z 1 , ..., Z n ) of n tuples Z =

Original: ( Y, A, S, X ) ∼ P forsome distribution P , outcome Y ∈ R , binary intervention
Revised: ( Y, A, S, X ) ∼ P for some distribution P , outcome Y ∈ R , binary intervention

Original: A ∈ { 0 , 1 } , sensitive feature S ∈ { 0 , 1 } ,and additional covariates X ∈ X ⊂ R d x for some compact subset X .
Revised: A ∈ { 0 , 1 } , sensitive feature S ∈ { 0 , 1 } , and additional covariates X ∈ X ⊂ R d x for some compact subset X .

Original: Throughout we assumelarger values of Y are preferred.
Revised: Throughout we assume larger values of Y are preferred.

Original: We let W =
Revised: We let W =

Original: ( S, X ) ∈ W represent the measured pre-interventionvariables and let Y a denote the potential outcome that would have been observed (possibly contraryto fact) under treatment or
Revised: ( S, X ) ∈ W represent the measured pre-intervention variables and let Y a denote the potential outcome that would have been observed (possibly contrary to fact) under treatment or

Original: intervention A
Revised: intervention A

Original: = a .
Revised: = a .

Original: A policy maker has to choose a treatment policy or atreatment regime 1 that is a function g : W → { 0 , 1 } to determine whether individuals with covariates
Revised: A policy maker has to choose a treatment policy or a treatment regime 1 that is a function g : W → { 0 , 1 } to determine whether individuals with covariates

Original: W will be assigned to the treatment 0 or 1 .
Revised: W will be assigned to the treatment 0 or 1 .

Original: For an arbitrary treatment regime g , we define the welfarefunction for which the treatment regime g ∈ G is applied to the population P by
Revised: For an arbitrary treatment regime g , we define the welfare or utility function for which the treatment regime g ∈ G is applied to the population P by

Original: [Equation]
Revised: [Equation]

Original: Throughout we assume the standard causal assumptions of consistency , no unmeasured confounding ,and positivity [e.g., 11, Chapter 12].
Revised: Throughout we assume the standard causal assumptions of consistency , no unmeasured confounding , and positivity [e.g., 11, Chapter 12].

Original: Under these assumptions, it is straightforward to show that theoptimal treatment regime leading to the largest value of R ( g ) is given by
Revised: Under these assumptions, it is straightforward to show that the optimal treatment regime leading to the largest value of U ( g ) is given by

Original: [Equation]
Revised: [Equation]

Original: where µ a ( W ) =
Revised: where µ a ( W ) =

Original: E [ Y | W, A = a ] , ∀ a ∈ { 0 , 1 } ; i.e., the optimal regime assigns the treatment thatyields the larger mean outcome conditional on the individual characteristics.
Revised: E [ Y | W, A = a ] , ∀ a ∈ { 0 , 1 } ; i.e., the optimal regime assigns the treatment that yields the larger mean outcome conditional on the individual characteristics.

Original: 2.2 Simple Motivating Example
Revised: 2.2 Simple Motivating Example

Original: Sometimes, efficient estimation of g ∗ in (1) alone can result in unfair treatment policies.
Revised: Sometimes, efficient estimation of g ∗ in (1) alone can result in unfair treatment policies.

Original: Consider thefollowing simple data-generating process
Revised: Consider the following simple data-generating process

Original: [Equation]
Revised: [Equation]

Original: where expit and Unif( l, u ) denote the inverse logit function and the uniform distribution over theinterval [ l, u ] .
Revised: where expit and Unif( l, u ) denote the inverse logit function and the uniform distribution over the interval [ l, u ] .

Original: Then the optimal treatment regime is 1 ( X > 0) .
Revised: Then the optimal treatment regime is 1 ( X > 0) .

Original: However, when we generate 100samples, as can be seen in Figure 1, a serious fairness problem is observed; under the optimaltreatment regime only less than 7 % of individuals with S = 0 are treated, while more than 95 % ofindividuals in the untreated group are S = 0 .
Revised: However, when we generatesamples, as can be seen in Figure 1, a serious fairness problem is observed; under the optimal treatment regime only less than 7 % of individuals with S = 0 are treated, while more than 95 % of individuals in the untreated group are S = 0 .

Original: Here, group S = 0 is discriminated by the estimated optimal treatment regime that is designed toresult in the greatest benefit overall in the population.
Revised: Here, group S = 0 is discriminated by the estimated optimal treatment regime that is designed to result in the greatest benefit overall in the population.

