Original: Calibrated Data-Dependent Constraints with Exact Satisfaction Guarantees
Revised: Calibrated Data-Dependent Constraints with Exact Satisfaction Guarantees

Original: Abstract
Revised: Abstract

Original: We consider the task of training machine learning models with data-dependent constraints.
Revised: We consider the task of training machine learning models with data-dependent constraints.

Original: Such constraints often arise as empirical versions of expected value constraints that enforce fairness or stability goals.
Revised: Such constraints often arise as empirical versions of expected value constraints that enforce fairness or stability goals.

Original: We reformulate data-dependent constraints so that they are calibrated : enforcing the reformulated constraints guarantees that their expected value counterparts are satisfied with a user-prescribed probability.
Revised: We reformulate data-dependent constraints so that they are calibrated : enforcing the reformulated constraints guarantees that their expected value counterparts are satisﬁed with a user-prescribed probability.

Original: The resulting optimization problem is amendable to standard stochastic optimization algorithms and we demonstrate the efficacy of our method on a fairness-sensitive classification task where we wish to guarantee the classifier’s fairness (at test time)
Revised: The resulting optimization problem is amendable to standard stochastic optimization algorithms and we demonstrate the efﬁcacy of our method on a fairness-sensitive classiﬁcation task where we wish to guarantee the classiﬁer’s fairness (at test time)

Original:  Motivation
Revised:  Motivation

Original: In machine learning (ML) practice, accuracy is often only one of many training objectives.
Revised: In machine learning (ML) practice, accuracy is often only one of many training objectives.

Original: For example, algorithmic fairness considerations may require a credit scoring system to perform comparably on men and women.
Revised: For example, algorithmic fairness considerations may require a credit scoring system to perform comparably on men and women.

Original: Here are a few other examples.
Revised: Here are a few other examples.

Original: Churn rate and stability The churn rate of an ML model compared to another model is the fraction of samples on which the predictions of the two models differ [21, 29].
Revised: Churn rate and stability The churn rate of an ML model compared to another model is the fraction of samples on which the predictions of the two models differ [21, 30].

Original: In ML practice, one may wish to control the churn rate between a new model and its predecessor because a high churn rate can disorient users and downstream system components.
Revised: In ML practice, one may wish to control the churn rate between a new model and its predecessor because a high churn rate can disorient users and downstream system components.

Original: One way of training models with small churn is to enforce a churn rate constraint during training.
Revised: One way of training models with small churn is to enforce a churn rate constraint during training.

Original: Precision, recall, etc.
Revised: Precision, recall, etc.

Original: Classification and information retrieval models must often balance precision and recall.
Revised: Classiﬁcation and information retrieval models must often balance precision and recall.

Original: To train such models, practitioners carefully trade off one metric for the other by optimizing for one metric subject to constraints on the other.
Revised: To train such models, practitioners carefully trade off one metric for the other by optimizing for one metric subject to constraints on the other.

Original: Resource constraints Practitioners sometimes wish to control how often a classifier predicts a certain class due to budget or resource constraints.
Revised: Resource constraints Practitioners sometimes wish to control how often a classiﬁer predicts a certain class due to budget or resource constraints.

Original: For example, a company that uses ML to select customers for a targeted offer may wish to constrain the fraction of customers selected for the offer.
Revised: For example, a company that uses ML to select customers for a targeted offer may wish to constrain the fraction of customers selected for the offer.

Original: Another prominent example of a stochastic optimization problem with resource constraints is the newsvendor problem, which we come back to in section 4.
Revised: Another prominent example of a stochastic optimization problem with resource constraints is the newsvendor problem, which we come back to in section 4.

Original: Unlike constraints on the structure of model parameters ( e.g. , sparsity), the constraints encoding the preceding training objectives are data-dependent .
Revised: Unlike constraints on the structure of model parameters ( e.g. , sparsity), the constraints encoding the preceding training objectives are data-dependent .

Original: This leads to the issue of constraint generalization : whether the constraints generalize out-of-sample.
Revised: This leads to the issue of constraint generalization : whether the constraints generalize out-of-sample.

Original: For example, if a classifier is trained to have comparable accuracy on two subpopulations in the training data, will it also have comparable accuracy on samples from the two subpopulations at test time?
Revised: For example, if a classiﬁer is trained to have comparable accuracy on two subpopulations in the training data, will it also have comparable accuracy on samples from the two subpopulations at test time?

Original: In this paper, we consider the out-of-sample generalization of expected-value constraints.
Revised: In this paper, we consider the out-of-sample generalization of expected-value constraints.

Original: To keep things simple, consider a stochastic optimization problem with a single expected-value constraint:
Revised: To keep things simple, consider a stochastic optimization problem with a single expected-value constraint:

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: where Θ is a (finite-dimensional) parameter space, f, g : Θ × Z
Revised: where ⇥ is a (ﬁnite-dimensional) parameter space, f, g : ⇥

Original: → R are (known) cost and constraint functions, and Z ∈ Z is a random variable that represents a sample.
Revised: ⇥ Z ! R are (known) cost and constraint functions, and Z 2 Z is a random variable that represents a sample.

Original: The distribution of Z is unknown, so we cannot solve (1.1) directly.
Revised: The distribution of Z is unknown, so we cannot solve (1.1) directly.

Original: Instead, we obtain IID training samples { Z i } ni =1 from the true underlying distribution P 0 and solve the empirical version of (1.1):(cid:98) θ n ∈
Revised: Instead, we obtain IID training samples { Z i } ni =1 from the true underlying distribution P 0 and solve the empirical version of (1.1):

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: The estimator (cid:98) θ n (of θ ⋆ ) is guaranteed to satisfy the empirical constraint ( i
Revised: The estimator b ✓ n (of ✓ ? ) is guaranteed to satisfy the empirical constraint ( i.e. , 1 n P ni

Original: [Equation]
Revised: [Equation]

Original: n (cid:80) ni =1 g ( (cid:98) θ n ; Z i ) ≤ 0 ), but it is unclear whether (cid:98) θ n satisfies the actual (population) constraint E P 0 (cid:2)
Revised: g ( b ✓ n ; Z i )  0 ), but it is unclear whether b ✓ n satisﬁes the actual (population) constraint E P⇥ g

Original: g ( θ ; Z ) (cid:3) ≤ 0 .
Revised: ( ✓ ; Z ) ⇤  0 .

Original: As we shall see, under standard assumptions on (1.1), (cid:98) θ n only satisfies the actual constraint with probability approaching 12 (see corollary 2.2).
Revised: As we shall see, under standard assumptions on (1.1), b ✓ n only satisﬁes the actual constraint with probability approaching 12 (see corollary 2.2).

Original: This is especially problematic for constraints that encode algorithmic fairness goals.
Revised: This is especially problematic for constraints that encode algorithmic fairness goals.

Original: For example, the 80% rule published by the US Equal Employment
Revised: For example, the 80% rule published by the US Equal Employment

Original: Opportunity Commission, interpreted in the machine learning context, requires the rate at which a classifier predicts the advantaged label in minority groups to be at least 80% of the rate at which the classifier predicts the advantaged label in the majority group [3].
Revised: Opportunity Commission, interpreted in the machine learning context, requires the rate at which a classiﬁer predicts the advantaged label in minority groups to be at least 80% of the rate at which the classiﬁer predicts the advantaged label in the majority group [3].

Original: In this paper, we propose a distributionally robust version of (1.2) that guarantees the actual constraint E 
Revised: In this paper, we propose a distributionally robust version of (1.2) that guarantees the actual constraint E P [Equation]

Original: P 0 (cid:2) g ( θ ; Z ) (cid:3) ≤ 0 will be satisfied with probability 1 − α :
Revised:  0 will be satisﬁed with probability 1  ↵ :

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: where D φ is a φ -divergence (see section 2 for details), (cid:98) P n is the empirical distribution of the training samples, and √ ρ α is the 1 − α quantile of a standard normal random variable. 
Revised: where D ' is a ' -divergence (see section 2 for details), b P n is the empirical distribution of the training samples, and p ⇢ ↵ is the 1  ↵ quantile of a standard normal random variable. More concretely, we show that [Equation]

Original: More concretely, we show that (cid:98) θ n achieves asymptotically exact constraint satisfaction
Revised: achieves asymptotically exact constraint satisfaction

Original: [Equation]
Revised: [Equation]

Original: Here the inner expectation is with respect to Z ; the outer probability is with respect to the training samples { Z i } ni =1 .
Revised: Here the inner expectation is with respect to Z ; the outer probability is with respect to the training samples { Z i } ni =1 .

Original: Three desirable properties of (1.3) areexact constraint satisfaction: If
Revised: Three desirable properties of (1.3) are

Original: the actual probability

Original: of constraint

Original:  satisfaction exceeds 1 − α , then the method is too conservative
Revised: exact constraint satisfaction: If the actual probability of constraint satisfaction exceeds 1  ↵ , then the method is too conservative.

Original: This may (unnecessarily) increase the cost of 
Revised: This may (unnecessarily) increase the cost of the model.

Original: the model. By picking ρ α in (1.3) carefully, constraints are satisfied with asymptotically exact probability 1 − α .
Revised: By picking ⇢ ↵ in (1.3) carefully, constraints are satisﬁed with asymptotically exact probability 1  ↵ .

Original: computationally efficiency: As we shall see, the computational cost of solving (1. is comparable to the cost of solving distributionally robust sample average approximation (SAA) problems.
Revised: computationally efﬁcient: As we shall see, the computational cost of solving (1.3) is comparable to the cost of solving distributionally robust sample average approximation (SAA) problems.

Original: pivotal: There are no nuisance parameters to estimate ( e.g. , asymptotic
Revised: pivotal: There are no nuisance parameters to estimate ( e

Original:  variances) in (1.3).
Revised: g. , asymptotic variances) in (1.3).

Original: The user merely needs to look up the correct quantile of the standard normal distribution for their desired level of constraint generalization.
Revised: The user merely needs to look up the correct quantile of the standard normal distribution for their desired level of constraint generalization.

Original: The rest of this paper is organized as follows.
Revised: The rest of this paper is organized as follows.

Original: In Section 2, we develop method, theory, and algorithm for stochastic optimization problems with single constraint.
Revised: In Section 2, we develop method, theory, and algorithm for stochastic optimization problems with single constraint.

Original: In Section 3, we extend our method, theory, and algorithm to stochastic optimization problems with multiple constraints.
Revised: In Section 3, we extend our method, theory, and algorithm to stochastic optimization problems with multiple constraints.

Original: In Section 4, we validate our theory by simulating a resource-constrained newsvendor problem.
Revised: In Section 4, we validate our theory by simulating a resource-constrained newsvendor problem.

Original: In Section 5, we demonstrate the efficacy of our method by using it to train an algorithmically fair income classifier.
Revised: In Section 5, we demonstrate the efﬁcacy of our method by using it to train an algorithmically fair income classiﬁer.

Original: In addition, we show how to apply our method to a fairness constrained learning problem and discuss two practical considerations for fair ML application scenarios.
Revised: In addition, we show how to apply our method to a fairness constrained learning problem and discuss two practical considerations for fair ML application scenarios.

