Distributionally Robust Performative Optimization
Authors: Zhuangzhuang Jia
Yijie Wang
Roy Dong
Grani A. Hanasusanto
Venue: NeurIPS 2025 poster ()
Keywords: distributionally robust optimization
decision-dependent uncertainty
performative stochastic optimization
Primary area: optimization

Abstract: In performative stochastic optimization, decisions can influence the distribution of random parameters, rendering the data-generating process itself decision-dependent. In practice, decision-makers rarely have access to the true distribution map and must instead rely on imperfect surrogate models, which can lead to severely suboptimal solutions under misspecification. Data scarcity or costly collection further exacerbates these challenges in real-world settings. To address these challenges, we propose a distributionally robust framework for performative optimization that explicitly accounts for ambiguity in the decision-dependent distribution. Our framework introduces three modeling paradigms that capture a broad range of applications in machine learning and decision-making under uncertainty. This latter setting has not previously been explored in the performative optimization literature. To tackle the intractability of the resulting nonconvex objectives, we develop an iterative algorithm named repeated robust risk minimization, which alternates between solving a decision-independent distributionally robust optimization problem and updating the ambiguity set based on the previous decision. This decoupling ensures computational tractability at each iteration while enhancing robustness to model uncertainty. We provide reformulations compatible with off-the-shelf solvers and establish theoretical guarantees on convergence and suboptimality. Extensive numerical experiments in strategic classification, revenue management, and portfolio optimization demonstrate significant performance gains over state-of-the-art baselines, highlighting the practical value of our approach.

Distributionally Robust Performative Optimization
Zhuangzhuang Jia', Yijie Wang', Roy Dong', Grani A. Hanasusanto"
Department of Industrial and Enterprise Systems Engineering
University of Illinois Trbana-Champpe 2School of Economics and Management, Tongji University
12j12, gat.h//ann..aahhhiiiii edu, yijiewangótongji.emm edu.e
Abstract
In performative stochastic optimization, decisions can influence the distribution random parameters, rendering the data-generating process itself decision-deeendent. In practice, decision-maker rarely have access to the true distribution map and must instead rely on imperfect surrogate models, which can lead severely subop- timal solutions under misspecification. Data scarcity or costly collection further exacerbates these challenges real-world settings. To address these challenges, we propose a distributionally robust framework for performative optimization that explicitly accounts for ambiguity in the decision-dependent distribution. Our framework introduces three modeling paradigms that capture a broad range of ap- plications machine learning and decision-making under uncertainty. This latter setting has not previously been explored in the performative optimization literature. To tackle the intractability of the resulting nonconvex objectives, we develop an iterative algorithm named repeated robust risk minimization, which alternates be- tween solving a decision-independee distributionally robust optimization problem and updating the ambiguity set based on the previous decision. This decoupling ensures computational tractability at each iteration while enhancing robustness model uncertainty. We provide reformulations compatible with off-the-shelf solvers and establish theoretical guarantees on convergence and suboptimality, Extensive numerical experiments in strategic classification, revenue management, and port- folio optimization demonstrate significant performance gains over state-of-the-art baselines, highlighting the practical value of our approach.
Introduction
Decision-maker's actions often have ripple effect on the external environment, which can lead to changes in the distribution of uncertain parameters. For instance, in the realm of portfolio management, institutional investors' decisions can have profound effect on stock prices. This is partly due to their substantial capital, which can propel stock prices to rise (or fall) when they are buying (or selling) [57], and partly because their actions shape market sentiment towards those particular stocks [2]. Similarly, in revenue management, airlines often make forecasts and design pricing strategies based on the historical patterns of passenger behavior. However, passengers are not static; they often modify their behaviors in reaction to the new pricing strategies of airlines, which in turn shifts the overall pattern of consumer behavior [
When the solution of stochastic optimization problem affects the distribution of the uncertain parameters, we call such a problem performative 440 15]. The primary goal of the decision maker within such a dynamic environment is to find a decision that minimizes the expected risk after the environment has reacted its deployment. natural way of introducing this decision- -deeeddet uncertainty is to construct a distributional map from the set of decisions to the space of distributions.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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However, the true underlying map is typically unknown in reality. Consequently, practitioners usually rely on a nominal or reference distribution map constructed from historical observations or domain expertise. While methods grounded in the reference distribution map might perform satisfactorily on the observed samples, they often fail to achieve acceptable performance in out-of- sample circumstances.
This paper aims to tackle this core deficiency by leveraging the ideas of Distributionally Robust Optimization. Unlike traditional approaches that assume single distribution map, distributionally robust performative optimization (DRPO) adopts a more flexible strategy: it establishes an ambiguity set of plausible distributions centered at the reference distribution. Next, the objective of the decision maker is derive an optimal decision that minimizes the worst-case expected risk, where the worst case is taken over all distribution maps from within this ambiguity set. By optimizing against this adversarial perspective within a neighborhood of the reference map, it mitigates the overfitting issue and improves the out-of-sample performance.
The difficulty in solving DRPO stems from the dependence of the distribution map on the decision, which prohibits the direct use of existing solution schemes from the robust and distributionally robust optimization literature. For instance, even under the simplest setting where the loss function is linear and the distribution map is linear in decisions, the resultant problem is a nonconvex bilinear program. We address this challenge by designing a repeated robust risk minimization algorithm. Specifically, the decision maker repeatedly obtains an optimal decision that minimizes DRPR risk using the reference distribution from the previous iteration. Finally, we conduct theoretical analysis of the algorithm, providing convergence and sub-optimality guarantees. Our main contributions can be summarized as follows:
1. General DRPO Framework: We propose distributionally robust framework based Wasser- stein distance for performative optimization. This approach enables safe decision-making when we lack full information about the underlying pacision-depender distributions and only have access to some reference distributions. It encompasses a wide class of loss functions and can be applied numerous machine learning and decision-making problems. As byproduct of our reformulations, we identify a relatively general setting where the distributionally robust model is equivalent to Tikhonov regularization.
We develop a repeated robust risk minimization algorithm for the problem that effectively mitigates the intractability of pecision-deeenden uncertainty. Our approach decouples the decisions associated with the ambiguity set from the expected loss, optimizing the latter while fixing the former to the previous decision in each iteration. This transforms the challenging DRPO problem into sequence of tractable conic programs, rendering the framework computationally feasible and amenable to solutions via off-the-shelf solvers.
Repeated Robust Risk Minimization Algorithm:
3. Rigorous Theoretical Guarantees: We provide convergence results of the repeated robust risk minimization algorithm the stable solutions for our proposed models. To our knowledge, the convergence of such an algorithm has not previously been established in the distributionally robust settings. Our results show that, in general, the distributionally robust models lead to faster convergence than non-robust schemes. We further prove that our stable solutions are near to the robust performatively optimal ones.
1.1 Related Work
Our study closely relates to body of research delving into endogenous uncertainty in stochastic optimization. Early examples include production planning problems with decision-dependent production costs (24) and oil planning problem with decision-dependent informa- tion discovery [8]. Their solution entails constructing a scenario-tree-ben
Stochastic Optimization:
stochastic programming model and implementing decomposition algorithm for resolution. Building on this, subsequent study [9 extends the methodology to the multi-stage setting, focusing on production scheduling that minimizes costs while fulfilling diverse product demands. More broadly, (54) explore general multi- stage stochastic optimization problems beset with endogenous uncertainty, devising a conservative solution framework by leveraging piecewise linear decision rule approximations.
Robust Optimization: In robust optimization, decision-dependent uncertainty is usually imposed directly on the uncertainty set. Specific applications include customized endogenous uncertainty

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sets for software partitioning [50] and robust scheduling, where the uncertainty set is constructed as a decision-dependers combination of simpler sets [56]. Decision-dependern sets have also been
designed for primitive uncertainties in control systems 663 The complexity analysis of the robust
selection problem with decision -deeeeddede information discovery is studied by [33] where they present polynomial complexity results for two special cases. For general settings, [37 show the problem is NP-complete and demonstrate the benefit of using endogenous uncertainty sets via shortest-path problem. Algorithmic approaches have been developed, including exact nested decomposition schemes for two-stage problems (38] and approximation methods based on decision rules for the multistage setting [62].
Our research expands upon the conventional framework of distributionally robust optimization by incorporating the influence of racision-making on the probability distribution. However, due to the problem difficulty, decision -deedddnnndeeeeentt ambiguity sets generally lead to intractable reformulations, which could be computationally intensive for large instances. [61] study a broad class of distributionally robust optimization problems with decision- dependent moment ambiguity sets and conduct stability analysis. For the two stage setting, [3] analyze a wide range of decision-dependent ambiguity sets and establish non-convex semi-infinite reformulations and [23] explore pecisio--dependet information discovery using the K-adaptabilityy approximation scheme. In multistage settings, [59] focus on moment- based ambiguity sets and derive mixed-integer semidefinite programming reformulation. This deccisionddpennde
Distributionally Robust Optimization:
DRO framework has been applied to a wide range of operations management problems, including nurse staffing [47], facility location [5], and retrofitting planning [14. In contrast to the existing approaches that require an explicit specification of the ambiguity set, our scheme relies only on reference distributions at finitely many decisions.
Performative Supervised Learning: Our study also belongs to an emerging class of research on performative supervised learning, where algorithmic predictions can actively mold the surrounding
environment and alter the underlying distributions of uncertain parameters [15] 20 40 32]. The
origins of performative learning can be traced back to studies on supervised learning under distribution
drifts 3 4 16). The foundational framework of performative prediction was introduced by [40] who designed retraining algorithm and analyze the convergence behaviors to stochastic performatively stable points. Unfortunately, the convergence results do not extend to the robust settings, as taking worst-case expectations introduces non-smoothness into the objective function. Gradient-hard basse methods have been developed in 40 32 15] for non-robust formulations, but they are inapplicable in robust settings due the intractability of computing gradients of worst-case expectations. (28 has extended the scope of performative prediction to decision-making via performative omniprediction. We refer readers who are interested in performative learning to a comprehensive review (21]
Finally, we would like to highlight the difference between our paper and two related works on distri- butionally robust performative prediction. [39 propose a distributionally robust performative model to promote machine learning fairness. However, their robust smoothness assumption on the objective and robust sensitivity assumption on the worst-case distributions unrcalistic. Additionally, unlike the Wasserstein ambiguity set adopted in our paper, their phi-divergence ambiguity set precludes any continuous distribution, so it cannot ensure true probability distribution coverage guarantee. 588 study distributionally robust performative prediction based on a KL-divergence ambiguity set. Similar to 39), such an ambiguity set may fail to provide sufficient protections against distributions whose scenarios do not coincide with those in the reference distribution. In the paper, the authors develop alternating minimization (coordinate descent) algorithm to solve the distributionally robust model. Unfortunately, such an algorithm may fail to converge, and the paper also does not provide any convergence guarantee. The algorithm also assumes the existence of solver that can solve performative risk-sensitive minimization problems; however, to our knowledge, there is no existing literature that studies such problems and provides convergent algorithms.
1.2 Notation and Terminology
We use R4 and R to denote the sets of nonnegative and strictly positive real numbers, respectively. The identity matrix is denoted by I. For any n E N, we define [n] as the index set [I....,n,} The Dirac measure concentrating unit mass & < = is denoted by &g- For any real-valued matrix A, its Schatten-q norm defined as ||||, = (tr(A A A///239, Random variables are designated with tilde signs (c.g., 2), while their realizations are denoted by the same symbols without tildes (c.g.. z).
3