Original: In data-driven decision-making, this kind ofalgorithmic bias can lead to critical issues in the real world as illustrated in the following examples.
Revised: In data-driven decision-making, this kind of algorithmic bias can lead to critical issues in the real world as illustrated in the following examples

Original: • Stop-and-Frisk: if A represents the policing practice of stop-and-frisk program, the established optimal treatment regime could be used as a recipe for discriminatory practice ofstop-and-frisk toward specific ethnic groups.
Revised: • Stop-and-Frisk: if A represents the policing practice of stop-and-frisk program, the established optimal treatment regime could be used as a recipe for discriminatory practice of stop-and-frisk toward specific ethnic groups.

Original: • Medical Resource Allocation: if A represents access to medical treatment or health careresources, many recent studies advocate not only cost-effectiveness but also other ethical values for rationing limited health resources [e.g., 8, 36].
Revised: • Medical Resource Allocation: if A represents access to medical treatment or health care resources, many recent studies advocate not only cost-effectiveness but also other ethical values for rationing limited health resources [e.g., 8, 36].

Original: 2.3 Proposed Framework
Revised: 2.3 Proposed Framework

Original: In this section, we lay out a framework for estimating optimal treatment regimes where we canminimize algorithmic unfairness below a particular level.
Revised: In this section, we lay out a framework for estimating optimal treatment regimes where we can minimize algorithmic unfairness below a particular level.

Original: Our strategy is to estimate each outcomeregression function µ a satisfying desired fairness criteria, and then plug back into the formula (1) sothat the same fairness criteria are also satisfied in the optimal regime.
Revised: Our strategy is to estimate each outcome regression function µ a satisfying desired fairness criteria, and then plug back into the formula (1) so that the same fairness criteria are also satisfied in the optimal regime.

Original: Specifically, we aim to estimate a functional approximation of µ a , defined by a projection onto afinite-dimensional parametric model subject to fairness constraints.
Revised: Specifically, we aim to estimate a functional approximation of µ a , defined by a projection onto a finite-dimensional parametric model subject to fairness constraints.

Original: Our target parameter can bereformulated as the following constrained stochastic optimization problemminimize β ∈B
Revised: Our target parameter can be reformulated as the following constrained stochastic optimization problem minimize β ∈B

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: for some δ j ≥ 0 and J = { 1 , ..., m } .
Revised: for some δ j ≥ 0 and J = { 1 , ..., m } .

Original: δ j is a prespecified tolerance for the maximum acceptable levelof unfairness.
Revised: δ j is a prespecified tolerance for the maximum acceptable level of unfairness.

Original: The solution of the above program corresponds to the coefficients of the estimatedbest-fitting function of µ a on the finite-dimensional model space spanned by the basis functionsb ( W ) =
Revised: The solution of the above program corresponds to the coefficients of the estimated best-fitting function of µ a on the finite-dimensional model space spanned by the basis functions b ( W ) =

Original: [ b 1 ( W ) , ..., b k ( W )]
Revised: [ b 1 ( W ) , ..., b k ( W )]

Original: ⊤ subject to m fairness constraints in C fair .
Revised: ⊤ subject to m fairness constraints in C fair .

Revised: C lin is a set of other deterministic linear constraints which could be used for penalization or incorporating prior information.

Revised: This can be generalized to the nonlinear constraints at the expense of stronger regularity conditions [20].

Original: Note that we do not assumeanything about the true functional relationship between Y a and W .
Revised: Note that we do not assume anything about the true functional relationship between Y a and W .

Original: This form of aggregated estimatorsare widely used in nonparametric regression [e.g., 12, 42].
Revised: This form of aggregated estimators are widely used in nonparametric regression [e.g., 12, 42].

Original: Following [28], we use the canonical form of fairness function g j : W × Y → R to accommodate abroad range of fairness measures.
Revised: Following [28], we use the canonical form of fairness function g j : W × Y → R to accommodate a broad range of fairness measures.

Original: For example, the criterion of independence that requires our modelto be independent of the sensitive feature can be applied by letting
Revised: For example, the criterion of independence that requires our model to be independent of the sensitive feature can be applied by letting

Original: [Equation]
Revised: [Equation]

Original: , which leads to (cid:12)(cid:12) E (cid:8) β ⊤ b ( W ) | S= 0 (cid:9) − [Equation] (cid:8) β ⊤ [Equation]
Revised: which leads to (cid:12)(cid:12) E (cid:8) β ⊤ b ( W ) | S

Original: [Equation] ) | S [Equation] 1 (cid:9)(cid:12)(cid:12) ≤ δ j .
Revised: = 0 (cid:9) − E (cid:8) β ⊤ b ( W ) | S = 1 (cid:9)(cid:12)(cid:12) ≤ δ j .

Original: We refer to [28, Section 3]for more examples.
Revised: We refer to [28, Section 3] for more examples.

Original: Similar projection approaches have also been used in causal inference [e.g., 19, 34, 39].
Revised: Similar projection approaches have also been used in causal inference [e.g., 19, 34, 39].