Original: Finally, we summarize our work in
Revised: Finally, we summarize our work in

Original: Section 6 and point out an interesting avenue of future work.
Revised: Section 6 and point out an interesting avenue of future work.

Original: 1.1 Related work
Revised: 1.1 Related work

Original: The closest work to our work is [26].
Revised: The closest work to our work is [27].

Original: They seek to pick a (data-dependent) uncertainty set U such that
Revised: They seek to pick a (data-dependent) uncertainty set U such that

Original: [Equation]
Revised: [Equation]

Original: This condition is stronger than necessary: we only require
Revised: This condition is stronger than necessary: we only require

Original: [Equation]
Revised: [Equation]

Original: where (cid:98) θ n is a (data-dependent) estimator (not necessarily (1.2) or (1.3)).
Revised: where b ✓ n is a (data-dependent) estimator (not necessarily (1.2) or (1.3)).

Original: [26] study (asymptotic) constraint satisfaction (1.4) for all deterministic objective functions (see [26], §1.1 for details).
Revised: [27] study (asymptotic) constraint satisfaction (1.4) for all deterministic objective functions (see [27], §1.1 for details).

Original: They advocate picking a KL divergence ball with radius that depends on the excursion probability of a certain χ 2 process.
Revised: They advocate picking a KL divergence ball with radius that depends on the excursion probability of a certain  2 process.

Original: Another closely related line of work is on data-splitting approaches for ensuring constraint generalization [36, 7].
Revised: Another closely related line of work is on data-splitting approaches for ensuring constraint generalization [37, 7].

Original: At a high level, they split the training data into a training and validation subsets and use the validation subset to tune models trained on the training subset so that they satisfy the constraints.
Revised: At a high level, they split the training data into a training and validation subsets and use the validation subset to tune models trained on the training subset so that they satisfy the constraints.

Original: Although (computationally) simple and intuitive, their approach does not allow users to precisely control the constraint violation probability.
Revised: Although (computationally) simple and intuitive, their approach does not allow users to precisely control the constraint violation probability.

Original: [26] is the latest in a line of work on distributionally robust optimization (DRO) that show the optimal value of DRO problems
Revised: [27] is the latest in a line of work on distributionally robust optimization (DRO) that show the optimal value of DRO problems

Original: [Equation]
Revised: [Equation]

Original: where U is a (data-dependent) uncertainty set of probability distributions, are upper confidence bounds for the optimal values of stochastic optimization problems.
Revised: where U is a (data-dependent) uncertainty set of probability distributions, are upper conﬁdence bounds for the optimal values of stochastic optimization problems.

Original: Common choices of uncertainty sets in DRO include uncertainty sets defined by moment or support constraints [6, 12, 22], φdivergences [4, 25, 30], and Wasserstein distances [33, 5, 18, 27, 34].
Revised: Common choices of uncertainty sets in DRO include uncertainty sets deﬁned by moment or support constraints [6, 12, 22], ' - divergences [4, 26, 31], and Wasserstein distances [34, 5, 18, 28, 35].

Original: This line of work is motivated by Owen’s seminal work on empirical likelihood [31].
Revised: This line of work is motivated by Owen’s seminal work on empirical likelihood [32].

Original: In recent work, [25, 15] show that the optimal value of DRO problems with empirical likelihood uncertainty sets leads to asymptotically exact upper confidence bounds for the optimal value of stochastic optimization problems ([15] consider more general φ -divergence uncertainty sets).
Revised: In recent work, [26, 15] show that the optimal value of DRO problems with empirical likelihood uncertainty sets leads to asymptotically exact upper conﬁdence bounds for the optimal value of stochastic optimization problems ([15] consider more general ' -divergence uncertainty sets).

Original: [5] establish similar coverage results for Wasserstein uncertainty sets.
Revised: [5] establish similar coverage results for Wasserstein uncertainty sets.

Original: Our work is also closely related to the work on the variance regularization properties of DRO [30], which uses DRO to approximate the variance regularization cost function (see (2.4)).
Revised: Our work is also closely related to the work on the variance regularization properties of DRO [31], which uses DRO to approximate the variance regularization cost function (see (2.4)).

Original: [20] establish similar results for Wasserstein DRO.
Revised: [20] establish similar results for Wasserstein DRO.

Revised: Lastly, we relate our work to the literature on chance constrained optimization (see [24] and the references therein).

Revised: The general goal of chance constrained optimization is to minimize a loss function subject to the probability of satisfying uncertain constraints is above a prescribed level.

Revised: While our methods reformulate expected value constraints and we show that the solution of the reformulated problem enjoys an asymptotically exact probabilistic guarantees of constraint satisfaction.

Revised: In addition, the data-dependent constraints in our work are also unknown in practice, which differs from the common setup in the chance constrained optimization literature.

Original:  Single expected value constraint
Revised:  Single expected value constraint

Original: We motivate (1.3) by considering a few alternatives.
Revised: We motivate (1.3) by considering a few alternatives.

Original: First, we note that the results later in this section show that (1.2) violates the actual constraint in (1.1) approximately half the time (see corollary 2.2).
Revised: First, we note that the results later in this section show that (1.2) violates the actual constraint in (1.1) approximately half the time (see corollary 2.2).

Original: The most straightforward modification of (1.2) to ensure (cid:98) θ n satisfies the (actual) constraint E 
Revised: The most straightforward modiﬁcation of (1.2) to ensure b ✓ n satisﬁes the (actual) constraint E P⇥ g [Equation]

Original: P 0 (cid:2) g ( θ ; Z ) (cid:3) ≤ 0 is to add a “margin” in (1.3); i.e. enforce the constraint
Revised: ✓ ; Z ) ⇤  0 is to add a “margin” in (1.3); i.e. enforce the constraint

Original: [Equation]
Revised: [Equation]

Original: in (1.2).
Revised: in (1.2).

Original: If we pick the slack term ϵ n such that
Revised: If we pick the slack term ✏ n such that

Original: [Equation]
Revised: [Equation]

Original: then it is not hard to check that the resulting (cid:98) θ n satisfies the (actual) constraint with probability greater than 1 − α [35, 28].
Revised: then it is not hard to check that the resulting b ✓ n satisﬁes the (actual) constraint with probability greater than 1  ↵ [36, 29].

Original: However, this approach is most likely conservative because the constraint is unnecessarily stringent for θ ’s such that 1 n(cid:80) ni
Revised: However, this approach is most likely conservative because the constraint is unnecessarily stringent for ✓ ’s such that 1 n

Original:  =1 g ( θ ; Z i ) is less variable.
Revised: P ni =1 g ( ✓ ; Z i ) is less variable.

Original: It is also not pivotal: ϵ n is often set using bounds from (uniform) concentration inequalities, which typically depend on unknown problem parameters.
Revised: It is also not pivotal: ✏ n is often set using bounds from (uniform) concentration inequalities, which typically depend on unknown problem parameters.

Original: To relax the empirical constraint in a way that adapts to the variability of the empirical constraints, we replace the uniform margin in (2.1) with a parameter-dependent margin:
Revised: To relax the empirical constraint in a way that adapts to the variability of the empirical constraints, we replace the uniform margin in (2.1) with a parameter-dependent margin:

Original: [Equation]
Revised: [Equation]

Original: where z α is the 1 − α quantile of a standard normal random variable and (cid:98) σ 2 ( θ ) is an estimate of the asymptotic variance of g ( θ ; Z ) .
Revised: where z ↵ is the 1  ↵ quantile of a standard normal random variable and b  2 ( ✓ ) is an estimate of the asymptotic variance of g ( ✓ ; Z ) .

Original: We recognize the (parameter-dependent) margin as (a multiple of) the standard error of the empirical constraint.
Revised: We recognize the (parameter-dependent) margin as (a multiple of) the standard error of the empirical constraint.

Original: It is possible to show that enforcing (2.2) achieves asymptotically exact constraint generalization (1.4) [26].
Revised: It is possible to show that enforcing (2.2) achieves asymptotically exact constraint generalization (1.4) [27].

Original: The main issue with this method is it is not amenable to standard stochastic optimization algorithms.
Revised: The main issue with this method is it is not amenable to standard stochastic optimization algorithms.

Original: In particular, even if the original constraint in (1.2) is convex, (2.2) is generally non-convex.
Revised: In particular, even if the original constraint in (1.2) is convex, (2.2) is generally non-convex.

Original: Another issue is that it is not pivotal: the user must estimate the asymptotic variance of g ( θ ; Z ) .
Revised: Another issue is that it is not pivotal: the user must estimate the asymptotic variance of g ( ✓ ; Z ) .

Original: To overcome these two issues, we consider a distributionally robust version of (1.2); i.e. enforcing
Revised: To overcome these two issues, we consider a distributionally robust version of (1.2); i.e. enforcing

Original: [Equation]
Revised: [Equation]

Original: where D φ ( P ∥ Q ) ≜ (cid:82) φ ( dPdQ ) dQ is a φ -divergence.
Revised: where D ' ( P k Q ) , R ' ( dPdQ ) dQ is a ' -divergence.

Original: Common choices of φ include φ ( t ) =
Revised: Common choices of ' include ' ( t ) =

Original: ( t − 1)(which leads to the χ 2 -divergence) and φ ( t ) =
Revised: ( t  1)(which leads to the  2 -divergence) and ' ( t ) =

Original: − log t + t − 1 (which leads to the Kullback-Leibler divergence).
Revised:  log t + t  1 (which leads to the Kullback-Leibler divergence).

Original: Although there are many other choices for the uncertainty set in (2.3), we pick an φ -divergence ball because (i) (2.3) with an φ -divergence ball is asymptotically equivalent to (2.2):
Revised: Although there are many other choices for the uncertainty set in (2.3), we pick an ' -divergence ball because (i) (2.3) with an ' -divergence ball is asymptotically equivalent to (2.2):

Original: [Equation]
Revised: [Equation]

Original: and (ii) it leads to pivotal uncertainty sets.
Revised: and (ii) it leads to pivotal uncertainty sets.

Original: For theoretical analysis, we always use φ ( t ) =
Revised: For theoretical analysis, we always use ' ( t ) =

Original: ( t − 1)and χ 2 -divergence in the remainder of this paper.
Revised: ( t  1)and  2 -divergence in the remainder of this paper.

Original: Before we state the asymptotically exact constraint satisfaction property of (1.3) rigorously, we describe our assumptions on the problem.
Revised: Before we state the asymptotically exact constraint satisfaction property of (1.3) rigorously, we describe our assumptions on the problem.

Original: smoothness and concentration: f and g are twice continuously differentiable with respect to θ , and f ( θ ⋆ ; Z ) , ∇ f
Revised: smoothness and concentration: f and g are twice continuously differentiable with respect to ✓ , and f ( ✓ ?

Original:  ( θ ⋆ ; Z ) , g ( θ ⋆ ; Z ) , ∇ g ( θ ⋆ ; Z ) are sub-Gaussian random variables.
Revised: ; Z ) , r f ( ✓ ? ; Z ) , g ( ✓ ? ; Z ) , r g ( ✓ ? ; Z ) are sub-Gaussian random variables.