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2 Preliminaries
In stochastic optimization, the goal of the decision maker is to find a decision E C R that minimizes the risk
R(Ø) Ep ((2,)));
(I)
where the vector e Z R comprises the random parameters, and IP denotes the underlying distribution. We assumee that the loss function is convex in 0 for any fixed & E Z. When the solution of stochastic optimization problem affects the distribution of the uncertain parameters, we call such problem performative 40 15 A natural way of introducing this decision-dedt -eeesidnndeeedeett uncertainty is through a map (--) from the set of decisions to the space of distributions. Hence, in performative stochastic optimization problems, the objective of the decision maker is to obtain decision 0 that minimizes the performative risk
PR(O) = ( ({(2, 0)},
where P(O) is the distribution of uncertain parameters given the choice of the decision O. Unfor- tunately, the true underlying distribution map PP)) is unknown to the decision makers and usually approximated using the reference distribution P() = E |) P. (0) where (2)(
represent plausible scenarios and (0)( (E[S) are their respective probabilities. This reference distribution could derived from observations or expert knowledge elicitation.
Although methods based on a reference distribution may perform well when the true distribution closely aligns with it, their performance often deteriorates when this assumption is violated. Distri- butionally robust optimization has proven effective in addressing such distributional ambiguity in non-performative stochastic optimization settings. Unlike traditional approaches that assume single distribution map, distributionally robust optimization adopts more flexible strategy: it establishes an ambiguity set IB (0)) of distributions that are close to the reference distribution. In this paper, we construct this ambiguity set based on the Wasserstein distance. formal definition of the Wasserstein distance is as follows.
Definition 1 (Wasserstein metric). For any r 1, let M( be the space of all probability distri- butions Q supported on Z satisfying Eg (((c(,==))] = Je c(,, (o)(((((z < co, where 2o E Z is some reference point and c(z, 20) a non non-negative, continuous and thus lower semi-continuous (55) reference metric on Z. The type-r (1 S r) Wasserstein distance between two distributions Q and Q2 is defined as
( ((zzı,z2)) 1
W, WQ1 (Q)
inf rET((U,
where IT(Q1 Q2) is the set of all joint probability distributions of random vectors 21 and ø2 with marginals Q1 and Q2, respectively.
The Wasserstein metric offers natural way of comparing two distributions when one derived from the other by small perturbations. The decision variable can be interpreted as transportation plan for moving a mass distribution denoted by Q to another one denoted by Q2. where the transportation cost between two points 2 and is given by c(21, 22). Therefore, the y-Wasserstein distance can be viewed as the r-th root of the minimum transportation cost between Q and Q2. An important advantage of the Wasserstein distance is its ability to handle distributions with non-overlapping supports, i... even when Q1 and Q2 have different supports, the Wasserstein distance provides a finite, meaningful value. By contrast, measures like KL-divergence become undefined under such scenarios. We now consider the following Wasserstein ambiguity set
B(P()) (Q E M(=) W, ((0)) sp}.
which neighborhood around the reference distribution P0()). The Wasserstein ambiguity set contains all the distributions whose r- Wasserstein distance from P((0) is less than or equal to
Equipped with the Wasserstein ambiguity set, the decision maker minimizes the distributionally robust performative risk
DRPR(0)
sup
Eq ((2
(c

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where Q is distribution from within the prescribed ambiguity set (0)). In other words, the model optimizes the expected risk over the worst-case distribution map, thereby mitigating overfitiing to the reference distribution and improving generalization performance to other plausible distributions.
We now introduce the following concepts regarding the optimality and stability of the solutions.
Definition (Robust performative optimality). A decision @RpO robust performatively optimal the following relationship holds:
ORPO E arg min sup Ee (),
B QEB(((0)
Definition 3 (Robust performative stability). A decision ORPs is robust performatively stable if the following relationship holds:
PRPS arg min sup 8c QEB(F(Onps))
Bg ((2
While distinct from the robust performatively optimal solution, a robust performatively stable solution constitutes a fixed point of the problem and is optimal with respect to the worst-case expected loss over the ambiguity set it induces.
Definition 4 (Robust decoupled performative risk). We define
(,) = sup
Bg 6)
QEB(P(n)
as the robust decoupled performative risk, separating the decision n associated with the ambiguity set and the decision O associated with the risk; then, fRPs arg ming Jones
3 Repeated Robust Risk Minimization Algorithm
In this section, we introduce the repeated robust risk minimization algorithm for solving the distribu- tionally robust performative risk minimization problem and investigate its fundamental properties. The algorithm starts with an initial solution 0, and for every t 2 0, the subsequent 0,+1 can be obtained by solving the following robust risk minimization problem:
min Je. (0). 0
(RRMP)
The algorithm addresses the computational challenges posed by decision-dependeed distributions by constructing the ambiguity set using the reference distribution based on the optimal decision from the previous iteration. Thus, it effectively decouples the current decision from the ambiguity set, simplifying the optimization process.
3.1 Models and Their Tractable Reformulations
We present three models that cover a broad spectrum of problems machine learning and decision- making under uncertainty. We first consider the case where the loss function is given by the composition of Lipschitz continuous function and a quadratic function. This class of problems is particularly relevant in machine learning and robust statistics, as it captures many commonly used models such as linear regression (36), logistic regression 22 and certain types support vector machines 448..
Model 1. Assume that the loss function is defined as
e(Z, -0 2z 0
(5)
where (Ý, 2.) includes random variables Ý E e & Rd and 00 E R. The function L(-) assumed be L-Lipschitz continuous.
We consider the 1- Wasserstein ball, where the ground cost function c is given by the Schatten-co norm. Under this setting, (RRMP) is equivalent to the Tikhonov regularized problem
[< 10 10Y+022 25 0 =0)] +L]((0,,))
inf E Eo, 00
5

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This result demonstrates that, under mild assumptions, {RMPP) can be reformulated as regularized risk minimization problem. This substantially generalizes the findings of [30], who established an equivalence to Tikhonov regularization for strongly convex quadratic loss functions and martingale- constrained Wasserstein ambiguity sets. Our result reveals that Tikhonov regularization can be obtained for broader class of loss functions without requiring complicating martingale constraints, thereby sharpening the theoretical understanding of the connection between distributional robustness and regularization.
The following model provides a general formulation that typically considered in the distributionally robust optimization literature [43 44 This model is pertinent to many applications in decision- making under uncertainty, including inventory management (29) and and energy systems [27]. To the best of our knowledge, such formulation has not previously been proposed in the performative optimization literature, which mainly focuses on prediction problems. Note that if the I--lipschitz continuous function L in ( is piecewise linear, then this model constitutes a generalization.
Model 2. Assume that the loss function defined as
((Z 0) = max Q2( JE[J]
where 2 = 22)2 includes random variables Y E SN, 2 E Rdd and 20 E RR. Each component (,,0) is quadratic finnciinn of the form
Q, ())) 20) z + c, (0)2²
(a)
with parameter-deeendent coefficients given by affine functions aj a0) aj A,0. by( b, ,,,, and ey (0) 2g0 + ē 0 for all j E [J], where J,, E IRN, 330 IR, A,, B, E Nxd and 22 E I...
Consider the I-Wasserstein ball with the Schatten-co norm ground cost. Then, the optimal value of the following exponential conic program provides an arbitrarily tight conservative approximation for {RMP).
inf
(0.)t + ts++
SEIS
s. t.
0 E ER
+1 Vs (S +1
Tej E R, (rej. M, So.j t.) E Kexp
A, Y A,O + (2a) ,A, + 22, B, + 22
(6)
ta y aa + 2b 2a + +222000 S Co.
Σ Taj Sp 4
Vs [S +1.
JEJ
where Y s+1 = pl, 2g41 = 0, *S+1 = and μr E 1. is the smoothing parameter.
20
The formulation (6) relies on the exponential smoothing techniques described in 7 Section 2.2]. The primary motivation for this approach lies in the need to handle the nonsmoothness incurred by the inner maximization over quadratic functions, which will hinder the convergence of our proposed algorithm. To address this challenge, we apply a log-sum-exp smoothing approximation with smoothing parameter ya > 0, which yields smoothed robust decoupled performative risk, denoted as Jb. And problem (6) is equivalent to infage Jeg (0) (see details in Appendix D).
As is standard in exponential smoothing, the smoothed objective serves uniform upper bound to the original nonsmooth function: Jo. (0) < J (0) AO E @, which ensures that optimization over the smooth surrogate does not underestimate the original objective. Importantly, Jh epi-converges to Je, as the smoothing parameter μl 0 [46 Thcorem 7. 17], which ensures convergence of optimal
solutions whenever + is compact 46 Theorem 7.33].
Finally, inspired by minimax (52] and adversarially robust optimization literature 49 we turn the setting where the loss function is convex-concave In this setting, we exploit the 2- Wassersstnin ambi- guity set to ensure the convergence of the repeated risk minimization algorithm. This model further allows one to impose additional structural support information that may reduce the overconservatism of the distributionally robust solutions. The following theorem provides a convex reformulation for the model, leveraging convex conjugate representations and support functions.

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Model 3. Let 2 C Th be nonempty, convex and closed set, and consider the 2- Wasserstein ball, where the ground cost с given by the Euclidean norm on R.. Suppose that for every 0, the function f( 0) is proper, concave, and upper- seneeeeeeiiooiuuuus in z. Then, the optimal value of the following finite convex program provides an arbitrarily tight conservative approximation for RRMP..
inf Σ (0.)
[1e S.. tzz((.) - -r 4A |1ra12)) + +
1
SE|S OEO, EER r, Rh as [S]; S. E IR" Vs [S]
where T E RR4+ a constant, [-(]" (E: supzER= E - i-((( 0)] denotes the conjugate of -I with respect to z, and oz (5( 2 & z is the support function of 2 E RR.
Note that in problem 7 the loss function (zz 0) and the support set Z enter the formulation through the convex conjugate of the negative loss [- ; 0). and the support function az respectively. Both transformations yield convex functions under the assumptions stated in Model Furthermore,
the term (1/4X) |r>1| is jointly convex in (A, r.)) [ section 3.2.6]. Consequently, all objective and constraint functions in problem 7 are convex, and the overall optimization problem is manifestly convex.
The reformulation in Model 3 introduces the dual variable A. which appears alongside the decision variable O. This motivates the use of an augmented vector (0, A) E x I2+ = O To ensure strong convexity in the joint variable 0 which important for the convergence guarantees the R algorithm, we introduce regularization term 7X2, where > 0. This leads to the regularized robust decoupled performative risk, denoted by Jy (0). And problem 7 is equivalent to inf T N7 (0). Notice that for any O E θ, infx20 Jz. (0, X) converges to Je, (0) as 0. If ((z, 0) is lower semicontinuous in 0, then so is Jo.- Hence, inf Jo, (t,A), epi-converges to Jl, (-) 46 Theorem
7.17), and the minimizer 0 of (7) converges a minimizer of (RRMP) whenever is compact 46 Theorem 7.33].
3.2 Convergence Analysis
We now establish convergence guarantees for the RRRM algorithm when applied the reformulations introduced in Models and 3 As noted earlier, we employ the smoothed objective Jg (0)
for (RMPP) in Model and the regularized objective J. (0) Model Each model requires
specific assumptions to ensure contraction of the risk map and hence convergence:
(A1) Model] The loss function satisfies the 7-strong convexity
B and 3-smoothness (B2
(A2) Model For all j E [J], A A, > and let 2 minjel]] Amin A, AA/A, Ay). In addition, the
feasible set O and the support of (0.) are bounded.
(A3) Model 3
The loss function satisfies the 7-strong convexity Additionally the support has finite diameter D sc.
B and B-jointly smoothness B2
Under these respective conditions, the algorithm converges linearly to stable point.
Theorem Suppose the loss functions in Models and Assumptions (Al) (A2) and (A3)
respectively, and that the distribution map (-)) satisfies the c-sensitivity condition (B Then:
(a) 10++1 - 0''41l1a S Exi|00, - 0''122 for all O., O2 E.
(b) en < the iterate 0: of (RRMP) converges linearly to a unique performatively stable point Ours:
0 ORPS|I2 A i (1 ex:
loog (l1O, et ORPS l2/
where A > is a predefined tolerance level. The fixed point 0RPS depends on the model and is denoted 0 frss under Model and furs under Model 3
Here, 8/(r + 2pL) for Model 4D) min for Model where ki, kig < oo are model-dependem constants.
Jaka (Nr
[1ikz +1) pax for Model 2 and (B
7