Original: There areseveral reasons why the above projection approach is preferred in our setting.
Revised: There are several reasons why the above projection approach is preferred in our setting.

Original: First, as will be seenshortly, the coefficients β may be estimated with flexible nonparametric methods while achieving the property of double robustness and tractable inference, and so does the target parameter g ∗ .
Revised: First, as will be seen shortly, the coefficients β may be estimated with flexible nonparametric methods while achieving the property of double robustness and tractable inference, and so does the target parameter g ∗ .

Original: It alsoprovides interpretability; it allows practitioners to understand and audit the resulting optimal regimesaccording to the specified level of unfairness.
Revised: It also provides interpretability; it allows practitioners to understand and audit the resulting optimal regimes according to the specified level of unfairness.

Original: Further, one may flexibly incorporate not only variousfairness constraints but also other practical constraints into estimation.
Revised: Further, one may flexibly incorporate not only various fairness constraints but also other practical constraints into estimation.

Original: Finally, the optimal solutionof ( P µ a ) can be readily estimated by solving the convex quadratic program that approximates ( P µ a ),which will be described in the following section.
Revised: Finally, the optimal solution of ( P µ a ) can be readily estimated by solving the convex quadratic program that approximates ( P µ a ), which will be described in the following section.

Original: Remark 1.
Revised: Remark 1.

Original: Another notable feature of our framework is that  we can consider a general setting whereonly a subset of covariates V ⊆ W can be used for predicting the counterfactual outcome Y a .
Revised: Another notable feature of our framework is that as in [20] we can consider a general setting where only a subset of covariates V ⊆ W can be used for predicting the counterfactual outcome Y a .

Original: Thisallows for runtime confounding, where some factors used by decision-makers are recorded in thetraining data (used to construct nuisance estimates) but are not available for prediction (see [6] andreferences therein).
Revised: This allows for runtime confounding, where some factors used by decision-makers are recorded in the training data (used to construct nuisance estimates) but are not available for prediction (see [6] and references therein).

Original: Notation.
Revised: Notation.

Original: Here we briefly introduce some notation used throughout this paper.
Revised: Here we briefly introduce some notation used throughout this paper.

Original: For any fixed vectorv , we let ∥ v ∥ q denote the L q -norm.
Revised: For any fixed vector v , we let ∥ v ∥ q denote the L q -norm.

Original: Let P n denote the empirical measure over ( Z 1 , ..., Z n ) .
Revised: Let P n denote the empirical measure over ( Z 1 , ..., Z n ) .

Original: Given asample operator h (e.g., an estimated function), we let P denote the conditional expectation over anew independent observation Z , as in P ( h )=
Revised: Given a sample operator h (e.g., an estimated function), we let P denote the conditional expectation over a new independent observation Z , as in P ( h )

Original:  P { h ( Z ) } = (cid:82) h ( z ) d P ( z ) 3 .
Revised: = P { h ( Z ) } = (cid:82) h ( z ) d P ( z ) 3 .

Original: Then we use ∥ h ∥ q, P todenote the L q ( P ) norm of h defined by ∥ h ∥ q, P = (cid:2)(cid:82)
Revised: Then we use ∥ h ∥ q, P to denote the L q ( P ) norm of h defined by ∥ h ∥ q, P = (cid:2)(cid:82)

Original: | h ( z ) | q d P ( z ) (cid:3) 1 q .
Revised: | h ( z ) | q d P ( z ) (cid:3) 1 q .

Original: Lastly, we let ≲ denote lessthan or equal to up to a nonnegative constant.
Revised: Lastly, we let ≲ denote less than or equal to up to a nonnegative constant.

Original:  Estimation and Inference
Revised:  Estimation and Inference

Original: ( P µ a ) is not directly solvable so we need to find an approximating program of the “true" program( P µ a ) .
Revised: ( P µ a ) is not directly solvable so we need to find an approximating program of the “true" program ( P µ a ).

Original: A complication arises since standard approaches to stochastic programming such as stochasticapproximation (SA) and sample average approximation (SAA) [e.g., 33, 40] are infeasible in oursetting, because i) the relevant sample moments and stochastic (sub)gradients depend on unobservedcounterfactuals, and ii) these approaches cannot incorporate efficient semiparametric estimators withcross-fitting [5, 35].
Revised: A complication arises since standard approaches to stochastic programming such as stochastic approximation (SA) and sample average approximation (SAA) [e.g., 33, 40] are infeasible in our setting, because i) the relevant sample moments and stochastic (sub)gradients depend on unobserved counterfactuals, and ii) these approaches cannot incorporate efficient semiparametric estimators with cross-fitting [5, 35].