Original: uniqueness: the stochastic optimization problem with a single expected value constraint (1.1) has a unique optimal primal-dual pair ( θ ⋆ , λ ⋆ ) , and θ ⋆ belongs to the interior of the compact set Θ .
Revised: uniqueness: the stochastic optimization problem with a single expected value constraint (1.1) has a unique optimal primal-dual pair ( ✓ ? ,  ? ) , and ✓ ? belongs to the interior of the compact set ⇥ .

Original: strict complementarity: λ ⋆ > 0 . 
Revised: strict complementarity:  ? > 0 . 4.

Original: positive definiteness: The Hessian of the Lagrangian evaluated at ( θ ⋆ , λ ⋆ ) is positive definite.
Revised: positive deﬁniteness: The Hessian of the Lagrangian evaluated at ( ✓ ? ,  ? ) is positive deﬁnite.

Original: The preceding assumptions are not the most general, but they are easy to interpret.
Revised: The preceding assumptions are not the most general, but they are easy to interpret.

Original: The smoothness conditions on f and g with respect to θ , the concentration conditions of f ( θ ⋆ ; Z ) and g ( θ ⋆ ; Z ) , and the uniqueness condition facilitate the use of standard tools from asymptotic statistics to study the large sample properties of the constraint value.
Revised: The smoothness conditions on f and g with respect to ✓ , the concentration conditions of f ( ✓ ? ; Z ) and g ( ✓ ? ; Z ) , and the uniqueness condition facilitate the use of standard tools from asymptotic statistics to study the large sample properties of the constraint value.

Original: The strict complementarity condition rules out problems in which the constraint is extraneous; i.e. problems in which the unconstrained minimum coincides with the constrained minimum.
Revised: The strict complementarity condition rules out problems in which the constraint is extraneous; i.e. problems in which the unconstrained minimum coincides with the constrained minimum.

Original: We are ready to state the asymptotically exact constraint satisfaction property of (1.3) rigorously.
Revised: We are ready to state the asymptotically exact constraint satisfaction property of (1.3) rigorously.

Original: The main technical result characterizes the limiting distribution of the constraint value.
Revised: The main technical result characterizes the limiting distribution of the constraint value.

Original: Theorem 2.1.
Revised: Theorem 2.1.

Original: Let (cid:98) θ n be an optimal solution of (1.3) converging in probability as n → ∞ to θ ⋆ .
Revised: Let b ✓ n be an optimal solution of (1.3) converging in probability as n ! 1 to ✓ ? .

Original: Under the standing assumptions, we have
Revised: Under the standing assumptions, we have

Original: [Equation]
Revised: [Equation]

Original: We translate this result on the constraint value to a result on constraint generalization.
Revised: We translate this result on the constraint value to a result on constraint generalization.

Original: Corollary 2.2.
Revised: Corollary 2.2.

Original: Let √ ρ α be the 1 − α quantile of a standard normal random variable.
Revised: Let p ⇢ ↵ be the 1  ↵ quantile of a standard normal random variable.

Original: Under the conditions of theorem 2.1, we have
Revised: Under the conditions of theorem 2.1, we have

Original: [Equation]
Revised: [Equation]

Original: where U ∼ N (0 , 1) is a standard Gaussian random variable.
Revised: where U ⇠ N (0 , 1) is a standard Gaussian random variable.

Original: From theorem 2.1 and corollary 2.2 (see proofs in Appendix A), we find that
Revised: From theorem 2.1 and corollary 2.2 (see proofs in Appendix A), we ﬁnd that

Original: picking ρ α = 

Revised: picking ⇢ ↵ =

Original: ( i
Revised: ( i

Original: e. , equivalently solving (1.2)) leads to a constraint violation probability that approaches 12 in the large sample limit.
Revised: e. , equivalently solving (1.2)) leads to a constraint violation probability that approaches 12 in the large sample limit.

Original: the relation between the mean and variance of the limiting distribution of the constraint value in Theorem 2.1 allows us to pick ρ α in a pivotal way ( i
Revised: the relation between the mean and variance of the limiting distribution of the constraint value in Theorem 2.1 allows us to pick ⇢ ↵ in a pivotal way ( i

Original: e. does not depend on nuisance parameters).
Revised: e. does not depend on nuisance parameters).

Original: 2.1 Stochastic approximation 
Revised: 2.1 Stochastic approximation for (1.3)

Original: for (1.3)

Original: In the rest of this section, we derive a stochastic optimization algorithm to solve (1.3) efficiently.
Revised: In the rest of this section, we derive a stochastic optimization algorithm to solve (1.3) efﬁciently.

Original: As we shall see, the computational cost of this algorithm is comparable to the cost of solving a DRO problem.
Revised: As we shall see, the computational cost of this algorithm is comparable to the cost of solving a DRO problem.

Original: The key insight is that the robust constraint function has a dual form (see Appendix J): 
Revised: The key insight is that the robust constraint function has a dual form (see Appendix J): [Equation] P

Original: [Equation]
Revised: [Equation]

Revised: D '

Original:  where φ ∗ ( s ) ≜ sup t { st − φ ( t ) } is the convex conjugate of φ .
Revised: [Equation] P k [Equation] where ' ⇤ ( s ) , sup t { st  ' ( t ) } is the convex conjugate of ' .

Original: As we use χ 2 -squared divergence and φ ( t )
Revised: As we use  2 -squared divergence and ' ( t )

Original: = ( t − 1) 2 , the corresponding φ ∗ ( s ) =
Revised: = ( t  1) 2 , the corresponding ' ⇤ ( s ) =

Original: s 2 4 + s .
Revised: s 2 4 + s .

Original: The Lagrangian of (1.3) is
Revised: The Lagrangian of (1.3) is

Original: [Equation]
Revised: [Equation]

Original: We see that evaluating the dual function inf θ L ( θ, λ ) (at a fixed λ ) entails solving a stochastic optimization problem that is suitable for stochastic approximation.
Revised: We see that evaluating the dual function inf ✓ L ( ✓ ,  ) (at a ﬁxed  ) entails solving a stochastic optimization problem that is suitable for stochastic approximation.

Original: This suggests a dual ascent algorithm for solving (1.3):
Revised: This suggests a dual ascent algorithm for solving (1.3):

Original: 1. evaluate the dual function at λ t by solving a stochastic optimization problem 2.
Revised: 1. evaluate the dual function at  t by solving a stochastic optimization problem.

Original:  update λ t with a dual ascent step.
Revised: 2. update  t with a dual ascent step.

Original: We summarize this algorithm in Algorithm 1.
Revised: We summarize this algorithm in Algorithm 1.

Original: The main cost of Algorithm 1 is incurred in the third line: evaluating the dual function.
Revised: The main cost of Algorithm 1 is incurred in the third line: evaluating the dual function.

Original: Fortunately, this step is suitable for stochastic approximation, so we can leverage recent advances in the literature to reduce the (computational) cost of this step.
Revised: Fortunately, this step is suitable for stochastic approximation, so we can leverage recent advances in the literature to reduce the (computational) cost of this step.

Original: The total cost of this algorithm is comparable to that of distributionally robust optimization.repeat
Revised: The total cost of this algorithm is comparable to that of distributionally robust optimization.

Original: Evaluatedual function:[Equation] µ
Revised: Evaluate function:

Original: [Equation](cid:111) [Table]
Revised: [Equation]

Original: 1 n [Table] ni = [Table]
Revised: Dual ascent update

Original: t φ

Original: [Equation]( g ( [Equation]
Revised: [Equation]

Original: ; Z i ) − [Equation] ν [Equation]
Revised: 5: until

Original: (cid:111)(cid:105) [Equation] [Table]  Multiple expected value constraints
Revised:  Multiple expected value constraints

Original: In this section, we extend the results from the preceding section to stochastic optimization problems with multiple data-dependent constraints.
Revised: In this section, we extend the results from the preceding section to stochastic optimization problems with multiple data-dependent constraints.

Original: Consider a stochastic optimization problem with K expected value constraints
Revised: Consider a stochastic optimization problem with K expected value constraints

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: Following the development in Section 2, we enforce the expected value constraints with robust versions of the sample average constraints:
Revised: Following the development in Section 2, we enforce the expected value constraints with robust versions of the sample average constraints:

Original: [Equation]
Revised: [Equation]

Original: subject to
Revised: subject to

Original: [Equation]
Revised: [Equation]

Original: where ρ = ( ρ 1 , . . . , ρ K ) ⊤ are uncertainty set radii for the constraints.
Revised: where ⇢ = ( ⇢ 1 , . . . , ⇢ K ) > are uncertainty set radii for the constraints.

Original: There are other approaches to enforcing multiple constraints that result in constraint generalization; we focus on (3.2) here because it allows the user to adjust the constraint generalization probability for different constraints.
Revised: There are other approaches to enforcing multiple constraints that result in constraint generalization; we focus on (3.2) here because it allows the user to adjust the constraint generalization probability for different constraints.

Original: First, we extend theorem 2.1 and corollary 2.2 to problems with multiple (expected value) constraints.
Revised: First, we extend theorem 2.1 and corollary 2.2 to problems with multiple (expected value) constraints.

Original: We assume
Revised: We assume

Original: smoothness and concentration: for k ∈ [ K ] , f, g k are twice continuously differentiable with respect to θ , and f ( θ ⋆ ; Z ) , ∇ f ( θ ⋆ ; Z ) , g k
Revised: smoothness and concentration: for k 2 [ K ] , f, g k are twice continuously differentiable with respect to ✓ , and f ( ✓ ? ; Z ) , r f ( ✓ ?

Original:  ( θ ⋆ ; Z ) , ∇ g k ( θ ⋆ ; Z ) are sub-Gaussian random variables.
Revised: ; Z ) , g k ( ✓ ? ; Z ) , r g k ( ✓ ? ; Z ) are sub-Gaussian random variables.

Original: uniqueness: the stochastic optimization problem with K expected value constraints (3.1) has a unique optimal primal-dual pair ( θ ⋆ , λ ⋆ ) , and θ ⋆ belongs to the interior of the compact set Θ .
Revised: uniqueness: the stochastic optimization problem with K expected value constraints (3.1) has a unique optimal primal-dual pair ( ✓ ? ,  ? ) , and ✓ ? belongs to the interior of the compact set ⇥ .

Original: strict complementarity: λ ⋆ ∈ int( R K + ) , i.e. , each component of λ ⋆ is strictly positive.
Revised: strict complementarity:  ? 2 int( R K + ) , i e. , each component of  ? is strictly positive.

Original: positive definiteness: The Hessian of the Lagrangian evaluated at ( θ ⋆ , λ ⋆ ) is positive definite.
Revised: positive deﬁniteness: The Hessian of the Lagrangian evaluated at ( ✓ ? ,  ? ) is positive deﬁnite.

Original: The strict complementarity constraint seems especially strong here because it requires all the constraints to be active.
Revised: The strict complementarity constraint seems especially strong here because it requires all the constraints to be active.