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This convergence result highlights several compelling advantages of our DRPO framework, particu- larly when contrasted with traditional approaches to performative optimization that necessitate strong convexity and smoothness for convergence. First, our DRPO framework establishes convergence guarantees for loss functions that are convex not necessarily strongly convex. Additionally, our exponential smoothing tricks applied in Model 2 enable the RRRM algorithm to converge for non-smooth loss functions. Finally, our DRPO framework accelerates the convergence rate compared with its non-robust counterpart. These advancements significantly broadens the scope of problems amenable to performative optimization and enhance the computational efficiency in practical problems.
3.3 Suboptimality Guarantees
Our next result demonstrates that the robust performatively stable solution ÓRPS close to the robust performatively optimal solution ORPo whenever the e-sensitivity of the distribution map is small, the loss function has large strong convexity parameter 7, or the distributionally robust model induces regularization with large strong convexity parameter p.
Theorem Suppose all conditions in Theorem hold. Additionally, assume that the loss function is L. L:-Lipschitz in z for both models, and Lo-Lipschitz in 0 for Model Then, the suboptimality gap between the robust performatively stable solution and the optimal solution under the true distribution is bounded as follows:
2..12
(a) Model
1
Jours (ORPs) - Jonro (ORPO) S
"42pL"
(b) Model
2
Jy
(Oirs) JORPO (ORPO) S 2(eha+2p' Jy
0kss
pix
(c) Model
3
Jo
@RPS) - Joaro (PRro) S 5TX²
(2+X+0' D² L'tth.th.)). miu(7,2r)
RPs
Here, μ² E [0, 1] is a constant that satisfies p S μ1 1' ORPOl12, and X is the upper bound on the optimal X as given in Lemma F.I
The suboptimality bound for Model 2 also highlights the advantages of our DRPO framework alongside the smoothing tricks. For non-smooth, convex but not strongly-conver functions, the suboptimality bound under the traditional performative prediction framework can be arbitrarily large (as + 0, p - 0) as suggested by our result.
4
Experiments
In this section, we present numerical experiments to evaluate the performance of our proposed models across three applications: strategic classification, revenue management, and portfolio optimization. All experiments were conducted on a laptop equipped with a 6-core, 2.3 GHz Intel Core i7 CPU and 16 GB of RAM. The optimization problems were implemented Python 3.11.
4.1 Strategic Classification
We consider a simulated strategic classification problem from 40 using a class-balanced subset of a Kaggle credit scoring dataset [25]. The dataset contains features ă E Rd about borrowers, such as their ages and the number of open loans. The outcomes ÿ E {-1, are equal to if the individual defaulted loan and -1 otherwise. The institution's objective is to predict whether an individual will default on their debt.
Under the strategic classification setting, individuals respond to the institution's 's classifier by altering their features to increase their likelihood of receiving a favorable classification. The institution employs logistic regression for classification, with z .ÿ. and the loss function is given by where L represents the logloss function with Lipschitz constant = 1, and the quadratic function T + 2z T e + simplifies to the affine function z 0. See Appendix G additional details.
log(1 + exp(-x 0y)). This setting aligns with Theorem |
We compare the performance of our robust models (with type-1 and type-2 Wasserstein ambiguity sets) against the alternating minimization algorithm under KL divergence ambiguity set (AMKL)

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from 158]. Additionally, we include non-robust model as a baseline for comparison. The training set is fixed at 200 samples, while approximately 3,600 data points are used for out-of-sample testing. This setup reflects realistic scenarios where data collection is costly or limited. In credit scoring, for example, obtaining labeled data often requires expensive evaluations, expert assessments, or lengthy observation periods. All models are trained for 40 iterations, with the robust parameter set to 0.1 for all robust variants.
0.70
Hokal topol
AMKL
Acoursyy
Hexal trged
0.60
0.55
0.30
0.45
0.01
Sensitivity, e
Figure 1: Out of sample performance of different approaches
Figure I displays the box plots of 50 independent trials. As shown, our robust models outperform the AMKL algorithm, which performs quite poorly. This may be due to its susceptibility to local optima, as well as the limitations of the KL divergence ambiguity set, which may not sufficiently guard against distributional shifts that not well represented in the reference distribution. All robust models significantly outperform the non-robust baseline, highlighting the effectiveness of distributional robustness. Finally, we find that the robust model with type-2 Wasserstein ambiguity set outperforms the one with a type-1 Wasserstein ambiguity set. This arises primarily due to the geometry of the ambiguity and the 2- 2-Wasserttein ball often provides better performance because it offers a wider range of radius values for which the robust solution outperforms its non-robust counterpart.
As discussed in [IO], selecting the optimal radius p is challenging,
4.2 Revenue Management
In this experiment, we address the rev- enue management problem where the decision-maker determines the unit price 0 for a fixed quantity of perishable products as hotel rooms or airplane seats, un- P(O),
30003
Optimality Gap (%)
Mean Profit
2000
20000
z,+. such
20000
der uncertain demand with higher prices inducing lower de- mand
22000
Opáanal Nescobusi Robsd
Robsat
400
sio
2ie
690
Quantity,
Quantity,
we model the price- dependent demands using an additive function E(0) at + + Here,
Following 41
Figure 2: Out-of- sample performance of pricing schemes
unknown parameters > 0 and b > 0 capture a deterministic linear demand curve, and ? is random variable with bounded support that is characterized by an unknown density function. Instead of the true model, we only have access to surrogate model -a0 + b, which deviates from the true model as the parameters and 1 do not accurately represent their counterparts. Under this additive demand setting, the associated loss function becomes piecewise quadratic function in 0, allowing us apply Model to formulate a robust version of the revenue management problem, which can be efficiently solved within seconds using an off- the-shelff commercial solver as it is convex problem.
Figure compares the out-of-sample performance of our robust scheme (orange) with benchmarks. Due to the non-smoothness of the loss function, the AMKL approach is not applicable in this setting. As shown in [1), when the true distribution map is available, the optimal price (green) can be derived in closed form, achieving the highest expected profit for each fixed quantity q in the left subplot. Similarly, the non-robust price (blue) can obtained by treating model. The confidence region in the left subplot, representing the 10th-90th percentile range of 100 independent tests, and the smaller optimality gap relative to the mean profit induced by the optimal price in the right subplot show that our robust price consistently outperforms the benchmark.
surrogate model as the true

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4.3 Demand Response Portfolio Optimization
We evaluate our robust scheme in a power system application, focusing on demand response (DR) portfolio optimization [51] 60] In electricity markets, consumers capable of lowering electricity consumption during certain periods are called DR resources. In this experiment, we consider DR aggregator (decision-maker) managing n. DR resources over planning horizon of I periods. The goal is to maximize the expected profit by determining commitment level 0, E RT to meet required deterministic demand reduction D, at each time E (7].
The challenge lies in the uncertainty of DR resource's 's performance, where the scheduled commitment level O₂ may significantly differ from the actual reduction level 0g due to random noise ž.. This noise is decision- -eeeeden,,, as larger commitments lead to higher variability. In this experiment, we model the actual reduction of each resource i as d,, = 0 2z. for all i E [n]. Here, the multiplicative noise follows a beta distribution whose parameters a and depend linearly on the commitment level: E,i ~ 2 B 3 a,Qr, + b,) with < O. This beta distribution has support of [0. 2] and a mean of 1, regardless of the value of a,0r., + bg. Therefore, the decision 0t,, only influences the distribution shape, with higher commitment levels leading to heavier tails, hence, higher variability of the actual reduction level Oz.i-
We consider three DR resources with distinct character- istics: Resource has high revenue but large variability, Resource 2 has low revenue but high predictability, and Resource offers balanced trade-off between the two.
Mean Profit
We follow the experiment setup in 12). including the loss function and the values of unit revenue, coeemmemmitment cost, and modeimcommmtment penalty. As their loss func- tion is piecewise-linear, the resulting optimization problem can formulated using Model 2
-1000
Non robust
Robist
and efficiently solved, For further details, we refer the reader to the original paper. We conduct two out-of-sample tests: low and high demand loads over the planning horizon, corresponding to small Dr and large Dr. respectively. Figure compares the mean profits of our robust scheme (orange) and the non-robust scheme (blue), showing that the robust approach outperforms the non-robust one. Notably, under high demand, the non-robust scheme often performs significantly poorly, resulting in losses in some cases.
Low
Hlgh
Demand Load, D;
Figure Out-of-sample performance of DR scheduling
5 Concluding Remarks
We have presented the first Wasserstein distributionally robust optimization framework for performa- tive optimization. In contrast to existing approaches, our framework accommodates a broader class of problems in eeeisiionmaaing under uncertainty, thereby extending the original scope of performative prediction. We proposed an efficient algorithm and established its convergence and suboptimality guarantees. To our knowledge, these theoretical results have not been previously established in the literature on robust performative prediction. Our experimental results demonstrate the superiority of approach over existing methods on the standard strategic classification benchmark, as well as in two decision-making applications: revenue management and demand response portfolio optimization.
Notably, as the ambiguity set radius P approaches zero, the robust objective coincides with the non-robust counterpart, which more directly targets the performative risk. This observation suggests a possible direction: designing algorithms that gradually shrink the ambiguity set over time, potentially trading robustness for improved approximation of the true performative risk as more information becomes available. Another promising direction is contextual performative optimization, where incorporating side information could further improve decision quality by enabling more accurate modeling of uncertainty.
Broader impacts. Our framework extends the scope of performative prediction beyond its original focus, enabling its application to wider range of ceeiisionmmainng problems. In high-stakes settings, adopting a distributionally robust optimization perspective allows our approach to prioritize safe and reliable deployment in the presence of uncertainty and potential adversarial conditions.
10

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Acknowledgments and Disclosure of Funding
Grani A. Hanasusanto is supported in part by the National Science Foundation (NSF) under Grants CCF-2343869 and ECCS- Roy Dong is supported in part by NSF under Grant CCF-2236484, Yijie Wang is supported in part by the Fundamental Research Funds for the Central Universities. We thank Hyuk Park for assistance with the numerical experiments and the anonymous reviewers for their constructive feedback that helped improve this work.
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Technical Appendices and Supplementary Material
A Preliminary Definitions
Definition 5 (Generalized strong convexity). We say that loss function {(z, 0) is y-strongly convex
in 0
(22,0) 2 (=,00) + Vol(2,0') 0¹) +
0
(B1)
for 0' E and 2 E 2. If 0, this condition reduces to the standard definition of convexity.
We will also use the following equivalent definition of strong convexity. A loss function {(2, 0) is y-strongly convex in 0 if the function
f(z,0) {(2 0) 21101
(B1')
convex for all E Z. Definition (Smoothness).
We say that loss function 6(z, 0) is 3-smooth if the gradient (z 0)
is 3-Lipschitz in 2, that is
[Vol(zz 0) To((2',0)1: S B 2'112>
(B2)
for all 2, 2 E Z. Definition (e-sensitivity). We say that a distribution map IP(() e-sensitive for 0, 0' E e Wi (P()), P(O))) s <| - 0'112.
(B3)
where W1 denotes the I-Wasserstein metric.
B Background on Exponential Smoothing
Let = (zI,...,ZZu) )n) denote finite support set. Given a loss function l θ 2 + R and distribution p An, where A, denotes the probability simplex. We define the exponentially smoothed objective as:
(E(0,2) zi) ))
(x(0) p plloog
Σ
pe exp
/1
where a >> 0 is smoothing parameter.
This function, also known the log-sum-exp function, widely used smooth approximation to the pointwise maximum function and has well-established properties in convex analysis [Bertsekas, 2015; Section 2.2]. This function provides smooth approximation max, E(0, 22) whenever p; for all i.
Properties.
The function (0) satisfies the following:
Approximation Bounds: The approximation error be precisely quantified:
max E(0, (0) < max ((), maxx((0,,, !
log
min, P.
provided p. 0 for all i. Thus, - 0, the smoothed objective (0) approaches the exact maximum, with the error vanishing linearly in up to a logarithmic multiplicative factor.
Convexity and Differentiability:: If each function {(0, 2:) is convex in A, then (0) is also convex, it is composition of convex functions closed under nonnegative weighted log-sum-exp operations. Moreover, (0) is continuously differentiable for all μ > 0, with gradient:
Vofu(() = Σ Tz(z;;0) Vol(O, 2i),
where the weight vector n. (=; E An defines softmax distribution:
P: exp (E(0, 21)un)
Ty (z1 0) Ej D3 exp (1(0) 21)/14
15