Original: We therefore build our estimators on the recent developments by [20, 21] wherecounterfactual components are estimated more flexibly.
Revised: We therefore build our estimators on the recent developments by [20, 21] where counterfactual components are estimated more flexibly.

Original: For convenience, define the following:
Revised: For convenience, define the following:

Original: [Equation]
Revised: [Equation]

Original: φ a is the uncentered efficient influence function for the parameter E { E [ Y | X, A
Revised: φ a is the uncentered efficient influence function for the parameter E { E [ Y | X, A

Original: = a ] } with a set ofthe nuisance components defined by η = { π a ( X ) , µ a ( X ) } [15].
Revised: = a ] } with a set of the nuisance components defined by η = { π a ( X ) , µ a ( X ) } [15].

Original: First, we provide influence-function-based semiparametric estimators for each component of ( P µ a ) .
Revised: First, we provide influence-function-based semiparametric estimators for each component of ( P µ a ).

Original: Following [5, 16, 38, 47], we propose to use sample splitting (or cross fitting ) to allow for arbitrarilycomplex nuisance estimators (cid:98) η .
Revised: Following [5, 16, 38, 48], we propose to use sample splitting (or cross fitting ) to allow for arbitrarily complex nuisance estimators (cid:98) η .

Original: Specifically, we split the data into K disjoint groups, each withsize n/K approximately, by drawing variables ( B 1 , ..., B n ) independent of the data, with B i = bindicating that subject i was split into group b ∈ { 1 , ..., K } .
Revised: Specifically, we split the data into K disjoint groups, each with size n/K approximately, by drawing variables ( B 1 , ..., B n ) independent of the data, with B i = b indicating that subject i was split into group b ∈ { 1 , ..., K } .

Original: Then the semiparametric estimators for
Revised: Then the semiparametric estimators for

Original: L MSE and each element in C fair based on the efficient influence function and sample splitting are givenby
Revised: L MSE and each element in C fair based on the efficient influence function and sample splitting are given by

Original: [Equation]
Revised: [Equation]

Original: [Equation] where we let P b n denote empirical averages only over the set of units { i : B i = b } in group band let (cid:98) η − B K denote the nuisance estimator constructed only using those units {
Revised: where we let P bn denote empirical averages only over the set of units { i : B i = b } in group b and let (cid:98) η − B K denote the nuisance estimator constructed only using those units {

Original: i : B i ̸ = b } .
Revised: i : B i ̸ = b } .

Original: Under weak regularity conditions, these sample-splitting-based semiparametric estimators attain theefficiency bound with the double robustness property, and thus allow us to employ flexible machine learning estimation methods while achieving the √ n -rate of convergence and valid inference [15] 4 .
Revised: Under weak regularity conditions, these sample-splitting-based semiparametric estimators attain the efficiency bound with the double robustness property, and thus allow us to employ flexible machine learning estimation methods while achieving the √ n -rate of convergence and valid inference [15] 4 .

Original: Consequently, our approximating program can be found as the following convex quadratic program 
Revised: Consequently, our approximating program can be found as the following convex quadratic program (QP) minimize

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original:  to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Revised: where (cid:98) C fair := (cid:8) β (cid:12)(cid:12) (cid:12)(cid:12)

Revised: P n (cid:8) g j ( φ a ( Z ; (cid:98) η − b ) , W ) β ⊤ b ( W ) (cid:9)(cid:12)(cid:12) ≤ δ j , j ∈ J (cid:9) .

Original:  P µ a ) can be  solved using off-the-shelf QP solvers.
Revised: ( (cid:98) P µ a ) can be readily solved using off-the-shelf QP solvers.

Original: Next, we introduce the following assumptions forour counterfactual component estimators.
Revised: Next, we introduce the following assumptions for our counterfactual component estimators.

Original: (A1) P ( (cid:98) π a ∈ [ ϵ, 1 − ϵ ]) = 1 for some ϵ >
Revised: (A1) P ( (cid:98) π a ∈ [ ϵ, 1 − ϵ ]) = 1 for some ϵ >(A2)

Original: [Equation]

Original: A2) [Equation] (cid:98) [Equation]
Revised: ∥ (cid:98) µ a − µ a ∥ 2 , P

Original: a
Revised: =

Original: [Equation] , P [Equation] o
Revised: o P (1) or ∥ (cid:98) π a − π a ∥ 2 , P =

Original: [Equation] 1) or ∥ [Equation]
Revised: o P (1) (A3) ∥ (cid:98) π a − π a ∥ 2 , P ∥ (cid:98) µ a − µ a ∥ 2 , P =

Original: [Equation]
Revised: o P ( n − 12 )

Original: Assumptions (A1) - (A3) are commonly used in semiparametric estimation in the causal inferenceliterature [14].
Revised: Assumptions (A1) - (A3) are commonly used in semiparametric estimation in the causal inference literature [14].