Original: It is possible (with extra notational overhead) to state the result in terms of just the active constraints.
Revised: It is possible (with extra notational overhead) to state the result in terms of just the active constraints.

Original: We refer to Section 5.1 for more information about the unknown active set.
Revised: We refer to Section 5.1 for more information about the unknown active set.

Original: Further, as long as the sample size is large enough, the active constraints in (3.2) coincide with the active constraints in (3.1).
Revised: Further, as long as the sample size is large enough, the active constraints in (3.2) coincide with the active constraints in (3.1).

Original: To keep things simple, we assume all the constraints are active.
Revised: To keep things simple, we assume all the constraints are active.

Original: Theorem 3.1.
Revised: Theorem 3.1.

Original: Let (cid:98) θ n be an optimal solution of (3.2) converging in probability as n → ∞ to θ ⋆ .
Revised: Let b ✓ n be an optimal solution of (3.2) converging in probability as n ! 1 to ✓ ? .

Original: Under the standing assumptions, we have
Revised: Under the standing assumptions, we have

Revised: [Equation]

Revised: [Equation]

Revised: CA

Original: [Equation]
Revised: [Equation]

Original: Corollary 3.2.
Revised: Corollary 3.2.

Original: Under the conditions of theorem 3. [Equation] we have
Revised: Under the conditions of theorem 3.1, we have

Original: [Equation]
Revised: [Equation]

Original: where √ ρ = ( √ ρ 1 , . .
Revised: where p ⇢ = ( p ⇢ 1 , . .

Original: . , √ ρ K ) ⊤ , and U is a Gaussian random vector with mean zero and covariance
Revised: . , p ⇢ K ) > , and U is a Gaussian random vector with mean zero and covariance

Original: [Equation]
Revised: [Equation]

Original: From theorem 3.1 and corollary 3.2 (see proofs in Appendix B and C), we find that the probability of constraint satisfaction decreases exponentially as the number of constraints increases.
Revised: From theorem 3.1 and corollary 3.2 (see proofs in Appendix B and C), we ﬁnd that the probability of constraint satisfaction decreases exponentially as the number of constraints increases.

Original: We also see that our method is no longer pivotal for multiple expected value constraints: the uncertainty set radii depends on the (unknown) correlation structure among the constraint values.
Revised: We also see that our method is no longer pivotal for multiple expected value constraints: the uncertainty set radii depends on the (unknown) correlation structure among the constraint values.

Original: Fortunately, it is not hard to estimate this correlation structure.
Revised: Fortunately, it is not hard to estimate this correlation structure.

Original: The most straightforward way is with the empirical correlation matrix.
Revised: The most straightforward way is with the empirical correlation matrix.

Original: Let (cid:98) Σ n be the empirical covariance matrix of the constraint values.
Revised: Let b ⌃ n be the empirical covariance matrix of the constraint values.

Original: The empirical correlation matrix is then given by(cid:98) R n
Revised: The empirical correlation matrix is then given by

Original: [Equation]− 12 .
Revised: [Equation]

Original: Finally, it is straightforward to extend the algorithm for solving (1.3) to (3.2).
Revised: Finally, it is straightforward to extend the algorithm for solving (1.3) to (3.2).

Original: The Lagrangian of (3.2) is
Revised: The Lagrangian of (3.2) is

Original: [Equation]
Revised: [Equation]

Original: where we recalled the dual form of the robust constraint function (2.5) in the second step.
Revised: where we recalled the dual form of the robust constraint function (2.5) in the second step.

Original: We see that evaluating the dual function inf θ L ( θ, λ ) (at a fixed λ ) entails solving a stochastic optimization problem that is suitable for stochastic approximation.
Revised: We see that evaluating the dual function inf ✓ L ( ✓ ,  ) (at a ﬁxed  ) entails solving a stochastic optimization problem that is suitable for stochastic approximation.

Original: This suggests a similar dual ascent algorithm for solving (1.3); we skip the details here (see Algorithm 2 in Appendix D).
Revised: This suggests a similar dual ascent algorithm for solving (1.3); we skip the details here (see Algorithm 2 in Appendix D) [Equation]

Original:  Simulations
Revised:  Simulations

Original: We simulate the frequency of constraint satisfaction for the following multi-item newsvendor problem:[Equation] subject
Revised: We simulate the frequency of constraint satisfaction for the following multi-item newsvendor problem:

Original: [Equation]
Revised: [Equation]

Original: where c ∈ R d + is the manufacturing cost, p ∈ R d + is the sell price, θ ∈ Θ = [0 , 100] d is the number of items in stock, Z ∈ R d is a random variable with probability distribution P 0 representing the demand, and there are d items in total.
Revised: where c 2 R d + is the manufacturing cost, p 2 R d + is the sell price, ✓ 2 ⇥ = [0 , 100] d is the number of items in stock, Z 2 R d is a random variable with probability distribution P 0 representing the demand, and there are d items in total.

Original: The distribution P 0 is unknown but we observe IID samples
Revised: The distribution P 0 is unknown but we observe IID samples

Original: Z 1 , . .
Revised: Z 1 , . .

Original: , Z n from P 0 .
Revised: , Z n from P 0 .

Original: All of the items have been partitioned into two groups so that the corresponding demand and stock can be written as Z
Revised: All of the items have been partitioned into two groups so that the corresponding demand and stock can be written as Z

Original: =
Revised: =

Original: ( Z (1) , Z (2) ) and θ = ( θ (1) , θ (2) ) .
Revised: ( Z (1) , Z (2) ) and ✓ = ( ✓ (1) , ✓ (2) ) .

Original: The constraints in the problem exclude stock levels that underestimate the demand too much for each group of items, where ε 1 , ε 2 > 0 indicate tolerance level of such underestimation.
Revised: The constraints in the problem exclude stock levels that underestimate the demand too much for each group of items, where " 1 , " 2 > 0 indicate tolerance level of such underestimation.

Original: The target of the problem is to maximize the profit while satisfying the constraints.
Revised: The target of the problem is to maximize the proﬁt while satisfying the constraints.

Original: It is easy to rewrite the maximization problem (4.1) as a minimization problem with expected value constraints in the form of (3.1) so that we can apply our method (3.2).
Revised: It is easy to rewrite the maximization problem (4.1) as a minimization problem with expected value constraints in the form of (3.1) so that we can apply our method (3.2).

Original: We pick P 0 as multivariate Gaussian with independent components so that the two constraints are generally uncorrelated with each other (see Appendix E for details).
Revised: We pick P 0 as multivariate Gaussian with independent components so that the two constraints are generally uncorrelated with each other (see Appendix E for details).

Original: Throughout the simulations, we solve (3.2) with ρ =
Revised: Throughout the simulations, we solve (3.2) with ⇢ =

Original: ( z α , z α ) ⊤ for α ∈ { 0 . 4 , 0 . 25 , 0 . 1 , 0 . 05 , 0 . 005 } .
Revised: ( z ↵ , z ↵ ) > for ↵ 2 { 0 . 4 , 0 . 25 , 0 . 1 , 0 . 05 , 0 . 005 } .

Original: As suggested by our asymptotic theory in Section 3, the nominal probability of constraint satisfaction is 1 − α for each constraint and (1 − α ) 2 for both constraints due to the independence setup.
Revised: As suggested by our asymptotic theory in Section 3, the nominal probability of constraint satisfaction is 1  ↵ for each constraint and (1  ↵ ) 2 for both constraints due to the independence setup.

Original: In Figure 1, we plot frequencies of constraint satisfaction for each constraint and both constraints, all of which are averaged over 1000 replicates.
Revised: In Figure 1, we plot frequencies of constraint satisfaction for each constraint and both constraints, all of which are averaged over 1000 replicates.

Original: As the sample size n grows, the frequency versus probability curve converges to the theoretical dashed line of limiting probability of constraint satisfaction, validating our theory in the large sample regime.
Revised: As the sample size n grows, the frequency versus probability curve converges to the theoretical dashed line of limiting probability of constraint satisfaction, validating our theory in the large sample regime.

Original: For more simulations ( e.g. , single constraint, two dependent constraints) we refer to Appendix E.[Figure] f r e q u e n c y o f s a t i s f y
Revised: For more simulations ( e.g. , single constraint, two dependent constraints) we refer to Appendix E.

Original: i n g 2 n d c o n s t r i a n t n =30 n =300 n = n

Original: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 limiting probability of satisfying both constraints 0.

Original: f r e q u e n c y o f s a t i s f

Original: y i n g b o t h c o n s t r i a n t s n =30 n =300 n = n

Original:  Application to fair machine learning
Revised:  Application to fair machine learning

Original: As ML models are deployed in high-stakes decision making and decision support roles, the fairness of the models has come under increased scrutiny.
Revised: As ML models are deployed in high-stakes decision making and decision support roles, the fairness of the models has come under increased scrutiny.

Original: In response, there is a flurry of recent work on mathematical definitions of algorithmic fairness [16, 23, 24] and algorithms to enforce the definitions [1, 10, 37].
Revised: In response, there is a ﬂurry of recent work on mathematical deﬁnitions of algorithmic fairness [16, 23, 25] and algorithms to enforce the deﬁnitions [1, 10, 38].

Original: A prominent class of fairness definitions is group fairness ; such definitions require equality of certain metrics ( e.g. false/true positive rates) among demographic groups.
Revised: A prominent class of fairness deﬁnitions is group fairness ; such deﬁnitions require equality of certain metrics ( e.g. false/true positive rates) among demographic groups.

Original: For example, consider a fair binary classification problem.
Revised: For example, consider a fair binary classiﬁcation problem.

Original: Let X ⊂ R d be the input space, { 0 , 1 } be the set of possible labels, and A be the set of possible values of the protected/sensitive attribute.
Revised: Let X ⇢ R d be the input space, { 0 , 1 } be the set of possible labels, and A be the set of possible values of the protected/sensitive attribute.

Original: In this setup, training and test examples are tuples of the form ( X, A, Y ) ∈ X ×
Revised: In this setup, training and test examples are tuples of the form ( X, A, Y ) 2 X ⇥

Original: A × Y , and a classifier is a map f : X → { 0 , 1 } .
Revised: A ⇥ Y , and a classiﬁer is a map f : X ! { 0 , 1 } .

Original: A popular definition of algorithmic fairness for binary classification is equality of opportunity [23].
Revised: A popular deﬁnition of algorithmic fairness for binary classiﬁcation is equality of opportunity [23].

Original: Definition 5.1 (equality of opportunity) .
Revised: Deﬁnition 5.1 (equality of opportunity) .

Original: Let Y = 1 be the advantaged label that is associated with a positive outcome and (cid:98) Y ≜
Revised: Let Y = 1 be the advantaged label that is associated with a positive outcome and [Equation] Y [Equation]

Original: f ( X ) be the output of the classifier.
Revised: f ( X ) be the output of the classiﬁer.