---

This smoothing mechanism not only ensures differentiability but also facilitates efficient computation of gradients for robust optimization objectives involving maxima.
C Auxiliary Lemmas
Lemma C.1 (First-order optimality condition; Section 4.2.3 in 19). Let f be convex function and let Zi be a closed convex set on which f differentiable, then
x² arg min f(æ)
and only if
ff(x*) (y 20) E. Lemma C.2 (26)). A distribution map P(.) is c-sensitive if and only for all 0, 0' E O we have
sup gEC
[0))) - En(o))((2)] S <ell0 - 02112.
where
C :o: R 1g(2) g(2')1 S L||= 2'12 Vz, E 2} is the space of all L-Lipschitz continuous functions. Lemma C.3. If fz1 0) is --strongly convex, then the worst case expectation
Jn sup Eg(((2.0))
QEB(P(n)
y-strongly convex
Proof. By the equivalent definition of y-strong convexity in convex 0. Hence, the worst-case expectation
B we have that E(z, 0) 21101|2 2 is
sup (EB(((())
Eq
((,
0)
convex in 0 since the expectation and the pointwise supremum operations preserve convexity. Thus, we have
())
sup
Eq
((( 0)
7 2
[011
2 1
QEB(F(n)
is 7-strongly convex in 0 by the definition
BT
Lemma C.4. Let x E Rd and define the smooth maximum function
fa(x) = loo
Σ
2s/P
ØE(J]
for any μ O. Then the following bounds hold:
æ. fp(a:) JEDJ
max 2y plog J.
jE(M]
Proof. Let M = maxjep] Σy. Then for each j [J], we have x, S M, and hence:
Σ
eiln Σ Min
Min.
JEM
JEM
Taking logarithms and multiplying by we obtain the upper bound:
Σ e0s/K ) s plog (JeMiN) = -clog + M.
fa(z) = plog
(8)
jE(M]
16

---

For the lower bound, observe that
Σ
Gssln 2 Miu
JELJ
since at least one term in the sum equals M/H, Thus:
La(x)) = aa log
Σ easl )
2 1 log (c M
(9)
JE
Combining both bounds in and 9
we have the desired result:
mnx Øj S fa(()) S max æj 4 log J. SE
JEl
D Deferred Proofs Related to Reformulations
D.1 Proof of Reformulation for Model
Proof. We begin by rewriting the random parameters Z
N+1
20
Next, we introduce the matrix variable
00 T 0 1
0
r
(10)
which allows us to rewrite the loss function E(Z,0) [((,, Z)). According to B Remark 1]. the worst-case expected loss over I-Wasserstein ambiguity set ((P(O.)). with cost induced by the Schatten-oo norm, is given by:
(zle(Z,0) 0)] = inf pd + Ep(o.) xER
sup C(T, Z)) A1| 211 Zes
sup (EB(((0,)
By applying Lemma 47], inner maximization problem can simplified as:
sup CT 2) 2 - Zes Hence, the worst-case expectation reduces to
C(TT, 2))
L| otherwise.
to0
Eg(e( 0)] || Fere.)L((() 2))
sup (eB()(O.))
To conclude, we observe that
Tee 0T
0
([eeT TT
0 ) ||(0, 1)
1
This completes the proof.
D.2
Proof of Reformulation for Model 2
Proof. Rewriting the random parameters Ž as
E N+N+
and introducing the matrix variable
(0) 6,(0)
b,()) (0)
r3
(11)
17

---

allow us to rewrite Q2(2,0) as (T Z). By 35 risk can be expressed as
Remark 6.6], the robust decoupled performative
(0)
sup QEB(H(.)
EC Ecle(Z,0)) =
Σ pa():)
max (2(2,,0)
max p 1 sE|j]
=E|S]
where
|ryl 0 (0) (0)
1 = -0((000,0))
by(0) ()
(0) ())
b;(0)] - a,(0)) a, + (0).
(()))
Applying the exponential smoothing techniques described smooth approximation of (0);
Section 2.2], we obtain the following
No (0) = Σ P. (0.) log SE
2,12..0//1 JE(J]
Σ
+ ps log
eell,lli/N (E(J]
Σ
(12)
where μ R is smoothing parameter. Introducing epigraphical variables, reformulate the objective function as the optimal value of the convex program
inf Σ
Po(Ot)t + ts+1
+E(S]
8. t.
to E R
Ys [S + Vs [8 + +
So.j E R
p| 111 S (S+LJ
Vj [J] Ys E [$ +1]j
(13)
Q, 1 0)
S Sen
r plog
Σ
enis/uu
St. Ys + +1].
jE|J]
The last constraint of 13 equivalent to Σje(s uesns/u-t// S a and can be reformulated using the exponential cone:
Σ
rag S (Fags - ta) E Keep E
nC[J]
where the exponential cone Kesp is defined as
Kexp = To complete the proof, substitute the expressions for T Vj [J] with
.i 2 02e 2//22 x2 > 0} [111, 0,0s),,)))))) :::ss::: :n 2 0.03 S 0).
0 A (2a) A, ビ 0 ब a. + Cjo,
and for (2, 0) Vs E [ E [J] with
, y 1 + (2a) y A + 22 B + 2/ 0 Y.aa 25, 2, + 2,0000
This completes the proof.
D.3 Proof of Reformulation for Model
Proof. By using Definition] 1 the robust decoupled performative risk Jo, (0) can be rewritten as (0) sup (EB(((0.)
Eq(e(2,0))
Σ (0:) ((z, (lz,0)Q.(dz)
sup Q.EM(Z)
#E[S]
t.
Σ (0.)
||2 -2.130. < 2
[E[S]
18

---

where 22 denotes the space of all probability distributions Q supported on 2 satisfying Eg Jz [= < sc. This reformulation follows from the law of total probabil- ity, where Q, represents the conditional distribution of given that the scenario (0,) is realized. Using the Lagrangian, we have
/ ((2, ((z.0)Q.(dz)
Jo, (0) sup inf Σ (.)) Q.EM(Z) AER dE[S]
/ || 22=-211(0.(a))
A
p²
Σ Po)
SC(S]
By the minimax theorem [53 which is valid under the assumption that é is upperssemicootiinuuus and concave 2, and the support set is convex, we can exchange the supremum and infimum to obtain:
X Σ (.) / (((2, A||= 2.112 (dz).
No. (0)
inf XE3+ ((M)
sup
sE[S]
From the fact that the space M(2) contains all the Dirac distributions supported on 2, we have
Jo. (0) = inf
p²A
Σ
(.,)) sup(e(z, 0) - A| 12
XER
SE(S]
=E2
Adding regularized term rX² where E I is positive constant, we have
(o) S inf
Σ #E[S]
(0,)
sup(e(z, 0) A||= 2.112 - +
AER_
=EZ
Therefore, minimizing the right-hand side provides an upper bound on Jo, (0). Next, we introduce auxiliary variables l, Vs [S], which yields the equivalent formulation for the right hand side of the above inequality
inf Σ (0,),, + p²> + 7X²
SE[S]
8. t.
E R to E RR [8] sup - A|I =211²) S t, Vs E [S].
(14)
*E2
By the definition of conjugate functions, we have
sup ((2 - A| 2. = [-t X2 + A/I2 (0)
=EZ
where x denotes the characteristic function of Based on results from 35 Theorem 4.2], [46
Theorem 11. 11.23], and (64 Lemma B.8). the conjugate functions of infimal convolutions and 2-norm balls given by
[-exx + A||= (0) inf (I- SS,, +zz(C.) +[]] - (-rr)) r..S,
with
[|| - 2.. sup(-r A|I= = 4A |ra122
v.
Substituting this back into the formulation 14 thus obtain that Je. (0) upper bounded by the optimal value of the following convex program:
inf Σd.00 (100 - S.. 0) + +oz((.)) rJa, 47 |ra113 + +
SEIS t. XER to ERRSS E (S]. TM Vs [S], C. Rd Vs [S].
Combining with outer minimization over 0 e O completes the proof.
19

---

E Deferred Proofs Related to Convergence
Lemma E.1. Consider the loss function defined Model 2 and assume O is bounded. We define the smoothed loss function
(z) μ poo
Σ EE(J]
(15)
where Z P(Oi) satisfies and || h1 for some constant k Then the gradient
Vol, (2,0 0) is 3-Lipschitz in Z for Jdki below.
k2 + 1)) for some constants 22, k3 < defined
Proof.
Since the coefficients (0), by (0), and (0) are affine for all j E [J] and θ bounded, there exist some constants l22 ka such that
[Ty S 22 for all j E [J],
V.. Fyll hg for all [d].j e (J].
By definition, we have the gradient of the smoothed loss
15
Volu(Z,0) = Σ
w w,(2200VoQ,(2.00)
JEM
where the softmax weights are
(2,(Z00//u
wy (2,0) = ExeLy e9.(2.0)/1)
and the gradient of Q with respect to 0 is given by
VoQ,(Z.0) 24, YA,O 2A, Y a, +B, x 2,,
To prove that Vol, is Lipschitz in Z, we examine || (eln(Z - (el, We first decompose the difference and apply the triangle inequality
Σ jEl
wy (2,20,0000
Σ Wy Wy (Z, OV, 0)
2
s
Σ
w. (,,2,,00000220.0
Σ
w (Z, (22, 0)
EEM
Σ
w OV.00 - Σ
w,(Z2, VV 0)
jE|o
Σ
0) -
0)
+ Σ
- -
jEM
jelI
(16)
Next, we bound the terms involved:
20

---

Weight difference:
e2(21,0)/2
|m, 0) -2,(Z2; 0)1
Eiely e9.(21.0)pp Eiely e9.122,0)/p
[Q, 0) QQ2 0)112
[[T), Z1) (T3 Z2) l
IA
[Ty|| Z Z2ll2
1 Z2ll2
where (a) comes from the Lipschitz continuity of the softmax function 17. and (b) uses the Cauchy-Sehwarz inequality.
Gradient:
(b) Vo.T,112 12 S kykyd,
(2,2(2
Σ 1VV II 21 sE(d]
iE(d]
where Vo,T, is a matrix whose (i1, 1-t matrix slice the gradient of the (2) component of the matrix with respect to A,.
Gradient difference:
([Ve (Z2,
S S d 1 Z2ll2 S dkis Zi Z2ll2.
Substituting the above bounds into (T6). we obtain
0)12 S Jdkyy 1ik2 +1 +1 || -Z2l22
Thus, the claim follows.
Lemma E.2. Consider the loss function defined in Model j E [J]. Define the smoothed loss function
2 and assume that A, Aj > O for all
gng ( = a plog
eol Phl
(17)
jE(M]
where F defined in 11 Then (0) is po-strongly convex in where
= min Amin (A, (A) Aj).
JC(J]
Proof. By definition, the gradient of the smoothed loss
is given by
teg #
= Σ (,)))),
SE[J]
where the softmax weights are defined as
(0)//0
wi (0)
Vje [J].
EiELJ @0//
and the gradient T with respect given by
Vor, 24, A, 22, a1 +ē,
21

---

Using the product rule and softmax identity:
+ 8²
(0) p
Σ
wy
(O)
wy [,(0) -
jE(j]
jE[J]
( A + p²
2p
Σ
wy( Vor, - Ø) vor, Ø]
/ JE[J]
jE|
where
9 = P Σ wy ( SE(J]
Notice that the first term is a convex combination of positive definite matrices A, A,, so it is positive definite. The second term is Gram matrix, hence positive semidefinite. Therefore, the minimum eigenvalue of the Hessian is lower bounded by the minimum eigenvalue the first term and we have
grep p
ž 2p min Amin A, A,)1.
JE[M
Thus, the claim follows.
Corollary E.1. Consider the setting of Model and define the following smoothed loss function
gZz (Z, e
where (Z( is the smoothed loss function defined (55 and E79 (0) defined in (17). Sup- pose that A, A > 0 for all j E [J]. Then g ( 0) is po-strongly convex in with a = 2 minje(J] Amiu A Aj), and the gradient Vog Z, 0) is B-Lipschitz in Z for B = Jakks [ik2 +1
for some constants k1, k2, kg < co.
Proof. By standard result in convex analysis, the function (x 0) is convex in 0, the log- sum-exp operator preserves convexity when applied to a collection of convex functions. Further- more, Lemma E.2 establishes that greg (0) pa-strongly convex, where
/
a min Amin (A, Aj).
As the sum of convex function and strongly convex function is strongly convex, g(Z,0) is pa-strongly convex.
The dependence of on Z comes only through fn (). Therefore, the Lipschitz continuity of the gradient Veg Z 0) with respect Z follows from Lemma 2, which provides the bound on the Lipschitz constant B. This concludes the proof.
Lemma E.3. Assume that the loss function f(z, 0) is concave in z, and that the set Z has a finite Define the function
diameter D sup,,''z ||z z1l12
g(v, A)) = sup {zz, 0) - A||= w113 + p 7
zEZ
Then the function is a-strongly convex in (0, A) where
min(1: 2T),
and the gradient (e (0,, is 4D) in
Proof. Define
(0, f(2, 0) A/|= -0ll + p²> + rx-
Then we have
g(v, (0, sup $((0, A), n2
=E2
22