Original: In the following theorem, we provide the large-sample properties of our proposedestimator.
Revised: In the following theorem, we provide the large-sample properties of our proposed estimator.

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: Let β ∗ and (cid:98) β denote the optimal solutions to ( P µ a ) and ( (cid:98) P µ a ) , respectively.
Revised: Let β ∗ and (cid:98) β denote the optimal solutions to ( P µ a ) and ( (cid:98) P µ a ) , respectively.

Original: If
Revised: If

Original: Assumptions (A1) and (A2) hold, then
Revised: Assumptions (A1) and (A2) hold, then

Original: [Equation]
Revised: [Equation]

Original: If we additionally assume (A3), uniqueness of β ∗ , and that the Linear Independence Constraint
Revised: If we additionally assume (A3), uniqueness of β ∗ , and that the Linear Independence Constraint

Original: Qualification (LICQ) and Strict Complementarity (SC) hold at β ∗ , then √ n ( (cid:98) β − β ∗ ) converges indistribution to a zero-mean normal random variable.
Revised: Qualification (LICQ) and Strict Complementarity (SC) hold at β ∗ , then √ n ( (cid:98) β − β ∗ ) converges in distribution to a zero-mean normal random variable.

Original: Further, (cid:98) β is efficient, meaning that there existno other regular asymptotically linear estimators that are asymptotically unbiased and have smallervariance.
Revised: Further, (cid:98) β is efficient, meaning that there exist no other regular asymptotically linear estimators that are asymptotically unbiased and have smaller variance.

Original: The above result immediately follows by Theorems 3.1 and 3.2 of [21], and gives conditions underwhich (cid:98) β is √ n -consistent and asymptotically normal.
Revised: The above result immediately follows by Theorems 3.1 and 3.2 of [21], and gives conditions under which (cid:98) β is √ n -consistent and asymptotically normal.

Original: Thus, asymptotically valid confidence intervalsand hypothesis tests can be constructed via the bootstrap.
Revised: Thus, asymptotically valid confidence intervals and hypothesis tests can be constructed via the bootstrap.

Original: LICQ and SC are regularity conditionscommonly found in the optimization literature [e.g., 40, 41]; see Appendix A for the formal definitions.
Revised: LICQ and SC are regularity conditions commonly found in the optimization literature [e.g., 40, 41]; see Appendix A for the formal definitions.

Original: The uniqueness of β ∗ simply requires that our basis functions are never perfectly collinear.
Revised: The uniqueness of β ∗ simply requires that our basis functions are never perfectly collinear.

Original: Once we obtain (cid:98) β 1 and (cid:98) β 0 through ( (cid:98) P µ a ), our proposed estimator for g ∗ is given by
Revised: Once we obtain (cid:98) β 1 and (cid:98) β 0 through ( (cid:98) P µ a ), our proposed estimator for g ∗ is given by

Original: [Equation]
Revised: [Equation]

Revised: Following the convention in the literature, we evaluate the performance of the above estimated treatment regime (cid:98) g in terms of the utility loss or regret relative to the maximum obtainable utility

Revised: U ( g ∗ ) , i.e., U ( g ∗ ) − U ( (cid:98) g ) , as will be analyzed in the following section in detail.

Original:  Regret Bounds and Fairness-Welfare Tradeoff
Revised:  Regret Bounds and Fairness-Welfare Tradeoff

Original: Here, we analyze thewelfare regret upper bounds and discuss its implication in incorporating fairnessinto optimal treatment regimes.
Revised: Here, we analyze the regret upper bounds and discuss its implication in incorporating fairness into optimal treatment regimes.

Original: To derive the upper bounds we require a margin condition, whichrestricts the probability that the two outcome regression functions get too close to each other in theneighborhood of µ 1 = µ 0 .
Revised: To derive the upper bounds we require a margin condition, which restricts the probability that the two outcome regression functions get too close to each other in the neighborhood of µ 1 = µ 0 .

Original: Definition 4.1 (Margin Condition) .
Revised: Definition 4.1 (Margin Condition) .

Original: For some α > 0 and for all t , we have that
Revised: For some α > 0 and for all t , we have that

Original: [Equation]
Revised: [Equation]

Original: The above margin condition is analogous to that used in [17, 22, 25, 26] as well as other problemsinvolving estimation of non-smooth parameters such as classification [2], clustering [24].
Revised: The above margin condition is analogous to that used in [17, 22, 25, 26] as well as other problems involving estimation of non-smooth parameters such as classification [2], clustering [24].

Revised: In the next lemma, adapting the comparison lemmas in [2], we give two useful inequalities between the regrets and the general L q risks of the corresponding outcome regression estimators proposed in the previous section under the margin condition.

Revised: Lemma 4.1.