Original: Equality of opportunity entails
Revised: Equality of opportunity entails

Original: P { (cid:98) Y = 1 | A = a, Y = 1 } = P { (cid:98) Y = 1 | A = a ′ , Y = 1 } for all a, a ′ ∈ A .
Revised: [Equation]

Original: Equality of opportunity, or true positive rate parity, means that the prediction (cid:98) Y =
Revised: Equality of opportunity, or true positive rate parity, means that the prediction b

Original:  h ( X ) conditioned on the advantaged label Y = 1 is statistically independent of the protected attribute A .
Revised: Y = h ( X ) conditioned on the advantaged label Y = 1 is statistically independent of the protected attribute A .

Original: Furthermore, an approximate version of equality of opportunity can be readily defined.
Revised: Furthermore, an approximate version of equality of opportunity can be readily deﬁned.

Original: We say 
Revised: We say that

Original: that (cid:98) Y = h ( X ) satisfies ε -equality of opportunity if P {(cid:98) Y = 1 |
Revised: [Equation] Y = h ( X ) satisﬁes " -equality of opportunity if P {

Original: A = a, Y = 1 } − P { (cid:98) Y = 1 | A = a ′ , Y = 1 } ≤ ε for for all a, a ′ ∈ A .
Revised: [Equation] for for all a, a2 A .

Original: In this case, ε > 0 represents a practitioner’s tolerance for fairness violations.
Revised: In this case, " > 0 represents a practitioner’s tolerance for fairness violations.

Original: Given a parametric model space H 
Revised: Given a parametric model space H =

Original: = { f θ ( · ) : θ ∈ Θ } and loss function ℓ : Θ × X × Y 
Revised: { f ✓ ( · ) : ✓ 2 ⇥ } and loss function ` : ⇥ ⇥ X ⇥ Y !

Original: → R + , an in-processing fair ML routine is to minimize the (empirical) risk E [ ℓ ( θ ; X, Y )] while satisfying some fairness constraints.
Revised: R + , an in-processing fair ML routine is to minimize the (empirical) risk E [ ` ( ✓ ; X, Y )] while satisfying some fairness constraints.

Original: Most commonly, definitions of group fairness (including equality of opportunity, demographic parity, and more) can be written as a special example of a general set of linear constraints [1, 2] of the form M µ ( θ ) ≤ c , where matrix M ∈ R K × T and vector c ∈ R K encode the constraints; µ ( θ ) : 
Revised: Most commonly, deﬁnitions of group fairness (including equality of opportunity, demographic parity, and more) can be written as a special example of a general set of linear constraints [1, 2] of the form M µ ( ✓ )  c , where matrix M 2 R K ⇥ T and vector c 2 R K encode the constraints; µ ( ✓ ) : ⇥ !

Original: Θ → R T is a vector of (conditional) moments µ t ( θ ) =
Revised: R T is a vector of (conditional) moments µ t ( ✓ ) =

Original: E [ h t ( X, A, Y, θ ) | E t ] for t ∈ [ T ] ; g t : X ×
Revised: E [ h t ( X, A, Y, ✓ ) | E t ] for t 2 [ T ] ; g t : X ⇥

Original: A × Y × Θ
Revised: A ⇥ Y ⇥ ⇥ !

Original: → R ; event E t is defined with respect to ( X, A, Y ) .
Revised: R ; event E t is deﬁned with respect to ( X, A, Y ) .

Original: This framework fits to our methodology if we note that each (conditional) moment can be written as
Revised: This framework ﬁts to our methodology if we note that each (conditional) moment can be written as

Original: [Equation]
Revised: [Equation]

Original: Here the indicator 1 {E t } takes value 1 if the event E t happens, and 0 otherwise.
Revised: Here the indicator 1 {E t } takes value 1 if the event E t happens, and 0 otherwise.

Original: Moreover, we use
Revised: Moreover, we use

Original: E t ( X, A, Y ) to emphasize that E t only depends on ( X, Y, A ) but not on θ in any way.
Revised: E t ( X, A, Y ) to emphasize that E t only depends on ( X, Y, A ) but not on ✓ in any way.

Original: Note that (5.1) is a ratio of expected values, which is a non-linear statistical functional of P 0 .
Revised: Note that (5.1) is a ratio of expected values, which is a non-linear statistical functional of P 0 .

Original: To use our method, we first replace the denominator of µ t ( θ ) with an estimator, such as the unbiased estimator (cid:98) P ( E t )
Revised: To use our method, we ﬁrst replace the denominator of µ t ( ✓ ) with an estimator, such as the unbiased estimator b

Original:  = 1 n(cid:80) ni
Revised: P ( E t ) = 1 n

Original:  =1 1 {E t ( X i , A
Revised: P ni =1 1 {E t ( X i , A

Original: i , Y i ) } .
Revised: i , Y i ) } .

Original: The resulting plug-in estimation of µ t ( θ ) then becomes linear in P 0 , allowing us to apply our method (see similar tricks in [8]).
Revised: The resulting plug-in estimation of µ t ( ✓ ) then becomes linear in P 0 , allowing us to apply our method (see similar tricks in [8]).

Original: We describe the application of our method to ε -equality of opportunity in Appendix F.
Revised: We describe the application of our method to " -equality of opportunity in Appendix F.

Original: 5.1 A two-stage method for unknown active set
Revised: 5.1 A two-stage method for unknown active set

Original: In practice, it is probable that only a subset of the constraints are active.
Revised: In practice, it is probable that only a subset of the constraints are active.

Original: Furthermore, we do not know beforehand whether or not a constraint is active in the true population problem.
Revised: Furthermore, we do not know beforehand whether or not a constraint is active in the true population problem.

Original: To handle this scenario, we propose a two-stage method:
Revised: To handle this scenario, we propose a two-stage method:

Original: At the first stage, we solve the sample average approximation (SAA) problem (3.2) with ρ = 0 K .
Revised: At the ﬁrst stage, we solve the sample average approximation (SAA) problem (3.2) with ⇢ = 0 K .

Original: By doing so, we identify the active set of the SAA problem.
Revised: By doing so, we identify the active set of the SAA problem.

Original: At the second stage, we solve (3.2) with ρ such that ρ k is a positive number only if the k -th constraint, k ∈{+
+} [ K ] , was identified as active at the first stage.
Revised: At the second stage, we solve (3.2) with ⇢ such that ⇢ k is a positive number only if the k -th constraint, k {+
+} [ K ] , was identiﬁed as active at the ﬁrst stage.

Original: In Appendix G, we show that the two-stage method also enjoys the calibration property (similar to Theorem 3.1 and Corollary 3.2) under standard assumptions ( i.e. , strict complementarity).
Revised: In Appendix G, we show that the two-stage method also enjoys the calibration property (similar to Theorem 3.1 and Corollary 3.2) under standard assumptions ( i.e. , strict complementarity).

Original: At a high level, the limiting probability of satisfying the true constraints depends solely on the correlation structure between active constraints and the uncertainty set radii for active constraints, as long as the
Revised: At a high level, the limiting probability of satisfying the true constraints depends solely on the correlation structure between active constraints and the uncertainty set radii for active constraints, as long as the

Original: SAA problem identifies active constraints with probability tending to 1 .
Revised: SAA problem identiﬁes active constraints with probability tending to 1 .

Original: 5.2 Proxy dual function for non-differentiable constraints
Revised: 5.2 Proxy dual function for non-differentiable constraints

Original: Constraint functions in fair ML are often non-differentiable.
Revised: Constraint functions in fair ML are often non-differentiable.

Original: For instance, fairness metrics are typically linear combinations of indicators that result in non-differentiable rate constraints [8–10].
Revised: For instance, fairness metrics are typically linear combinations of indicators that result in non-differentiable rate constraints [8–10].

Original: This prevents the use of any gradient-based optimization algorithms.
Revised: This prevents the use of any gradient-based optimization algorithms.

Original: Fortunately, only the dual function evaluation step in Algorithm 1 requires access to gradients.
Revised: Fortunately, only the dual function evaluation step in Algorithm 1 requires access to gradients.

Original: Therefore, we can modify the algorithm by: (1) introducing proxy dual function, which uses a differentiable surrogate ˜ g instead of the non-differentiable g in the dual function evaluation step; (2) keeping g in the dual ascent step.
Revised: Therefore, we can modify the algorithm by: (1) introducing proxy dual function, which uses a differentiable surrogate ˜ g instead of the non-differentiable g in the dual function evaluation step; (2) keeping g in the dual ascent step.

Original: For an indicator function h ( t )
Revised: For an indicator function h ( t )

Original: = 1 { t > 0 } , one can replace it by sigmoidal function h 1 ( t ) =
Revised: = 1 { t > 0 } , one can replace it by sigmoidal function h 1 ( t ) =

Original: (1+ e − at ) −or hinge upper bound h 2 ( t )
Revised: (1+ e  at ) or hinge upper bound h 2 ( t )

Original: =
Revised: =

Original: max { 0 , t + 1 } to produce smooth surrogates for non-differentiable rate constraints [11, 17, 9].
Revised: max { 0 , t + 1 } to produce smooth surrogates for non-differentiable rate constraints [11, 17, 9].

Original: We summarize the proxy dual ascent algorithm in Appendix H.
Revised: We summarize the proxy dual ascent algorithm in Appendix H.

Original: 5.3 Adult experiments
Revised: 5.3 Adult experiments

Original: We compare the frequency of constraint satisfaction (at test time) of the sample average approximation and our methods with nominal probability 0 .
Revised: We compare the frequency of constraint satisfaction (at test time) of the sample average approximation and our methods with nominal probability 0 .

Original: 95 using the Adult dataset from UCI [13].
Revised: 95 using the Adult dataset from UCI [13].

Original: The classification task is to predict whether an individual’s income per year is higher than $50 K.
Revised: The classiﬁcation task is to predict whether an individual’s income per year is higher than $50 K.

Original: The fairness goal is ε -demographic parity ( ε -DP): | P ( (cid:98) Y = 1 | A = 1) − P ( (cid:98) Y = 1 | A = 0)
Revised: The fairness goal is " -demographic parity ( " -DP): | [Equation]

Original: | ≤ ε , where A = 1 for male is the advantaged group and A = 0 for female is the disadvantaged group.
Revised: , where A = 1 for male is the advantaged group and A = 0 for female is the disadvantaged group.

Original: We use a logistic regression model for classification and techniques in this section for implementation.
Revised: We use a logistic regression model for classiﬁcation and techniques in this section for implementation.

Original: In Figure 2, we have line plots for frequency of constraint satisfaction and box plots for classification error rate, all of which are summarized over 100 replicates.
Revised: In Figure 2, we have line plots for frequency of constraint satisfaction and box plots for classiﬁcation error rate, all of which are summarized over 100 replicates.

Original: The left panel shows that solving (3.1) directly leads to one half chance of constraint violation, while our method’s constraint satisfaction frequency matches its nominal value.
Revised: The left panel shows that solving (3.1) directly leads to one half chance of constraint violation, while our method’s constraint satisfaction frequency matches its nominal value.

Original: The price of a higher chance of test-time fairness satisfaction is an increase in classification error rate as shown in the right panel.
Revised: The price of a higher chance of test-time fairness satisfaction is an increase in classiﬁcation error rate as shown in the right panel.