---

Since (), is 2t-stnnggly concave in v, the maximizer
z' =(), v) - arg max (), 1), v,
ZEZ
is unique. By Danskin's theorem 6 Section 6.11]. the function is differentiable, and its gradient with respect to (0 is
V (o.)))(v, (0.))) V (,,)),
- Vol(2" 1
-11=" - 211 + p² + 27
where x² o)) The Hessian is Vooyooo, (0,
v, x²) V (o)))((0,,))) 1),
Vgl(z" 0)
(18)
2T =
all.
0
where a = min(y,27) This proves strong convexity in (0, A).
Before we move to the Lipschitz smoothness, we first prove several inequalities. Fix 21, 22 2, we have
|||= 02) - v213 | ((,,)), 01) - in 11122112
(a)
[[]==((0, A), 02) 02l12 + |0 ()) v1) v)) -1112) x [[|= A), 02) 02l12 - >= |= A), v1) 01) - -n (2)
(19)
2DI ((0, 02) =" ((0, v, 11) + 2DI122 - 01l12
where (a) uses the difference of squares, and (b) uses the triangular inequality and the boundness of Z. Next we prove =0 *(0, v) is 1-Lipschitz continuous in v as follows.
2A|1= ((0, v1) - x 200 A), 22)
K V ( A), U2 20 ((0.A), v2)) + V=0((0, A), 02 x" ((0, 1), v1)),
z" z'( A), U2) -= =00
(0)
V ( A), U2 02, = (0, v1)), ((0, U2) - A(),)),
V V A), U2, ((0,)). - V20((0, (), Un x' ((0 A), v1)).
2" =0( A), U2) - z" (0,1)
(2)
22Al|21 - 22 (00,
- x² "(0 A), f1)ll
where (a) uses strong concavity of (0, A), v. )) (b) and (c) come from the first order optimality conditions for z" ((0, A), m1)) and = 00, A), 02)): VV ( (0, A), v, x (0, A), v)), x= ='((0, A), v) < 0,
and (d) uses Schwarz inequality. Hence.
((0, A), v1) =' ((0, A), 02)112 || S || 22
(20)
Finally we show (o (0, ))) is Lipschitz in v. consider the gradient difference
V Vie,))i(1,,00 (0) - Vo ())l2 (O.
Vol(="((0,))) (), [1= 1). u2) 0 = ((0, ((0,)), A), 1113
2
((,, 0) Vel(= 02)
+ ((0, - 221 - || ((, 21) -
- ~00((0 A), 22)122
(00
= ((0, A), Un +2 01l12
(8 4D) 022
vil1a -
where (a) comes from the triangular inequality, (b) uses the B-jointly smoothness of 6(2, 0) and
and (c) uses 20
Hence the gradient Ve,gg(v, (0, A)) is (8 + -ipsschchittz in
23

---

E.1 Proof of Theorem 1
Proof. Let G( denote an optimal solution of RRMP iteration t, i.e.,
01+1 = G(O,) E arg min Ne, (0).
00
where 0, is the current solution and 0++ denotes the optimal solution for next iteration. We first prove the convergence result for Model
| Observe that
Jo. (0) (ite)ll((2,) 0)
where - pT 1)
Fix 77 n @. Since (-) (7 + -stronnyy convex, where 2pT. comes from the strong convexity of the regularization term p (0, 1). we have
GD + 7 2pL
J, (,(())
[[((n) G(n')11
2
J, J,)(()))) 7 2pL.
2 [[G(n) G(n'))l
where the second inequality follows from the fact that
(G(n') - G(n)) VJ,((((n) 2
in view of the first-order optimality condition in Lemma C.1 since G(n) E a Combining the two inequalities, we obtain
(7 2pL))lG((m - G
<G(((()) Gu,
(21)
I (G(n) - (((())-Gr)))) -
where the second inequality follows from the fact that (G(n) VV (G(n'( > 0 in view of the first-order optimality condition G(n'). Next, we will upper bound using Cauchy- Schwarz inequality, as follows:
(G(n) - (G,)
<l((G(n) (( VJ,(G)))))))
[G(()) - G(n (V (Vl Em (V (V((2
[G(n) G(r')ll eBIln 1/1122
Here, (a) follows from the representation of the loss function, while (b) uses the Kantorovich- Rubinstein Lemma C.2 since the loss function is 3-jointly smooth from Lemma E.3 and the map P(0) is e-sensitive. Combining this bound with (21). we get
[[G(T) - G( <
+ 2pL 11
T7 1/'122
Our claim (a) then established by simply performing the change of variables 17 0 and r7' + 0'. To prove claim (b), we observe that 0, = G(O,-1) by the definition of (RRMP), and ORPs = G(Onrs) by the definition of stability. Applying the result of the claim (a) yields
(,
c3
00 0 0RPS||2 S ? 2pL 0. 0RPS|12 S
( 2pL
100
0RPSl12-
Setting the right-hand side expression to be at most 6 and solving for t completes the proof for Model For Model Observe that
J (0) = Ep,o,)l9(Z 0)).
where (Z,00) (2 ( and gre(()) is defined in (17 From Corollary E.
(0). Here (,, 0) is the smoothed loss function defined in (5).
know that g( Z, 0) is pa-strongly convex

---

in 0 with 2 [i]] Amin (A, Aj), and the gradient Veg is B-Lipschitz in Z for 8
Jk3 k2 + 1) for some constants k1, k2, k3 < co. Using the same techniques as in the proof of Model
1 can therefore establish the desired result for Model 2
Finally, for Model one can observe that
Jo, ( = 1 (g(
where
g (22 0) = sup {(z 0) - ^|l= 2 + p² +
ZEZ
know that g (Z, is a-strongly convex in 0 with a min(7; 2r), and that its gradient V " Z, is (3 + 4D)-Lipschiz continuous in Z. Hence, by applying the same arguments as in the preceding analysis, we obtain the desired result.
From Lemma E.3
F Deferred Proofs Related to Sub-optimality Guarantees
F1 Proof of Sub-optimality Guarantee for Model
Theorem Suppose the loss function Model 7 y-strongly corvex in 0 (BI and is 3-smooth (B2).
Furthermore, assume that the loss function {(z, 0) is L Lipschitz in 2. Then, the following subopti- mality bound holds:
2e
Toges (ORps) JØRPO (ORPO)
7 + 2pT.
Proof. Since JøRrs (0) is (7 2pL)-strongly convex in 0, we have
7 + 2pL.
|RRO Ours 2 - Joars (Onro) Joars (fars)
2
since (ORPo- ORPS) V.Jeps (ORPs) > 0 by the optimality of Orrs. Using the fact that Joars (ORps) 2 JoRP (ORro), we can further upper bound the right-hand side
Jones (ORPO) Joars (ORps)
< Jonns (ORoO) - Jonro (ORPO) eL. ORPS @RPO
(22)
where the second inequality holds due the e-sensitivity of the distribution map P²()) and the Ly- Lipschitz continuity the loss function in z. In summary, we obtain
2c1.z ORRs ARPOLL2 S + + 2pL
(23)
Next, we derive a bound on the suboptimality of the robust performatively stable solution ORPS. We have
JøRrs (ORPs) - JeRro (ORPO) S JøRrs (ORPO) - JØgro (Orpo)
< cL, |Orrs - 0nrol12
2 I
<
7 + 2pL
where the first inequality follows from the suboptimality of ORPO in Jenus (0), the second inequality from 22). and the last inequality from (23). This completes the proof.
F.2 Proof Sub-optimality Guarantee for Model
Theorem 4. Consider the loss function {(2, 0) from Model 2 Suppose that A, A, 0 for all j E [J], and that Y 2 for all [S]. Additionally, assume that f(Z, 0) is L Lipschitz in Z.
25

---

Then, the suboptimality of the stable point RPS of the smoothed robust objective satisfies:
(RRPs) - -eRRO (ORPO) < 2(ELs - 2p log J)
o RPS
(24)
pa
where a 2 minnjee jE(J] min A Aj), pL E [0, 1] a constant that satisfies μ S 1 ORPO 2. To (0) the smoothed robust objective defined in 12. and Oro denotes the minimizer of the
original robust objective Jo, (0).
Proof. By Lemma C.4 we have for any 0,
Te, (0) S To (0) le. + 2/ log J.
(25)
Additionally, from Corollary
E.1 the function Jh (0) is pa-strongly convex, with
2 minje[[] Amin (A, A,). By the definition of strong convexity, we have:
pa ||ORpo k S Kps
(ORPo)
J kps
(0 kPs))
(26)
where the inequality follows from the first-order optimality condition of 0 firs i.e.,
RRO
0
RPs)
Viu
(f 2 0.
We next bound the right-hand side of 26 Using the fact that Jøaro (ORPO) S Jon (ORps) S (Riss), we have
T
firs
(ORPO) - Oars
(0Rps)
I J (ORPO) - Jonro (ORPO)
RPS
I ohns (ORpO) JRRPo (ORpO) + 2/ log J
(27)
cl 10urs Onro l| + 2/ log J.
where the second inequality comes from 25 and the last inequality holds due the Lipschitz
continuity of ((Z, 0) in Z. Substituting (7 into the strong convexity inequality 26
gives:
pa |ORPo 0 0ess 12 <l= ORPOLL2 + 24g J.
(28)
Assuming / S 1 RPS ORPOlLa 222 where nt E [0, 1], divide both sides by |10 kis PRPOlL2 to obtain:
1 - PRPOLL2 1 2(eT.z + 2/ log
(29)
pa
Finally, we derive a bound on the suboptimality of the robust performatively stable solution 0
JH 0
(RRs))
Jonro (ORPO) S Johrs (QRPo) - Jenro (ORPO)
S (eT,g + 2p log J) - ARPOll2
< 2(c + 2/ log J)²
where the first inequality uses suboptimality of ORpO in J (0), the second follows from 27 and
the last inequality uses the bound in (29). This concludes the proof.
F33 Proof of Sub-optimality Guarantee for Model
Theorem 5. Suppose that Z has a finite diameter D supx el - < oo and O is
bounded. Assume that the loss function f(2, 0) in Theorem (3) is La Lipschitz continuous in O and b L.-lipschitz in z. Let Jž (0) denote the objective defined in problem 7 and let Uurs denote robust performative stable point under this objective. Then the following suboptimality bound holds:
(2rX x² + D2 + Lo cL2)2cLz
J (0 -eRPO S 7X² >x²²
(30)
where a = min(y, 2T) the strong convexity parameter, and X is the upper bound on the optimal
as given in Lemma Fl
26