Original: The following result shows that the welfare regret of the propose estimator (cid:98) g in (2) depends on boththe nuisance estimation accuracy and the level of fairness which we would like to attain.
Revised: Assume that the margin condition (3) holds with the margin exponent α > 0 , and let ∆ ≡ ∆( W ) = µ 1 ( W ) − µ 0 ( W ) and (cid:98) ∆ ≡ (cid:98) ∆( W ) = [Equation] β ⊤ 1 b ( W ) − (cid:98)

Revised: β ⊤ 0 b ( W ) .

Revised: Then we have

Revised: [Equation]

Revised: Further, for any 1 ≤ q < ∞ , we have

Revised: [Equation]

Revised: Based on the above lemma, the next theorem gives the upper bounds of the utility regret for our proposed estimator (cid:98) g in (2).

Original: Our result isasymptotic in the sample size n .
Revised: Our results are asymptotic in the sample size n .

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Assume  that the margin condition  holds with the margin exponent  αin (3) and∥ µ a ( W ) ∥ ∞ < ∞ .
Revised: Assume (A1) and (A2) and that the margin condition (3) holds with the margin exponent 0 < α < ∞ .

Original: Then under Assumptions (A1) and (A2), we have the asymptotic regret bounds
Revised: Also let

Original: [Equation]
Revised: [Equation]

Original: where
Revised: and define the remainder terms

Original: [Equation]
Revised: [Equation]

Original: and λ j ≥ 0 is the Lagrange multiplier associated with the j -th fairness constraint in ( P µ a ) .
Revised: where λ j ≥ 0 is the Lagrange multiplier associated with the j -th fairness constraint in ( P µ a ) [Equation]

Original: A sketch of the proof is given in Appendix B. Note that the same bound derived in the above theoremalso holds for P { (cid:98) g  ( W )̸
Revised: Then we have (i) U ( g ∗ ) − U ( (cid:98) g ) ≲ max a [Equation] a ( W )

Revised: [Equation]

Revised: [Equation]

Revised: (ii) Pr { (cid:98) g

Revised: ( W ) ̸

Revised: = g ∗ ( W ) } ≲

Revised: [Equation]

Revised: a

Revised: [Equation]

Revised: µ a ( W ) − (cid:98)

Revised: β ⊤ a

Revised: [Equation]

Revised: [Equation]

Revised: [Equation]

Revised: (iii) U ( g ∗ ) − U ( (cid:98) g )

Revised: ≲ max

Revised: [Equation]

Revised: (cid:13) (cid:13)(cid:13) µ

Revised: [Equation]

Revised: ( W )

Revised: [Equation]

Revised: ⊤ a

Revised: [Equation]

Original:  = g ∗ ( W ) } , a probability that (cid:98) g differs from the true optimal treatment policyg ∗ over a new observation.
Revised: ( W ) [Equation] A sketch of the proof is given in Appendix C. In (ii), Pr { (cid:98) g ( W ) ̸ = g ∗ ( W ) } indicates a probability that (cid:98) g differs from the true optimal treatment policy g ∗ over a new observation.

Revised: Theorem 4.1 shows that the utility regret depends on both the nuisance estimation accuracy and the level of fairness which we would like to attain.

Original: The bound  in Theorem 4.1 consists of three terms.
Revised: Specifically, each bound listed in Theorem 4.1 consists of three terms.

Original: The first term is an un-avoidable modelingerror minimized through least square estimation, which will vanish if µ a ( · ) lies in the functionspace spanned by the basis functions b ( · ) . 
Revised: The first term is an unavoidable modeling error minimized through least square estimation, which will vanish if µ a ( · ) lies in the function space spanned by the basis functions b ( · ) . From 4, one may further bound this modeling error to obtain a more interpretable form by noticing that

Original:  The second term, R 1 ,n , is a doubly robust second-orderterm that will be small if either π or µ a are estimated accurately.
Revised: [Equation] The second term, R 1 ,n , is a doubly robust second-order term that will be small if either π or µ a are estimated accurately.

Original: In nonparametric modeling, thecondition ∥ (cid:98) π − π ∥ 2 , P ∥ (cid:98) µ a − µ a ∥ 2 , P = O P ( n − 12 ) substantially lowers the bar for the nuisance estimatorconvergence rate, which allows much more flexible methods to be employed while still achieving √ nrates; for example, it suffices that these nuisance functions are estimated consistently at n 14 rates.
Revised: In nonparametric modeling, the condition ∥ (cid:98) π − π ∥ 2 , P ∥ (cid:98) µ a − µ a ∥ 2 , P = O P ( n − 12 ) substantially lowers the bar for the nuisance estimator convergence rate, which allows much more flexible methods to be employed while still achieving √ n rates; for example, it suffices that these nuisance functions are estimated consistently at n 14 rates.