Original: From the baseline to 95% chance of fairness satisfaction, we basically trade off 2% increase in error rate.
Revised: From the baseline to 95% chance of fairness satisfaction, we basically trade off 2% increase in error rate.

Original: We refer to Appendix I and K for details and more experiments.
Revised: We refer to Appendix I and K for details and more experiments.

Original: [Figure]
Revised: [Figure]

Original:  Summary and discussion
Revised:  Summary and discussion

Original: We explore the problem of exact constraint satisfaction probability in stochastic optimization with expected-value constraints.
Revised: We explore the problem of exact constraint satisfaction probability in stochastic optimization with expected-value constraints.

Original: We propose a distributionally robust reformulation of data-dependent constraints and provide a theoretical guarantee of constraint satisfaction with an asymptotically exact probability specified by the user.
Revised: We propose a distributionally robust reformulation of data-dependent constraints and provide a theoretical guarantee of constraint satisfaction with an asymptotically exact probability speciﬁed by the user.

Original: For solving the reformulated problem, a scalable dual ascent algorithm and its variants are proposed.
Revised: For solving the reformulated problem, a scalable dual ascent algorithm and its variants are proposed.

Original: The computational cost of our algorithm is comparable to that of a standard distributionally robust optimization problem.
Revised: The computational cost of our algorithm is comparable to that of a standard distributionally robust optimization problem.

Original: Our theory on exact constraint satisfaction probability is validated via simulations on the resource-constrained newsvendor problem.
Revised: Our theory on exact constraint satisfaction probability is validated via simulations on the resource-constrained newsvendor problem.

Original: The efficacy of our methods is empirically demonstrated on fair machine learning applications.
Revised: The efﬁcacy of our methods is empirically demonstrated on fair machine learning applications.

Original: Some data-dependent constraints are by nature non-linear in the underlying probability measure.
Revised: Some data-dependent constraints are by nature non-linear in the underlying probability measure.

Original: For example, (5.1) is a ratio of expected values.
Revised: For example, (5.1) is a ratio of expected values.

Original: An intriguing direction for future research is to generalize the methods and theory developed in this work to constraints on non-linear functions of expected values.
Revised: An intriguing direction for future research is to generalize the methods and theory developed in this work to constraints on non-linear functions of expected values.

Original: Such forms of constraints are known as statistical functionals in statistics literature [19].
Revised: Such forms of constraints are known as statistical functionals in statistics literature [19].

Original: The non-linear dependence of the constraint function on the probability measure precludes the stochastic approximation as a general way of evaluating the dual function, as the constraint function no longer admits a dual form (2.5), calling for the development of a new algorithm.
Revised: The non-linear dependence of the constraint function on the probability measure precludes the stochastic approximation as a general way of evaluating the dual function, as the constraint function no longer admits a dual form (2.5), calling for the development of a new algorithm.

Original: Acknowledgments and Disclosure of Funding
Revised: Acknowledgments and Disclosure of Funding

Original: This paper is based upon work supported by the National Science Foundation (NSF) under grants no.and 2113373. A Proofs of Theorem 2.1 and Corollary 2.
Revised: This paper is based upon work supported by the National Science Foundation (NSF) under grants no.

Original: Note that Theorem 3.1 implies Theorem 2.1 and Corollary 3.2 implies Corollary 2.2 by letting K = 1 .

Original: Therefore, it is sufficient to prove Theorem 3.1 and Corollary 3.2, whose proofs can be found in

Original: Appendix B and C respectively.

Original: □

Original: B Proof of Theorem 3.

Original: Consider a stochastic optimization problem with K expected value constraints ( P 0 ) :

Original: [Equation]

Original: ∈ [ K ] } .

Original: Our proposed robust constraint method solves

Original: [Equation]

Original: n

Original: [Equation]

Original: where ρ = ( ρ 1 , . . . , ρ K ) ⊤ is the collection of critical radii of uncertainty sets.

Original: Here we denote the empirical distribution (cid:98) P n by P n for notation simplicity.

Original: As a special case of our robust method, the sample average approximation (SAA) or empirical risk minimization (ERM) solves

Original: [Equation]

Original: We denote

Original: [Equation]

Original: and

Original: [Equation]

Original: G k ( θ ) =

Original: E Z ∼ P n [ g k ( θ ; Z )] and (cid:98)

Original: G kn ( θ ) =

Original: sup D φ ( Q ∥ P n )

Original: [Equation]

Original: k /n

Original: E Z ∼ Q [ g k ( θ ; Z )] for k ∈

Original: [ K ] , and

Original: G ( θ ) =  

Original: G 1 ( θ ) .

Original: G K

Original: [Equation]

Original: θ )

Original:   and (cid:98) G n ( θ ) =   (cid:98) G 1 n ( θ ) .

Original: (cid:98) G

Original: Kn ( θ )   .

Original: Note that (cid:98) F n ( · ) and (cid:98) G kn ( · ) ’s are random functions serving as approximations to F ( · ) and G k ( · ) ’s.

Original: Consider the Lagrangian functions

Original: [Equation]

Original: and

Original: [Equation]

Original: of the programs ( P 0 ) and ( P n ) respectively.

Original: Lemma B.1 (Theorem 6.6.2 in [32]) .

Original: Suppose that:

Original: (i)

Original: The functions F ( θ ) and G k ( θ ) , k ∈ [ K ] , are twice continuously differentiable.

Original: (ii)

Original: The true program ( P 0 ) has a unique optimal solution θ ⋆ and a unique vector λ ⋆ of the Lagrange multipliers with θ ⋆ being an interior point of Θ .

Original: (iii)

Original: The Hessian matrix ∇ 2 L ( θ ⋆ , λ ⋆ ) is positive definite.

Original: (iv) The random functions (cid:98) G kn ( θ ) , k ∈

Original: [ K ] , are Lipschitz continuous in a neighborhood of θ ⋆ and differentiable at θ ⋆ with probability 1.

Original: (v)

Original: ∥ ∆ in ( θ ⋆ ) ∥

Original: [Equation]

Original: O p

Original: [Equation]

Original: n

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: and there is a neighborhood U of θ ⋆ such that

Original: [Equation]

Original: Here we define random

Original: mappings ∆ 1 n

Original: θ ) =

Original: ∇ (cid:98) F n ( θ ) − ∇ F ( θ ) , ∆ 2 n ( θ ) = (cid:98)

Original: G

Original: n θ )

Original: −

Original: G (

Original: θ

Original: and ∆ 3 n ( θ ) = ∇ (cid:98) G n

Original: θ

Original: ) − ∇

Original: G (

Original: θ ) .

Original: (vi) Random vectors √ n ( ∇ (cid:98) L n ( θ ⋆ , λ ⋆ ) , (cid:98) G n ( θ ⋆ )) converge in distribution as n → ∞ to a random vector Y =

Original: ( Y 1 , Y 2 ) .

Original: Let (cid:98) θ n be an optimal solution of ( P n ) converging in probability as n → ∞ to θ ⋆ .

Original: Then

Original: [Equation]

Original: where ¯ x = ¯ x ( Y ) is the optimal solution to the quadratic programming problem

Original: [Equation]

Original: to

Original: [Equation]

Original: From now on, we use φ ( t ) =

Original: ( t − 1) 2 , which gives the χ 2 -divergence.

Original: Lemma B.1 is adapted from

Original: Theorem 6.6.2 in [32] under the strict complementarity assumption.

Original: Recall the standing assumptions, (i) , (iv) , (v) are guaranteed by the smoothness and concentration assumption, (ii) is postulated by the uniqueness assumption, and (iii) is ensured by the positive definiteness assumption.

Original: Now we derive the limiting distribution required in (vi) .

Original: Let B denote the ℓ 2 -ball of radius 1 in R d .

Original: According to Lemma 24 in [14], for each k ∈ [ K ] there exists ϵ k > 0 such that, with probability 1, there exists an N k such that n ≥ N k implies

Original: [Equation]

Original: G

Original: [Equation]

Original: n for all θ ∈ θ ⋆

Original: [Equation]

Original: B .

Original: Taking ϵ 0 = min { ϵ k : k ∈

Original: [ K ] } and N 0 = max { N k : k ∈

Original: [ K ] } , we have the following uniform expansion holds, that is,

Original: [Equation]

Original: all k

Original: [Equation]

Original: and θ ∈

Original: [Equation]

Original: ϵ

Original: [Equation]

Original: P 0 -almost surely given n ≥ N 0 .

Original: Therefore, for sufficiently large n , for k ∈ [ K ] we have

Original: [Equation]

Original: and ∇

Original: [Equation]

Original: For the objective function and its empirical counterpart, we have

Original: [Equation]

Original: and

Original: [Equation]

Original: Now we derive the the limiting distribution of random vectors √ n ( ∇ (cid:98) L n ( θ ⋆ , λ ⋆ ) , (cid:98) G n ( θ ⋆ )) .

Original: [-For simplicity of notations, we denote P n m = E P n [ m ( θ ⋆ ; Z )] and P 0 m = E P 
-]
Revised: [-For simplicity of notations, we denote P n m = E P n [ m ( θ ⋆ ; Z )] and P 0 m = E P 
-]

Original: [ m ( θ ⋆ ; Z )] for any random function m ( θ ; Z ) , Cov( ∇ g k , g k ) =

Original: Cov P 0 ( ∇ g k ( θ ⋆ ; Z ) , g k

Original: ( θ ⋆ ; Z )) and Var[ g k ]

Original: =

Original: [-Var P 
-]
Revised: [-Var P 
-]

Original: [ g k ( θ ⋆ ; Z )] .

Original: Then we have

Original: [Equation]

Original: 

Original: [Equation]

Original:  

Original: [Equation]

Original: 

Original: [Equation]

Original: 

Original: [Equation]

Original:   

Original: [Equation]

Original: o P (cid:18) 1 √ n (cid:19) .

Original: By central limit theorem, √ n

Original: [Equation]

Original: 

Original: [Equation]

Original: , where Σ 11 =

Original: Var P

Original: [Equation]

Original: Σ

Original: [Equation]

Original: Var P

Original: [Equation]

Original: G

Original: [Equation]

Original: ⋆

Original: [Equation]

Original: Z

Original: [Equation]

Original: [Equation]

Original: K

Original: [Equation]

Original: K

Original: [Equation]

Original: By Slutsky’s theorem,

Original: [Equation]

Original: where

Original: [Equation]

Original: Therefore

Original: [Equation]

Original: we conclude that the limiting distribution of √ n ( ∇ (cid:98) L n ( θ ⋆ , λ ⋆ ) , (cid:98) G n ( θ ⋆ )) is

Original: [Equation]

Original: By Lemma (B.1), we have

Original: [Equation]

Original: where ¯ x is given by the linear system

Original: [Equation]

Original: or

Original: [Equation]

Original: B

Original: [Equation]

Original: [Equation]

Original: which implies √ n ( (cid:98) θ n − θ ⋆ )

Original: d −→ ¯ x ∼ N (¯ µ, ¯Σ) for some ¯ µ and ¯Σ determined by (B.1).