---

Proof. From Lemma E.3
the objective function Jú, (0) is a-strongly convex in 0 (0, with min(7, 2r). Applying the strong convexity inequality, we obtain:
a
0 RRO @RPS|
||2 S Je Jones (RRO) - Jokps (ORPs):
2
where the inequality follows from (RRoO- - nnss) Vornes (RRs) 2 0 by the optimality of RRS
Using the fact that J Aro (ORPO) S J6 Jops (ORPs), we can further upper bound the right-hand side
Tokes (ORPO)
Jokps (ORPS)
S J Rps
0 RPO) J6 RPO
(RRO))
< cL, RPs - p
(31)
S cL. ORp
i RPO 2
where the second inequality holds due the e-sensitivity of the distribution map IP(-) and the La- Lipschitz continuity of the loss function in In summary, obtain
0 RRsS Rpoll2 -
(32)
Now consider the suboptimality decomposition:
Jo oirs
@ØRps) Joaro (RRPO)
(33)
J Joaps RRRS) Jorro (PRPO)
Jōāro (ORPO) JORPO (Rro))
We now bound each term:
1. First term. Before start, we first provide bound on J (Rrs)
Jā kps
(
kps
From Lemma the function f'Z, 0) is Lipschitz in 0. Hence:
J hrs Curs) J0 (RRoo)
Eecoars) (T ARPS + f(Z, f(2,0kPs) RPS)] - E Eeco [T RPO + f(Z 0 RPO))
(34)
=r|(Anps + Anro) Anro)] + Ep [f(2,0Rps) f(z, RROO)
5(2T)+ + p² + DR ARPOL + L ORrlll2.
where Ā the upper bound on the optimal X defined in (38 Now we provide bound on the first term, using the decomposition:
Jo (Rrs) Jō aro (TRo))
kps
J
Rps 0 RRPS)
Aps (0 (RRO))
Jō (RRO) Jărro @RRO)
(2r + 0² + +D2) |RRSS Aarol + Lo R O 12 + cL. 10 - 0kpoll2
(2r + p² + +D2) ARPS ARPOl + (Lg + eL, 0 krolle
(2T) + 0² + D2 + Lg 22EL
(35)
where the first inequality comes from
31 and 34). the last inequality holds because of
(32
2. Second term. Using the fact that J7, 0 augments (O) by 722:
J T (nro) Jonro @RRO)
< J enro (ORpo) Jāaro @RPO) S xX²
(36)
Substituting these into we obtain
+ Lo + cLg)2eL2 Tiss 0 (urs) Jenro (OROo) STX (2rX p² +
This concludes the proof.
27

---

Lemma F.1. Suppose that 2 has a finite diameter D = sup2, 1 < oc and O is bounded. Assume that the loss function E( 0) is Le Lipschitz continuous in and L -ippschhi in 2. Then there exist f, l such that for all (z, 0) x o
& ((. S.
Consider the following univariate minimization problem
inf p² + T2² + XER+
0,1
(37)
where
lc(2s (appl(zz.)) A11 2..12). xE2
Then, the problem
37 admits minimizer
X X, where
A 1r 4T ( 1p2
(38)
2r
Proof. First, we observe that for fixed 2.) the function la(z, 0, A) is convex and lower semi- continuous in A, it is the supremum of functions affine in X. Since lower semi-continuity is preserved under expectation over discrete distribution P())). the objective function in (37 is convex and lower semi-continuous in A.
Next, establish lower bound. Note that for all À 0,
fa A) 2 {(2.. 0) 26
since the supremum is attained at 2 2.. Hence, we can bound the objective from below:
p²> 7 T + (e (lo(, 0 A)] 2 p²> x rx2 +
Thus, the infimum of problem have:
37 is attained for some X E [0, +00). By optimality of X", we then
E ((00))((Es, 0)] <p²x + + 2 + Ex(0 Le, 0, p²x0 + T~²
$ B (2. 0,0)
S o (sup f(z, 0)] S 7
=E2
where the second inequality comes from evaluating the objective we obtain:
Rearranging the inequality,
x² + +2* < E [((E,, ((2,,)) 0)] <
Solving the quadratic inequality yields the upper bound
4T E) p²
X s =
2T
This concludes the proof.
Lemma F22. Suppose has finite diameter D sup supx,'e2 ||2 2'112 < oc and the loss function
f(z, 0) Lg Lipschitz O. For 2. E Z, define the function
f(2,, 0
sup h(z, 2, 0),
(39)
zE2
where 0 (0, e R4 and
h(2, 2., 0 = f(zz(( 0) A/I - 2.13 + p²>
Then for any 0,6 E θ, the function f satisfies the Lipschitz bound:
If f(e., 0 - (p² + XII + Lo 10000000000 0'1l2-
28

---

Proof. We start with the absolute difference between the two evaluations of f:
( 0) f(2,, 0') = max ( 0) [(2,, ((,,, 0') s(2,, 0)}
Next, we bound the first term; the second is symmetric. Let
A f( 0) f( 0') sup h(z, 2. 0) sup h(z', 2, 00
=62
z'e2
Using the inequality
supa(z) supb(z) sup(a(z) - b(z)),
obtain
h(z, 2 0) h(z, e, 8'1
A sup
xE2
[z(z, 0) [(z 0¹) (A - -x)ll= -2,1 p X)]
sup
=E2
S sup [[(z, 0) [(z, 0') + sup xe2
10² ||2 - 2 12)((A X)1 sEZ
For the first term, since f is Lg- Lg-Lipschitz in 0:
|((2, 0) [(z, 0') S Lo|10 02112-
For the second term, observe that || - -2x112 - D implies
10² - ||= 2=1121 S p² + D>2
Thus,
100 - 102-
- X'I S (0² + Dー) -|- 1.) -
By symmetry, the same bound holds for f(2., 0') f(2x, 0). This concludes the proof.
F44 Proof of Theorem 2
The proof of Theorem follows directly from the results of Theorem 3 Theorem 4
and Theorem 5
G Experiment Details
G.1 Strategic Classification
Following 40 32,. we assume that individuals have linear utilities () x and quadratic
costs c( -)== = - " T where is a positive constant regulating the cost of altering features and thus the sensitivity of the distribution map. In other words, individuals aim to minimize their assigned probability of default but are unable to change their true outcome ÿ. We select 5 C [P - 1] strategic features, such as the number of open credit lines. Each time an individual manipulates their strategic features as depicted in [40] Section 5], the best response for individual results in the update
a's = ēs -
where i's, -s. 0s E IR!S)
Robust type-1. Consider the I-Wasserstein ball, it follows from Theorem T.. at each time t, we can solve the following Tikhonov regularization problem
Jo, (0) inf (Ee,.) [log (1 + exp(-x 0ii) + pL]](0, 1)12. oce
29

---

Robust type-2. Consider the 2- Wasserstein ball, it follows from Proposition T at each time t, we can solve the following problem
a² + +
Jo, (0) inf xER
aá, 0g. xE|s) ae(-
Σ sup
42
where h(a) (a + 1) log(1 + a) a llo((-a).
Next, we introduce auxiliary variables I, Vs E [S] and combine with outer minimization over 0 E O which yield the equivalent formulation for the right hand side of the above inequality
inf
5 4 p²>
Σ
SESS
t. E O,
A te E As E [S]
(40)
a² 4
ai. 0ú. o h((a) S to Vs [S].
sup aE(-1,0)
To handle the last constraints, we discretize over some finite set Sox i.e.,
a² Ooj h(aa)= S t, Va E Sa As E [S].
at_ Oú. +
4X
For the experiment we choose Sa = 0-9, ... -013) Proposition Consider the logistic loss
(z 0) = log (1 + exp( - x (To))
with ()) and O Rd. Consider the 2- 2-Wasserstein ball, where the ground cost e is given by the Euclidean norm on R.d. Assume the support set X' is convex and closed, then we have
Jo. (0) = inf AER4
1
Σ sup +E[S]
a Oū. + a² 2 -maaa p2>, 4X
S
aE(-1,0)
where M(x))) (a + 1) log(1 + a) log(
Proof. We follow the argument in the proof of Theorem For fixed O,, we have
(0) inf XER
1
Σ
sup [((z, 0) - >| 22 p²>|
S
sez
SE
where & = E.... Note that label üs is fixed, we can simplify the inner supremum
sup [[(x 0) - A|Iz 2.113 + p²>]
#ex
Next, substitute the logistic loss
[(zg.. 0) = log (1 + exp(-x' TOg.))
Using the Fenchel conjugate dual formulation 45 of the logistic loss:
log(1 + +)= sup {ar heu--hi h(a),, where l(())) = (a + 1) log(1 + + a lolo(-a)) aE(-1,0)
we rewrite the inner supremum as:
sup log(1 (1 + exp(- TTo.)))))) Oğ.)) A||x 2..12
nEX
= sup sup fax Oūj h(a)) - 21|x - 12113}
#EX aE(-1,0)
sup (oar Oÿ. A|Iac 1,113) h(a)) x
sup aE(-1,0)
30

---

We now compute the inner supremum over z. For fixed a, Ds. and 0, the expression
0: Oú.- || -2.12
is concave quadratic in The optimum is achieved at: Substituting x* back in yields:
x x. + a 21
0i..
at Oÿ. a 0 T0..
41
Therefore, the robust objective becomes:
Ja, (0) inf AER;
1 S
Σ *E(S) aE(-1,0)
sup
02. 0ÿ, + 4A - - + p²A.
0
Alternative minimization under KL divergence ambiguity set (AMKL) Consider the Kull- back-Leibiee (KL) divergence ambiguity set. According to [58] Proposition 3.1], the distributionally robust optimization problem under KL divergence can be reformulated as:
min inf (ulogEroo, p20
[ exp(- "0i) 1/1
up)
where p 0 the radius of the ambiguity set. To solve this optimization problem. (58] propose an alternating minimization approach, which we refer to as AMKL. The procedure alternates between the following two steps:
1. 0-step: Fix the robustness parameter and minimize the objective with respect to 0.
2. μ-step: Fix the model parameter 0, and minimize the objective with respect to ja.
This iterative process is repeated until convergence. In the O-step, apply the repeated risk minimization algorithm, following the suggestion in (58) that this subproblem can be addressed using any suitable performative risk minimization algorithm.
G.2 Impact of the Robust Parameter
We investigate the impact the robust parameter p on out-of-sample performance. Specifically, we consider the revenue management problem described in section under different quantities of perishable products, q E (200, 300, 500).
Scharl
2096
300
1000
Robunt PaEATTATT,
Robust Parmiter,
Robust Parameter,
(a) 200
(b) 300
(c) g 500
Figure 4: Out-of sample performance as a function of the robust parameter p and estimated on the basis of 100 simulations.
Figure 4 iillustrates the optimal mean profit as function of p. averaged over 100 independent simula- tion runs. We observe that the oft-of-sample performance improves as p increases up to critical Wasserstein radius p", beyond which begins to deteriorate. Notably, the non-robust counterpart over wide range of values. This pattern was consistently observed across all simulation settings and provides an empirical justification for adopting a distributionally robust approach.
robust model outperforms
31

---

NeurIPS Paper Checklist
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Question:
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Answer: [Yes]
Justification: Our claims are supported by both theoretical analysis and empirical results. We present experiments the standard strategic classification benchmark, well as two decision-making problems revenue management and demand response portfolio optimization.
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The authors are encouraged to create separate "I "Limitations" section in their paper.
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The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
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While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and prool's
Question: For each theoretical result, does the paper provide the full of assumptions and complete (and correct) proof?
32