Original: The third term, R 2 , has particularly important implications.
Revised: The third term, R 2 , has particularly important implications.

Original: It measures the imbalances in covariatedistributions with respect to the sensitive feature, which is closely related to the level of unfairness inthe optimal treatment policy; the larger the imbalances, the more likely the estimated optimal policiesare unfair.
Revised: It measures the imbalances in covariate distributions with respect to the sensitive feature, which is closely related to the level of unfairness in the optimal treatment policy; the larger the imbalances, the more likely the estimated optimal policies are unfair.

Original: If we use small values of the tolerance level δ j so that the optimum β ∗ is constrainedby the j -th fairness constraint, then the corresponding Lagrange multiplier, λ j , is positive.
Revised: If we use small values of the tolerance level δ j so that the optimum β ∗ is constrained by the j -th fairness constraint, then the corresponding Lagrange multiplier, λ j , is positive.

Original: On thecontrary, if we loosen the standard by using large values of δ j so that the j -th fairness constraint doesnot constrain β ∗ , λ j is set to zero.
Revised: On the contrary, if we loosen the standard by using large values of δ j so that the j -th fairness constraint does not constrain β ∗ , λ j is set to zero.

Original: Therefore, our attempts toward making optimal treatment policiesmore fair may lead to an additional welfare loss (regret) relative to the universally maximum feasiblewelfare R ( g ∗ ) .
Revised: Therefore, our attempts toward making optimal treatment policies more fair may lead to an additional welfare loss (regret) relative to the universally maximum feasible utility U ( g ∗ ) .

Original: In other words, there is a tradeoff between fairness in the optimal treatment regimeand the maximum obtainable welfare.
Revised: In other words, there is a tradeoff between fairness in the optimal treatment regime and the maximum utility.

Original: In short, Theorem 4.1 implies that although the proposed approach has considerably reduced theburden on nuisance estimation, regardless how accurately we estimate the nuisance components thereis a price that comes with imposing fairness constraints for the optimal treatment regime to achievethe desired fairness level.
Revised: In short, Theorem 4.1 implies that although the proposed approach has considerably reduced the burden on nuisance estimation, regardless how accurately we estimate the nuisance components there is a price that comes with imposing fairness constraints for the optimal treatment regime to achieve the desired fairness level.

Original:  Discussion
Revised:  Discussion

Original: We propose a new framework for fair and robust estimation of optimal treatment regimes.
Revised: We propose a new framework for fair and robust estimation of optimal treatment regimes.

Original: Our methodis easily implementable and allows practitioners to flexibly incorporate various fairness constraints tomeet the desired level of fairness.
Revised: Our method is easily implementable and allows practitioners to flexibly incorporate various fairness constraints to meet the desired level of fairness.

Original: This affords new opportunities to leverage the recent developmentin algorithmic fairness for optimal treatment regimes.
Revised: This affords new opportunities to leverage the recent development in algorithmic fairness for optimal treatment regimes.

Original: There are two important messages in our regret bound analysis.
Revised: There are two important messages in our regret bound analysis.

Original: First, the proposed estimator is robustagainst model misspecification and allow to use more flexible nonparametric methods while stillachieving √ n convergence rates to the maximum obtainable welfare.
Revised: First, the proposed estimator is robust against model misspecification and allow to use more flexible nonparametric methods while still achieving √ n convergence rates to the maximum utility.

Original: Second, there is a tradeoffbetween fairness and the maximum obtainable welfare of the 
Revised: Second, there is a tradeoff between fairness and the maximum utility, which is independent of accuracy of the nuisance estimation.

Original:  A Formal Definitions of the Regularity Conditions
Revised: APPENDIX A Formal Definitions of the Regularity Conditions

Original: First, for a feasible point ¯ β ∈ C fair we define the active index set.
Revised: First, for a feasible point ¯ β ∈ C fair we define the active index set.

Original: Definition A.1 (Active set) .
Revised: Definition A.1 (Active set) .

Original: For ¯ β ∈ C fair , we define the active index set J 0 by
Revised: For ¯ β ∈ C fair , we define the active index set J 0 by

Original: [Equation]
Revised: [Equation]

Original: In what follows, we define LICQ and SC with respect to ( P µ a ).
Revised: In what follows, we define LICQ and SC with respect to ( P µ a ).

Original: Definition A.2 (LICQ) .
Revised: Definition A.2 (LICQ) .

Original: Linear independence constraint qualification (LICQ) is satisfied at ¯ β ∈ S ifthe vectors ∇ β g j (¯ β ) , j ∈ J 0 (¯ β ) are linearly independent.
Revised: Linear independence constraint qualification (LICQ) is satisfied at ¯ β ∈ S if the vectors ∇ β g j (¯ β ) , j ∈ J 0 (¯ β ) are linearly independent.