Original: By delta method

Original: [Equation]

Original: we have

Original: [Equation]

Original: Now we calculate ∇ G ( θ ⋆ )

Original: ⊤ ¯ µ and ∇ G ( θ ⋆ )

Original: ⊤ ¯Σ ∇ G ( θ ⋆ )) .

Original: For notation simplicity, we denote ∇ 2 L = ∇ 2 L ( θ ⋆ , λ ⋆ ) , ∇ G = ∇ G ( θ ⋆ ) and H = ( ∇ 2 L ) − 1 ∇ G [ ∇ G ⊤ ( ∇ 2 L ) − 1 ∇ G ] − 1 .

Original: By block matrix inversion, we have

Original: [Equation]

Original: By (B.1), we have

Original: [Equation]

Original: Note that ∇ G ⊤ H

Original: [-= I K and ∇ G ⊤ (cid:8) ( ∇ 2 L ) − 1 − H ∇ G ⊤ ( ∇ 2 L ) − 1 (cid:9) = 
-]
Revised: [-= I K and ∇ G ⊤ (cid:8) ( ∇ 2 L ) − 1 − H ∇ G ⊤ ( ∇ 2 L ) − 1 (cid:9) = 
-]

Original: K × K .

Original: We have ∇ G

Original: [Equation]

Original: ⊤

Original: [Equation]

Original: G ⊤

Original: [Equation]

Original: and

Original: [Equation]

Original: Therefore

Original: [Equation]

Original: we conclude that

Original: [Equation]

Original: Hence we complete the proof of Theorem 3.

Original: [Equation]

Original: □

Original: C Proof of Corollary 3.

Original: Recall that D

Original: =

Original: [-diag(Var P 
-]
Revised: [-diag(Var P 
-]

Original: [ g 1 ( θ ⋆ ; Z )] , . . .

Original: [-, Var P 
-]
Revised: [-, Var P 
-]

Original: [ g K ( θ ⋆ ; Z )]) .

Original: According to Theorem 3.

Original: [Equation]

Original: we have

Original: [Equation]

Original: feasible

Original: [Equation]

Original: θ n ; Z )] .

Original: [-E P 
-]
Revised: [-E P 
-]

Original: [ g K ( (cid:98) θ n ; Z )] 

Original:  ∈ − R K

Original: +  

Original: =

Original: lim n →∞ P   √ n  

Original: [-E P 
-]
Revised: [-E P 
-]

Original: [ g 1 ( (cid:98) θ n ; Z )] .

Original: [-E P 
-]
Revised: [-E P 
-]

Original: [ g K ( (cid:98) θ n ; Z )] 

Original:  ≤ 0 K  

Original: = P  

Original: N   − 

Original: [- ( ρ 1 Var P 
-]
Revised: [- ( ρ 1 Var P 
-]

Original: [ g 1 ( θ ⋆ ; Z )])...

Original: [-( ρ K Var P 
-]
Revised: [-( ρ K Var P 
-]

Original: [ g K ( θ ⋆ ; Z )])  , Var P  g 1 ( θ ⋆ ; Z ) ...

Original: g K ( θ ⋆ ; Z ) 

Original:    ≤ 0 K   =

Original: P    

Original: N  

Original: [Equation]

Original: D

Original: [Equation]

Original: [-   ( ρ 1 Var P 
-]
Revised: [-   ( ρ 1 Var P 
-]

Original: [ g 1 ( θ ⋆ ; Z )])...

Original: [-( ρ K Var P 
-]
Revised: [-( ρ K Var P 
-]

Original: [ g

Original: [Equation]

Original: ( θ ⋆ ; Z )]) 1   , D − 12 Var   g 1 ( θ ⋆ ; Z ) ...

Original: g K ( θ ⋆

Original: [Equation]

Original: Z ) 

Original:  D −   ≤ D − 12 0 K     = P  

Original: N   −   √ ρ... √

Original: ρ K 

Original:  , Corr P  g 1 ( θ ⋆ ; Z ) ...

Original: g K ( θ ⋆ ; Z ) 

Original:    ≤ 0 K  

Original: = P  

Original: N   0 K , Corr P  g 1 ( θ ⋆ ; Z ) ...

Original: g K ( θ ⋆ ; Z ) 

Original:    ≤   √ ρ... √

Original: ρ K 

Original:    .

Original: Hence we complete the proof of Corollary 3.2.

Original: □

Original: D Dual ascent algorithm for (3.2)

Original: We summarize the dual ascent algorithm for solving (3.2) in Algorithm 2.

Original: Similar to Algorithm 1, the main cost of Algorithm 2 is incurred in the evaluation of dual function.

Original: The dual function evaluation step is still suitable for stochastic approximation.

Original: Therefore, the total cost of Algorithmis comparable to that of a standard distributionally robust optimization problem.

Original: [Table]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: E Simulations: details and more

Original: In this section, we provide details for simulations on the multi-item newsvendor problem with independent constraints (which we present in the main text), and conduct more simulations on: (1) multi-item newsvendor problem with dependent constraints, and (2) single-item newsvendor problem with a single constraint.

Original: E.1 Multi-item newsvendor problem

Original: First recall that in Section 4, we simulate the frequency of constraint satisfaction for the following multi-item newsvendor problem:

Original: [Equation]

Original: subject to

Original: [Equation]

Original: where c ∈ R d + is the manufacturing cost, p ∈ R d + is the sell price, θ ∈ Θ = [0 , 100] d is the number of items in stock, Z ∈ R d is a random variable with probability distribution P 0 representing the demand, and there are d items in total.

Original: The distribution P 0 is unknown but we observe IID samples

Original: Z 1 , . .

Original: , Z n from P 0 .

Original: All of the items have been partitioned into two groups so that the corresponding demand and stock can be written as Z

Original: = ( Z (1) , Z (2) ) and θ = ( θ (1) , θ (2) ) .
Revised: 1916271, 2027737, and 2113373.

Original: The constraints in the problem exclude stock levels that underestimate the demand too much for each group of items,

Original: where ε 1 , ε 2 > 0 indicate tolerance level of such underestimation.

Original: The target of the problem is to maximize the profit while satisfying the constraints.

Original: We can rewrite the maximization problem (E.1) as a minimization problem with expected value constraints in the form of (3.1), that is,

Original: [Equation]

Original: [-subject to E P 
-]
Revised: [-subject to E P 
-]

Original: [( ∥ Z (1) ∥ 22 − ∥ θ (1) ∥ 22 ) + − ε 1 ] ≤

Original: [Equation]

Original: (E

Original: [Equation]

Original: [Equation]

Original: so that we can apply our method (3.2).

Original: We generate Z 1 , . . .

Original: , Z n IID from multivariate normal distribution N ( µ, Σ) .

Original: In addition, we set the number of items d = 4 , the mean of the normal distribution µ = (10 , 10 , 10 , 10) ⊤ , the cost c = (1 , 1 , 1 , 1) ⊤ , the price p = (2 , 2 , 2 , 2) ⊤ , and the tolerance level of underestimation ( ε 1 , ε 2 ) =

Original: Moreover, we partition the items into the group of the first two items and the group of the last two items.

Original: We solve the empirical problem with robust constraint:

Original: [Equation]

Original: subject to sup

Original: [Equation]

Original: using samples of size n ∈ { 30 , 300 , 3000 } .

Original: For the covariance of the multivariate normal distribution, we consider a exchangeable correlation structure

Original: [Equation]

Original: We solve (E.3) with ρ =

Original: ( ρ, ρ ) ⊤ =

Original: ( z α , z α ) ⊤ for α ∈ { 0 . 4 , 0 . 25 , 0 . 1 , 0 . 05 , 0 . 005 } , of which the corresponding √ ρ ∈ { 0 . 253 , 0 . 674 , 1 . 281 , 1 . 644 , 2 . 575 } .

Original: [-Independent constraints In Section 4, we consider r = 
-]
Revised: [-Independent constraints In Section 4, we consider r = 
-]

Original: so that the two constraints are generally uncorrelated with each other.

Original: As suggested by the asymptotic theory in Section 3, the nominal probability of constraint satisfaction is 1 − α for each constraint and (1 − α ) 2 for both constraints due to the independence of two constraints.

Original: The results are discussed in Section 4.

Original: Dependent constraints Now we consider r = 0 .

Original: 6 so that the two constraints are generally correlated with each other.

Original: As suggested by the asymptotic theory in Section 3, the nominal probability of constraint satisfaction is 1 − α for each constraint.

Original: The nominal probability of constraint satisfaction for both constraints is no longer (1 − α ) 2 due to the dependence of two constraints, but the probability is given by

Original: [Equation]

Original: In Figure 3, we plot frequencies of constraint satisfaction for each constraint and both constraints, all of which are averaged over 1000 replicates.

Original: As the sample size n grows, the frequency versus probability curve converges to the theoretical dashed line of limiting probability of constraint satisfaction, validating our theory in the large sample regime.

Original: We note that in this example, the frequency of constraint satisfaction is higher than that of the experiments with independent constraints, for each ρ .

Original: [Figure]

Original: E.2 Single-item newsvendor problem

Original: In this subsection

Original: [Figure]

Original: we consider the following single-item newsvendor problem:

Original: [Equation]

Original: subject

Original: [Equation]

Original: where c > 0 is the manufacturing cost, p ≥ c is the sell price, θ ∈ Θ = [0 , 100] is the number of items in stock, and Z is a random variable with probability distribution P 0 representing the demand.

Original: The distribution P 0 is unknown but instead we observe IID samples Z 1 , . . .

Original: , Z n from P .

Original: The constraint in the problem excludes stocking levels that underestimate the demand too much, where ε > 0 indicates tolerance level of such underestimation.

Original: The target of the problem is to maximize the profit while satisfying the constraint.

Original: Note that the problem is equivalent to

Original: [Equation]

Original: which is a particular case of (1.1).

Original: We generate Z 1 , . . .

Original: , Z n IID from exponential distribution with mean 10 .

Original: In addition, we set the cost c = 1 , the price p = 2 , and the tolerance level of underestimation ε = 1 .

Original: We solve the empirical problem with robust constraint:

Original: [Equation]

Original: using samples of size n ∈ { 300 , 3000 , 30000 } .

Original: In Figure 4, we plot frequencies of constraint satisfaction which are averaged over 1000 replicates.

Original: As the sample size n grows, the frequency versus probability curve converges to the theoretical dashed line of limiting probability of constraint satisfaction, validating our theory in the large sample regime.

Original: [Figure]

Original: F Application of our method to ε -equality of opportunity

Original: Continuing with Section 5.1, we demonstrate how to apply our method to enforce ε -equality of opportunity.

Original: To keep things simple, we assume there are only two demographic groups; i.e. |A| = 2 .

Original: Without loss of generality, we refer to one group as advantaged ( A = 1 ) and the other as disadvantaged ( A = 0 ).

Original: First we estimate rates P { A = 1 , Y = 1 } and P { A = 0 , Y = 1 } consistently by (cid:98) p 1 = 1 n

Original: [Equation]

Original: ni =1 1 { A i = 1 , Y i = 1 } and (cid:98) p 0 = 1 n (cid:80) ni

Original: =1 1 { A i = 0 , Y i = 1 } .