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Reviews:
[{"review_id": "mrlY5K8ias", "reviewer": "Reviewer_MXSY", "summary": "This paper tackles performative optimization where decisions influence the distribution of uncertain parameters (e.g., algorithmic predictions affecting user behavior). The key challenge is that practitioners only have access to imperfect reference distributions rather than true decision-dependent distribution maps, leading to poor performance under model misspecification. The authors propose a distributionally robust performative optimization (DRPO) framework using Wasserstein ambiguity sets to optimize against worst-case distributions within a neighborhood of the reference distribution. The core innovation is a decoupling strategy that separates the decision variable in the ambiguity set from the objective function, transforming an intractable bilevel optimization into manageable conic programs.", "questions": "While the authors mention that 2-Wasserstein seems \"limited to machine learning settings,\" I wonder if there might be alternative mathematical approaches that could potentially overcome this limitation? It would be helpful to better understand how significant this restriction might be in practice.", "limitations": "Yes. While the authors acknowledge some limitations (e.g., current focus on 1-Wasserstein, 2-Wasserstein restrictions), authors should include a brief discussion of how experimental scale might affect conclusions and the need for larger-scale validation.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1.This paper appears to make a contribution by systematically introducing distributionally robust optimization into the performative optimization framework, which seems to address an important gap in this topic.\n2.The decoupling strategy appears to be  mathematically elegant with reasonable theoretical justification.\n3.The three proposed models seem to cover a reasonably broad spectrum of applications, demonstrating the potential versatility of the framework.\nWeaknesses\n1.The convergence guarantees appear to depend critically on the ε-sensitivity assumption $W_1(P(\\theta)- P(\\theta)^`)\\leq ε||\\theta - \\theta^` ||_2$. However, it seems that the paper might benefit from providing more guidance on how practitioners could estimate or verify ε in real applications. \n2.The strategic classification experiment appears to use 200 training samples, which might seem modest by current ML standards.\n3.The paper might benefit from a clearer explanation of why existing DRO methods cannot be directly adapted to the performative setting. Perhaps a more detailed comparison could help readers better appreciate the unique challenges that necessitate the proposed approach\n4.While mentioned in limitation, the significance of restricting 2-Wasserstein to machine learning settings deserves deeper discussion.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the careful, detailed, and constructive feedback, which we will duly consider and integrate into our revised manuscript. We hope the review can reevaluate our paper based on our response below:\n\n$\\textbf{Weekness}$\n1. We thank the reviewer for the clarifying question. We would like to clarify that $\\epsilon$ is the property of the underlying distribution mapping and is not a parameter. Unlike prior work on decision-dependent distribution shift [Perdomo et al. 2020, Mendler-Dünner et al. 2020], where the mapping P itself is known, our assumption is much weaker. In practice, given two decisions $\\theta$ and $\\theta'$, along with their corresponding empirical reference distributions $\\hat P(\\theta)$ and $\\hat P(\\theta')$ based on observed samples, one can compute the 1-Wasserstein distance $W_1(P(\\theta), P(\\theta'))$ via linear programming. This allows for an estimate of $\\epsilon$ using $\\frac{ W_1(P(\\theta), P(\\theta'))}{\\|\\theta - \\theta' \\|_2}$.\n2. We thank the reviewer for pointing this out. While the number of training samples in the experiment may seem modest, this choice reflects many real-world scenarios where data collection is expensive or limited. For example, in settings like credit scoring, obtaining labeled examples can require costly evaluations, expert judgments, or long delays. Importantly, one of the motivations for our distributionally robust framework is to address such small-sample effects. By explicitly accounting for uncertainty in the data-generating distribution, DRO-based methods can improve generalization and robustness even when the available data is limited. We will clarify this motivation in the revision.\n3. We thank the reviewer for the constructive sugggestion. Traditional DRO methods typically assume a fixed but unknown data-generating distribution and aim to minimize the worst-case risk over a predefined ambiguity set. In contrast, the performative setting introduces a unique challenge: the data distribution itself depends on the model parameters. This endogenous dependence means that standard DRO formulations—which are designed to handle exogenous uncertainty—cannot directly account for the feedback loop between decisions and the data-generating process.\n\n   Moreover, a key technical challenge in the performative setting lies in identifying structural assumptions that guarantee the convergence of the proposed algorithm, despite this decision-dependent distributional shift. We will revise the paper to include a more explicit comparison between our framework and traditional DRO approaches, and to better highlight the novel challenges posed by the performative setting.\n4. We thank the reviewer for the careful feedback.  In this context, \"machine learning\" refers to regression and classification problems previously studied in the performative prediction literature [Perdomo et al. 2020, Mendler-Dünner et al. 2020, Jagadeesan et al. 2022], where the objective typically involves minimizing a loss over data influenced by the model’s predictions. In contrast, decision-making problems often involve economic costs as well as risk-sensitive objectives, such as the maximum of several functions to capture risk aversion or worst-case scenarios. These formulations introduce additional challenges under the performative setup. We note that Model 2 in our paper is designed to accommodate such settings, providing a more flexible framework that can handle these more complex, decision-oriented objectives. We will clarify this distinction in the revised version to better highlight the generality of our approach beyond standard machine learning formulations. \n\n$\\textbf{Questions}$\n\nWe thank the reviewer for this excellent question. Our results in Model 3 rely on Danskin's theorem, which requires the loss function to be concave in the uncertainty in order to get a well-defined derivative of the worst-case objective. Extending the analysis to settings where the objective involves the maximum of multiple such loss functions poses significant challenges. In particular, taking a maximum over concave functions generally destroys concavity. Addressing this would require new analytical tools. Exploring these directions is an interesting avenue for future work.\n\n$\\textbf{Limitations}$\n\nWe thank the reviewer for the careful feedback. We would like to emphasize that the limited data regime we consider is not just a practical constraint, but a characteristic feature of the performative optimization setting. Unlike traditional machine learning, where data is assumed to be drawn from a fixed or evolving distribution, performative optimization requires sampling from distributions that are themselves induced by the decisions. That is, in order to evaluate or improve a policy, one must actively deploy a decision (e.g., hiring policy, pricing policy) and then observe the resulting data distribution, which can be expensive, time-consuming, or infeasible at scale [Milli et al. 2019, Mendler-Dünner et al. 2020]. Our method is designed to be data-efficient in this regime, and we aim to achieve robust and stable performance even with limited samples per iteration. We will clarify this in the revised manuscript. \n\n$\\textbf{References}$\n\n- S. Milli, J. Miller, A. D. Dragan, and M. Hardt. The social cost of strategic classification. In Proceedings of the conference on fairness, accountability, and transparency, pages 230–239, 2019\n- J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020\n- C. Mendler-Dünner, J. Perdomo, T. Zrnic, and M. Hardt. Stochastic optimization for performative prediction. Advances in Neural Information Processing Systems, 33:4929–4939, 2020\n- M. Jagadeesan, T. Zrnic, and C. Mendler-Dünner. Regret minimization with performative feedback. In International Conference on Machine Learning, pages 9760–9785. PMLR, 2022"}, {"review_id": "Vhf0oDcN8m", "reviewer": "Reviewer_zEFG", "summary": "The paper introduces a distributionally robust framework for performative optimization. It proposes a tractable iterative algorithm with theoretical convergence and suboptimality guarantees. Empirical results on strategic classification, pricing, and energy applications demonstrate improved robustness over baselines.", "questions": "Q1: As mentioned in W1, the paper focuses on distributionally robust performative risk (DRPR), but many applications care about performance with respect to the true performative risk (PR). Is it possible to provide any theoretical guarantees or bounds on the PR of the proposed method?\n\nQ2. Could the authors elaborate more on why \"While methods grounded in the reference distribution map might perform satisfactorily on the observed samples, they often fail to achieve acceptable performance in out-of-sample circumstances\" in the literature? Specifically, if the reference distribution is close to the true distribution, wouldn't those methods still perform reasonably well? If the main challenge is in estimating a good reference distribution, how does the proposed framework help mitigate that?\n\nQ3: How to choose \\epsilon in practice?\n\nQ4: What will be the problem-specific parameters, such as alpha, rho, beta..., in the numerical problems? Will the proposed method still work if those parameters are bad in terms of the bound in Theorem 2?", "limitations": "NA", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\nS1: The paper proposes a new tractable method for performative optimization that incorporates distributional robustness to address model uncertainty.\n\nS2: The method is supported by both theoretical analysis and empirical evaluation across multiple applications.\n\nWeaknesses:\nW1: While the theoretical results focus on minimizing distributionally robust performative risk (DRPR), it would strengthen the work to connect these guarantees more directly to the performative risk (PR), which is often the ultimate objective.\n\nW2: The paper claims to address the out-of-sample limitations of methods relying on reference distributions. However, it is unclear how the proposed method helps when the reference distribution itself is poorly specified (see Q2 for more details).\n\nW2: Models 1 and 2 seem restrictive. It would be helpful to clarify whether these models cover a broad class of problems studied in prior performative optimization literature.", "was_revised": true, "final_justification": "Thank the authors for the detailed explanations, addressing my concerns.", "rebuttal": "We sincerely appreciate the reviewer’s thoughtful and constructive feedback. We will incorporate these insights into the revised manuscript and hope that the responses below help clarify our contributions and address the reviewer's concerns during the reevaluation process.\n\n$\\textbf{Weekness}$\n\n1. We really appreciate the constructive feedback from the reviewer. We agree that connecting our theoretical guarantees for minimizing the distributionally robust performative risk (DRPR) to the performative risk (PR) is important, especially since PR is often the ultimate objective.\n    \n    However, optimizing PR directly relies heavily on modeling the distribution map that captures how deployed decisions influence future data distributions. In practice, this map is often unknown or misspecified, which can result in poor estimates of the true PR. Our framework is specifically designed to address this challenge by optimizing a robust surrogate (DRPR) that mitigates the impact of such model misspecification.\n    \n    Unlike existing work [Perdomo et al. 2020, Mendler-Dünner et al. 2020] in performative optimization literature that typically assumes access to the true distribution map, our approach relies on much weaker assumptions. Moreover, we would like to emphasize that the limited data regime we consider is not simply a practical constraint but an inherent characteristic of the performative setting. In contrast to traditional machine learning, where data can be passively observed from a fixed or evolving distribution, performative optimization requires active data collection under deployed decisions (e.g., hiring, pricing), making large-scale sampling expensive or infeasible [Milli et al. 2019, Mendler-Dünner et al. 2020].\n\n    Our method is designed to be data-efficient and robust in this setting, offering reliable performance even with a small number of samples per iteration. We will clarify this distinction and expand on the connection between DRPR and PR in the revised manuscript.\n2. We apologize for the confusion. To ensure out-of-sample performance guarantees, a standard assumption in distributionally robust optimization is that the true data-generating distribution is contained in the ambiguity set. Leveraging the concentration inequality for the Wasserstein distance between the empirical and true distributions, we can ensure this with high confidence by appropriately choosing the radius. As shown in~\\cite[Theorem 3.4]{mohajerin2018data}, given $N$ sample data points, we have the data distribution $P$ lies within a Wasserstein ball around the empirical reference distribution $\\hat P$ of radius $\\rho_N(\\beta)$ with confidence level $1-\\beta$ for some prescribed $\\beta \\in (0,1)$, more specifically,\n    \\begin{align*}\n        P^n[W(P, \\hat P) \\leq \\rho_N(\\beta)] \\geq 1 - \\beta.\n    \\end{align*}\n    Now, if there is a distribution shift between the empirical reference distribution and the test distribution, and we train the algorithm based on samples from the reference distribution. Then, a typical assumption is that the magnitude of the distribution shift is known in terms of Wasserstein distance $\\tau$. Then, by the triangle inequality for Wasserstein distance, the radius of the ambiguity set is given by $\\tau$ plus the concentration inequality above. We will clarify this in the revision and more clearly explain how it enables out-of-sample performance guarantees even in the presence of moderate distribution shift.\n3. We apologize for the confusion and appreciate the reviewer's comment. We would like to clarify that both models are designed to encompass a broad class of problems studied in the existing performative optimization literature. For example, Model 1 captures widely used predictive models such as strategic classification [Perdomo et al. 2020], strategic regression [Harris et al. 2021]. Model 2 accommodates problems in decision-making problems including inventory management [Lee et al. 2021] and energy systems [Kim and Powell, 2011]. \n\n    Additionally, we would like to highlight that Model 3, presented in Appendix F, further generalizes the framework to accommodate problems studied in the minimax and adversarially robust optimization literature. In light of Reviewer 4’s comment, we will move the description of Model 3 and its associated results into the main text. We will also clarify how the three models relate to and extend existing approaches in the literature. \n\n$\\textbf{Questions}$\n\n1. We thank the reviewer for the great question. As we noted in response to Weakness 1, our method is designed to be both data-efficient and robust in settings where the distribution map is unknown or potentially misspecified.