Original: Definition A.3 (SC) .
Revised: Definition A.3 (SC) .

Original: Let L ( β, γ ) be the Lagrangian.
Revised: Let L ( β, γ ) be the Lagrangian.

Original: Strict Complementarity (SC) is satisfied at¯ β ∈ S
Revised: Strict Complementarity (SC) is satisfied at ¯ β ∈ S

Original: if, with multipliers ¯ γ j ≥ 0 , j ∈ J 

Revised: if, with multipliers ¯ γ j ≥ 0 , j ∈ J

Original: (¯ β ) , the Karush-Kuhn-Tucker (KKT) condition
Revised: (¯ β ) , the Karush-Kuhn-Tucker (KKT) condition

Original: [Equation]
Revised: [Equation]

Original: is satisfied such that ¯ γ j > 0 , ∀ j ∈ J 0 (¯ β ) .
Revised: is satisfied such that ¯ γ j > 0 , ∀ j ∈ J 0 (¯ β ) .

Original: LICQ is arguably one of the most widely-used constraint qualifications that admit the first-ordernecessary conditions.
Revised: LICQ is arguably one of the most widely-used constraint qualifications that admit the first-order necessary conditions.

Original: SC means that if the j -th inequality constraint is active then the correspondingdual variable is strictly positive, so exactly one of them is zero for each 1 ≤ j ≤ m .
Revised: SC means that if the j -th inequality constraint is active then the corresponding dual variable is strictly positive, so exactly one of them is zero for each 1 ≤ j ≤ m .

Original: SC is widelyused in the optimization literature, particularly in the context of parametric optimization [e.g., 40, 41].
Revised: SC is widely used in the optimization literature, particularly in the context of parametric optimization [e.g., 40, 41].

Original: BSketch of the Proof of Theorem 4. 
Revised: B Proof of Lemma 4.

Original: Proof.
Revised: Proof.

Revised: The proof mimics the proofs of Lemma 5.1 and Lemma 5.2 in [2].

Revised: To show the first inequality, note that

Revised: [Equation]

Revised: [Table]

Original: Under the margin condition, by Lemma 1 of [18] and the Cauchy–Schwarz inequality, wehave
Revised: where the first inequality follows by Lemma 1 of [18] and the last by the margin condition [Table] Next, for any t > 0 we have [Table]

Original: [Equation]
Revised: [Equation]

Revised: [Table]

Revised: [Equation]

Revised: [Table]

Revised: [Equation]

Revised: [Table]

Revised: [Equation]

Revised: where the third inequality follows by the Hölder and Markov inequalities and the last by the margin condition.

Original: We let β ∗ a denote an optimal solution to the following unconstrained optimization problem
Revised: Now, the RHS in the last display is minimized when t = O [Equation] ∆

Original: [Equation] 
Revised: [Equation] , yielding [Equation] C Proof of Theorem 4.1 (Sketchy)

Original: and let (cid:101) β
Revised: Proof.

Original: a be an optimal solution to ( P µ a ).
Revised: It suffices to show the results in the part (i).

Original: Then ∀ a ,
Revised: By the first inequality in Lemma 4.1, we have

Original: [Equation]
Revised: [Equation]

Original: The unavoidable modeling error can be upper bounded in a more interpretable form by noticing
Revised: Recall that β ∗ a denotes an optimal solution to the following unconstrained optimization problem

Original: [Equation]
Revised: [Equation]

Original: from the definition of the least square estimator. Further
Revised: and let (cid:101) β a be an optimal solution to ( P µ a ). Then ∀ a , by the triangle and Cauchy–Schwarz inequalities,

Original: [Equation]
Revised: [Equation]

Original:  one may show that
Revised: Next, one may show that

Original: [Equation]
Revised: [Equation]

Original: by noticing that (4) and ( P µ a ) can be viewed as the same form of a parametrized program (afterwriting (4) as a Lagrange dual form) and then applying the stability results [41, Chapter 6].
Revised: by noticing that (4) and ( P µ a ) can be viewed as the same form of a parametrized program (after writing (4) as a Lagrange dual form) and then applying the stability results [41, Chapter 6].

Original: Finally,by Theorem 3.1, it follows that
Revised: Further, by Theorem 3.1, it follows that

Original: [Equation]
Revised: [Equation]

Original: Putting the pieces together, we obtain the desired result.
Revised: Since 0 < α < ∞ , we obtain the desired result by putting the pieces together due to Minkowski’s inequality.

Original: Finally, it immediately followsfrom Theorem 4.1 by noting that
Revised: Then the part (ii) immediately follows by the fact that

Original: [Equation]
Revised: [Equation]

Original: where the first inequality follows by Lemma 1 of [18].