Original: Then, we construct a robust constraint

Original: [Equation]

Original: or equivalently

Original: [Equation]

Original: G Two-stage method for unknown active set

Original: In this section, we show that the two-stage method in Section 5.1 also has the calibration property (similar to Theorem 3.1 and Corollary 3.2) if the true program ( P 0 ) , i.e. (3.1), is not ill-behaved.

Original: First we recall the two-stage method:

Original: At the first stage, we solve the program ( P n ) with ρ = 0 K , that is, the sample average approximation (SAA) problem.

Original: We identify that (cid:98) J + ,n ⊂ [ K ] is the active set of the SAA problem, that is, the j -th constraint of the SAA problem is active if and only if j ∈ (cid:98) J + ,n .

Original: At the second stage, we solve the program ( P n ) with ρ such that ρ j is some positive number if j ∈ (cid:98) J + ,n .

Original: This means that at the second stage we replace the sample mean approximation to the constraint by its distributionally robust counterpart if such constraint was identified as active at the first stage.

Original: The index set [ K ] can be partitioned into three parts:

Original: [Equation]

Original: where J

Original: + is the active set with positive Lagrange multipliers, J 0 is the active set with zero Lagrange multipliers, and J − is the inactive set.

Original: We assume the strict complementarity holds in the sense that

Original: J 0 ( θ ⋆ , λ ⋆ ) = ∅ .

Original: Proposition G.1 (Preservance of active constraints) .

Original: Assume the strict complementarity holds.

Original: We have (cid:98) J + ,n = J

Original: + ( θ ⋆ , λ ⋆ ) with probability converging to 1 as n → ∞ .

Original: Proof of Proposition G.1.

Original: Let (cid:98) θ (SAA) n be a solution to the sample average approximation problem at the first stage.

Original: Let (cid:98) λ n be the associated Lagrange multiplier.

Original: It is known by the consistency of SAA [32] that

Original: [Equation]

Original: and

Original: [Equation]

Original: Therefore, with probability converging to 1 , (cid:98) λ k,n > 0 , k ∈ J + ( θ ⋆ , λ ⋆ ) and E P [ g k ( (cid:98) θ (SAA) n ; Z )] < 0 , k ∈ J − ( θ ⋆ , λ ⋆ ) .

Original: Hence we complete the proof of Proposition G.1.

Original: □

Original: Theorem G.2.

Original: Suppose the true program ( P 0 ) has m active constraints with positive Lagrange multipliers and K − m inactive constraints (without loss of generality we let J + = { 1 , . . .

Original: , m } and

Original: J − = { m +1 , . . .

Original: , K } ).

Original: Let (cid:98) θ n be the two-stage estimator.

Original: Under the standing assumptions, we have

Original: [Equation]

Original: This result shows that the limiting probability of satisfying the true constraints only depends on the correlation structure between active constraints and the uncertainty set radii for the active constraints.

Original: Proof of Theorem G.2.

Original: At the first stage, we identify J ⊂

Original: [ K ] as active set with probability p J .

Original: Here the randomness is introduced by the data samples Z 1 , . . .

Original: , Z n .

Original: Let 2 [ K ] be the power set of [ K ] .

Original: We have

Original: [Equation]

Original: By Proposition G.1, as n → ∞ we have p

Original: [Equation]

Original: [-and p J → 0 for any J ∈ 
-]
Revised: [-and p J → 0 for any J ∈ 
-]

Original: [ K ] and J ̸

Original: = J + .

Original: By Corollary 3.2, we have lim n

Original: [Equation]

Original: θ n is feasible | identify J + as active set (cid:111)

Original: =

Original: P  

Original: N  

Original: [Equation]

Original: , Corr P  g

Original: [Equation]

Original: g

Original: [Equation]

Original: 

Original:    ≤   √ ρ... √ ρ m 

Original:    .

Original: Therefore,

Original: P

Original: [Equation]

Original: θ n is feasible

Original: [Equation]

Original: = (cid:88)

Original: [Equation]

Original: [Equation]

Original: [Equation]

Original: θ n is feasible | identify J as active set (cid:111) P { identify J as active set }

Original: [Equation]

Original: θ n is feasible | identify J

Original: [Equation]

Original: as active set

Original: [Equation]

Original: identify J as

Original: [Equation]

Original: Hence we complete the proof of Theorem G.2.

Original: □

Original: H Proxy dual ascent algorithm for non-differentiable constraints

Original: We summarize in Algorithm 3 the proxy dual ascent algorithm for solving a stochastic optimization problem with single non-differentiable constraint.

Original: The difference between Algorithm 3 and 1 is in the step of evaluating (proxy) dual function: Algorithm 3 uses a differentiable surrogate ˜ g (highlighted in orange) instead of the non-differentiable g in Algorithm 1 in this step.

Original: Algorithm

Original: [-1: Input: starting dual iterate λ 
-]
Revised: [-1: Input: starting dual iterate λ 
-]

Original: ≥2: repeat 3:

Original: Evaluate proxy dual function:

Original: [Equation]

Original: [Table]

Original: update: λ

Original: [Equation]

Original: (cid:104)

Original: [Equation]

Original: [Table]

Original: Algorithm 4 summarizes the proxy dual ascent algorithm for solving a stochastic optimization problem with multiple non-differentiable constraints.

Original: In contrast to Algorithm 2, Algorithm 4 replaces the non-differentiable functions g k ’s by their differentiable surrogates ˜

Original: g k ’s (highlighted in orange) in the (proxy) dual function evaluation step.

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: [Table]

Original: I Real data experiments: details and more

Original: In this section, we provide details for experiments on Adult dataset with ε -demographic parity as the fairness goal (which we present in the main text), and conduct additional experiments on Adult dataset with ε -false positive rate parity as the fairness goal.

Original: I.1 Adult with ε -demographic parity

Original: Recall that in Section 5.3 the fairness goal is ε -demographic parity ( ε -DP):

Original: [Equation]

Original: where A = 1 for male is the advantaged group and A = 0 for female is the disadvantaged group.

Original: We use a logistic regression model for classification by predicting (cid:98) Y = 1 { θ ⊤ X ≥ 0 } and training such model parameterized in θ by logistic loss.

Original: We implement the two-stage method and proxy dual ascent algorithm by replacing indicator function by the sigmoidal function h 1 ( t ) =

Original: (1 + e − at ) − 1 for a = 2 .

Original: We do bootstrap evaluation by treating the full dataset as the true probability measure and sample such probability measure with replacement as the training distribution (with sampling rate 50% ).

Original: I.2 Adult with ε -false positive rate parity

Original: Similar to the preceding subsection, now we consider the fairness goal to be ε -false positive rate parity:

Original: [Equation]

Original: where A = 1 for male is the advantaged group and A = 0 for female is the disadvantaged group.

Original: In Figure 5, we have line plots for frequency of constraint satisfaction and box plots for classification error rate, all of which are summarized over 100 replicates.

Original: The patterns shown in the left and right panels are similar to that in Figure 2.

Original: We observe more variation in frequencies of constraint satisfaction across different false positive rate tolerance designs due to the label and demographic attribute imbalance of Adult dataset.

Original: [Figure]

Original: J Dual form of the robust constraint function (2.5)

Original: In this section, we provide a standard derivation for the dual form of the robust constraint function (2.5).

Original: We introduce the likelihood ratio L ( Z )

Original: =

Original: dP ( Z ) /d (cid:98)

Original: P n ( Z ) .

Original: By change of variable, we can rewrite the robust constraint function (2.5) as

Original: [Equation]

Original: where the supremum takes over measurable functions.

Original: This gives us a constrained optimization problem.

Original: Let µ ≥ 0 be the Lagrange multiplier for E (cid:98) P n (cid:2) φ ( L ( Z )) (cid:3) ≤ ρ and ν ∈ R be the Lagrange multiplier for E (cid:98) P n (cid:2)

Original: L ( Z ) (cid:3) = 1 .

Original: The corresponding Lagrangian is

Original: [Equation]

Original: For regular φ -divergence

Original: [Equation]

Original: we have

Original: [Equation]

Original: Here the last equality holds according to the definition of the convex conjugate φ ∗ ( · )

Original: [Equation]

Original: K Experiments on additional baseline and dataset

Original: In this section

Original: [Equation]

Original: we conduct more experiments using two-dataset approach of [8] as an additional baseline and default of credit card clients dataset from UCI [13] as an additional dataset.

Original: K.1 Adult with ε -demographic parity (continued)

Original: We continue the experiments on Adult dataset with ε -demographic parity in Section 5.3 and I.1 by adding two-dataset approach of [8] as an additional baseline.

Original: The two dataset approach of [8] splits the training set into two parts, one for updating the model parameters and the other for updating the Lagrangian multipliers, with the goal to improve the generalization of fairness constraints.

Original: Figure 6 demonstrates that the two-dataset approach marginally improves the frequency of constraint satisfaction over the SAA.

Original: The cost of such an improvement is increased variation in classification error rate.

Original: Due to the fact that the two-dataset approach does not use the entire training set to update the model parameters, statistical efficiency is sacrificed.

Original: Although the two-dataset approach outperforms the SAA in terms of constraint satisfaction frequency, both are inferior to our methods, which achieve the user-prescribed level of frequency

Original: [Figure]

Original: K.2 Adult with ε -false positive rate parity (continued)

Original: We continue the experiments on Adult dataset with ε -false positive rate parity in Section I.2 by adding two-dataset approach of [8] as an additional baseline.

Original: Figure 7 demonstrates similar patterns as Figure 6.

Original: The two-dataset approach improves constraint satisfaction frequency over the SAA, but has a worse classification error rate or one comparable to it.

Original: Our methods achieve desired frequency of constraint satisfaction at the user-prescribed level.

Original: [Figure]

Original: K.3 Credit with ε -demographic parity

Original: Predicting whether or not an individual defaulted on a loan is the classification task of the UCI default of credit card clients (Credit) dataset.

Original: Membership in the demographic group is determined by an individual’s level of education: A = 1 if a person has earned a graduate degree; otherwise, A = 0 .

Original: We consider the fairness goal to be ε -demographic parity:

Original: [Equation]

Original: where A = 1 for individuals with a graduate degree is the advantaged group

Original: [Figure]

Original: As depicted in Figure

Original: [Figure]

Original: our methods achieve the level of constraint satisfaction frequency specified by the user at the expense of a slight increase in classification error.

Original: [Figure]

Original: K.4 Credit with ε -true positive rate parity

Original: Similar to the preceding subsection, now we consider the fairness goal to be ε -true positive rate parity:

Original: [Equation]

Original: where A = 1 for individuals with a graduate degree is the advantaged group

Original: [Figure]

Original: The patterns shown in Figure 9 are similar to that in Figure 8.

Original: The SAA has the lowest error rate but the worst generalization of constraint satisfaction.

Original: The two-dataset approach increases the SAA’s frequency of constraint satisfaction at the cost of an increase in classification error rates.

Original: Our methods can achieve the desired high probability of constraint satisfaction

Original: [Figure]

Original: whereas neither of the two baselines can

Original: [Figure]