\n    \n    A key challenge in this setting is that, unlike traditional distribution robust optimization, which we typically assume a single known true distribution [Esfahani and Kuhn, 2018], performative optimization involves a distribution that changes with the decision variable. This makes it significantly more difficult to relate the robust objective directly to the true performative risk.\n    \n    That said, we note that as the ambiguity set radius $\\rho$ tends to zero, the robust objective coincides with the non-robust counterpart, which more directly targets the PR. This observation suggests that one possible direction is to design algorithms that gradually shrink the ambiguity set over time, potentially trading robustness for improved approximation of the true PR as more information becomes available. Developing theoretical guarantees for such adaptive schemes would require new technical tools and is an interesting direction for future work. We will add a discussion of this in the revised version.\n2. We apologize for the confusion. Please refer to our response to Weakness 2.\n3. We thank the reviewer for the question. We would like to clarify that $\\epsilon$ is the property of the underlying distribution mapping and is not a parameter. Unlike prior work on decision-dependent distribution shift [Perdomo et al. 2020, Mendler-Dünner et al. 2020], where the mapping P itself is known, our assumption is much weaker. In practice, given two decisions $\\theta$ and $\\theta'$, along with their corresponding empirical reference distributions $\\hat P(\\theta)$ and $\\hat P(\\theta')$ based on observed samples, one can compute the 1-Wasserstein distance $W_1(P(\\theta), P(\\theta'))$ via linear programming. This allows for an estimate of $\\epsilon$ using $\\frac{ W_1(P(\\theta), P(\\theta'))}{\\|\\theta - \\theta' \\|_2}$.\n4. We apologize for the confusion. Here, the parameters such as $\\alpha,\\beta,\\gamma, L_z, L, \\rho, \\epsilon$ are problem-specific constants that characterize properties of the loss function, the distributional map, and the ambiguity set. These are not tunable hyperparameters, and their values are not required for implementing our algorithm. Instead, they appear in the convergence analysis to establish theoretical guarantees. In practice, for example, in the strategic classification task, $L = 1$ for the logistic loss function [Gao and Pavel 2017] and the logloss objective is known to be $\\frac{1}{4n}\\sum_{i=1}^n \\|x_i\\|_2^2+ 2\\rho L$ smooth [Shwartz and David 2014]. We varied $\\epsilon$ as shown in Figure 1. The radius $\\rho$ of the Wasserstein ball was fixed to 0.1 across all robust models. We emphasize that our method is robust to the lack of explicit knowledge of these constants and remains effective in practice. We will clarify this in the revised manuscript.\n\n$\\textbf{References}$\n\n- J. H. Kim and W. B. Powell. Optimal energy commitments with storage and intermittent supply. Operations Research, 59(6):1347–1360, 2011.\n- S. Shalev-Shwartz and S. Ben-David. Understanding machine learning: From theory to algorithms. Cambridge University Press, 2014\n- B. Gao and L. Pavel. On the properties of the softmax function with application in game theory and reinforcement learning. arXiv preprint arXiv:1704.00805, 2017\n- P. Mohajerin Esfahani and D. Kuhn. Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1):115–166, 2018\n- S. Milli, J. Miller, A. D. Dragan, and M. Hardt. The social cost of strategic classification. In Proceedings of the conference on fairness, accountability, and transparency, pages 230–239, 2019\n- J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020\n- C. Mendler-Dünner, J. Perdomo, T. Zrnic, and M. Hardt. Stochastic optimization for performative prediction. Advances in Neural Information Processing Systems, 33:4929–4939, 2020\n- M. Jagadeesan, T. Zrnic, and C. Mendler-Dünner. Regret minimization with performative feedback. In International Conference on Machine Learning, pages 9760–9785. PMLR, 2022\n- K. Harris, H. Heidari, and S. Z. Wu. Stateful strategic regression. Advances in Neural Information Processing Systems, 34:28728–28741, 2021\n- S. Lee, H. Kim, and I. Moon. A data-driven distributionally robust newsvendor model with a wasserstein ambiguity set. Journal of the Operational Research Society, 72(8):1879–1897, 2021"}, {"review_id": "v1reFq75YR", "reviewer": "Reviewer_1Jfc", "summary": "This paper explores performative stochastic optimization, where decisions affect the underlying distribution of uncertain parameters, focusing on robust solutions when the true decision-dependent distribution is unknown. The authors propose a distributionally robust performative optimization (DRPO) framework using Wasserstein ambiguity sets, covering both machine learning and more general decision-making under uncertainty. The paper presents theoretical convergence and suboptimality guarantees and reformulations compatible with standard solvers. Experiments in strategic classification, revenue management, and portfolio optimization indicate consistent performance gains over baselines.", "questions": "In Figure 1, the performance gap between the robust type-1 and type-2 Wasserstein ball models is shown, but the practical difference is not deeply discussed: is the difference mainly due to the geometry of the ambiguity set, problem-specific characteristics, or hyperparameter tuning?", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n1. The authors introduce a distributionally robust performative optimization (DRPO) framework that is broader in scope than previous works, notably supporting a wider variety of decision-making beyond traditional prediction tasks.\n\n2. This paper theoretically establishes convergence guarantees for the repeated robust risk minimization algorithm with both smooth and some nonsmooth loss functions, which broadens applicability.\n\n3. This paper conducts experimental evaluations in three distinct application domains. In each, the proposed approach consistently outperforms non-robust and relevant existing rivals. \n\n## Weaknesses\n\n1. While the experimental section spans three domains, the treatment of each is somewhat brief. For example, for the revenue management and portfolio optimization tasks (Section 4.2), quantitative comparisons are mostly limited to a single robust method versus a standard baseline; only the strategic classification task benchmarks multiple methods, as shown in Figure 1.\n\n2. More extensive ablation or sensitivity analyses (e.g., systematically varying ambiguity set sizes, number of iterations, etc.), particularly for the portfolio and revenue management use cases, would strengthen the empirical story.\n\n3. This paper claims the reformulations are “compatible with off-the-shelf solvers,” but fails to provide concrete runtime comparisons or scalability analyses to contextualize practical tractability.\n\n4.  Section 1.1(Related Work) positions the work well within the field but could benefit from a more explicit contrast with the limitations and strengths of direct alternatives, for instance, quantifying where/when the paper’s approach provides superior robustness or convergence.", "was_revised": true, "final_justification": "The authors' rebuttal helps to address some of my concerns and clarify the contribution of this paper, and additional experiments will be added in the future revision. Therefore, I keep my score unchanged. (4: Borderline accept)", "rebuttal": "We are grateful to the reviewer for their detailed and insightful comments. We have addressed each point carefully below and will reflect these improvements in the revised manuscript. We kindly invite the reviewer to reassess the paper in light of our responses.\n\n$\\textbf{Weekness}$\n\n1. Thank you for your constructive comment. We would like to clarify that, to the best of our knowledge, unlike in machine learning problems, there are no existing baselines applicable to the decision-dependent distributionally robust setting we consider for the revenue management and portfolio optimization tasks. While there are related works in the stochastic programming literature (e.g., Fonseca and Junca 2023, Noyan et al. 2022) that study decision-dependent uncertainty, such as models where the decision affects the radius of the ambiguity set, these approaches are not directly applicable to our framework. We will revise the manuscript to make this distinction clearer and emphasize that our aim in these experiments is to demonstrate the flexibility and applicability of our approach to realistic decision-making problems beyond strategic classification.\n2. Thank you for your suggestion. In our current experiments, we fixed the size of the ambiguity set across all robust models to be 0.1 in the strategic classification task. However, due to space constraints, we were unable to include more sensitivity analyses for the portfolio and revenue management tasks in the main paper. In the revised version, we will include additional results that explore the sensitivity analysis.\n3. We apologize for the confusion. Our claim that the reformulations are \"compatible with off-the-shelf solvers\" refers to the fact that after applying the reformulations in Models 1–3, the resulting optimization problems can be cast as convex programs (e.g., exponential cone programs or smooth convex problems) that are solvable using standard solvers such as Gurobi, Mosek, or CVXPY-based backends.\n\n    We would like to emphasize that the limited data regime we consider is not just a practical constraint, but a characteristic feature of the performative optimization setting. Unlike traditional machine learning, where data is assumed to be drawn from a fixed or evolving distribution, performative optimization requires sampling from distributions that are themselves induced by the decisions. That is, in order to evaluate or improve a policy, one must actively deploy a decision (e.g., hiring strategies, pricing policy) and then observe the resulting data distribution, which can be expensive, time-consuming, or infeasible at scale [Milli et al. 2019, Mendler-Dünner et al. 2020]. Our method is designed to be data-efficient in this regime, and we aim to achieve robust and stable performance even with limited samples per iteration. We will clarify this distinction and expand on the computational aspects in the revised manuscript.\n4. Thank you for the suggestion. To the best of our knowledge, there are only a few existing works that address distributionally robust performative prediction. For example, [Peet et al. 2022] propose a distributionally robust performative model to promote fairness in machine learning. However, their framework relies on a robust smoothness assumption on the objective and a robust sensitivity assumption on the worst-case distributions, both of which may be unrealistic in practice. Additionally, their use of a $\\phi$-divergence ambiguity set excludes continuous distributions, thereby limiting its ability to provide coverage guarantees over the true distribution. Similarly, [Xue and Sun 2024] study distributionally robust performative prediction based on a KL-divergence ambiguity set. Like [Peet et al. 2022], such divergence-based ambiguity sets may fail to protect against distributional shifts involving novel or unseen scenarios outside the reference distribution. Moreover, their proposed scheme lacks convergence guarantees.\n    \n    By contrast, our work adopts a Wasserstein ambiguity set, which provides coverage guarantees for continuous distributions, and we establish provable convergence guarantees under weaker assumptions. We will revise Section 1.1 to make this comparison more explicit.\n\n$\\textbf{Questions}$\n\n1. We thank you for the clarifying question. The performance gap observed between the robust type-1 and type-2 Wasserstein ball models in Figure 1 arises primarily due to the geometry of the ambiguity set. As discussed in [Byeon 2025], selecting the optimal radius $\\rho$ is challenging, and the 2-Wasserstein ball often provides better performance because it offers a wider range of radius values for which the DRO solution outperforms its non-robust counterpart. In our experiments, we set the robustness parameter to 0.1 across all robust models to ensure comparability. We will clarify this in the revised manuscript and expand the discussion of how the ambiguity set geometry influences performance.\n\n$\\textbf{References}$\n\n- D. Fonseca and M. Junca. Decision-dependent distributionally robust optimization. arXiv preprint arXiv:2303.03971, 2023\n- N. Noyan, G. Rudolf, and M. Lejeune. Distributionally robust optimization under a decision-dependent ambiguity set with applications to machine scheduling and humanitarian logistics. INFORMS Journal on Computing, 34(2):729–751, 2022\n- S. Milli, J. Miller, A. D. Dragan, and M. Hardt. The social cost of strategic classification. In Proceedings of the conference on fairness, accountability, and transparency, pages 230–239, 2019\n- C. Mendler-Dünner, J. Perdomo, T. Zrnic, and M. Hardt. Stochastic optimization for performative prediction. Advances in Neural Information Processing Systems, 33:4929–4939, 2020\n- L. Peet-Pare, N. Hegde, and A. Fyshe. Long term fairness for minority groups via performative distributionally robust optimization. arXiv preprint arXiv:2207.05777, 2022.\n- S. Xue and Y. Sun. Distributionally robust performative prediction. Advances in Neural Information Processing Systems, 37:55030–55052, 2024\n- G. Byeon. Comparative analysis of two-stage distributionally robust optimization over 1-wasserstein and 2-Wasserstein balls. arXiv preprint arXiv:2501.05619, 2025"}, {"review_id": "aZdMOHk1aM", "reviewer": "Reviewer_PzxX", "summary": "This paper extends the distributionally performative robust framework to ambiguity sets captured by Wasserstein balls. The authors show a robust version of the repeated risk minimization algorithm linearly converges to a performatively robust stable fixed point in varying settings. These theoretical results are supplemented by experiments demonstrating the efficacy of their results on robust tasks.", "questions": "> * [Line 142] The term random parameters is ambiguous, is this the data point? \n> * [Lines 151-152] The empirical reference distribution is a bit vague. \n> * What are the primary difficulties in broadening the scope of the assumption to include all strongly convex functions? \n> * Are the positive results tight, that is if any of the assumptions are replaced the convergence results will not hold?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "> Strengths\n> * The paper is well written with clear exposition, ample motivation and a useful recap of related work. \n> * This paper extends the established framework of distributionally robust performative optimization to include the more general case of Wasserstein balls as ambiguity sets rather than the KL divergence balls which fail when the support is not identical\n> * This paper provides the first known convergence results for distributionally robust performative optimization.\n\n> Weaknesses \n> * The assumptions on the loss function in Model 1 are constraining, as the set of all quadratic and lipschitz combinations is a small subset of all strongly convex functions. A more general assumption would be expanding the quadratic function to be all strongly convex functions. \n> * Beyond the reformulations in the models, the proofs of convergence are quite standard in the performative literature. \n>  * Some additional discussion concerning the decision dependent nature of the experiments would be appreciated. \n>  * The comparison lemma between performatively optimal and stable points is useful, but it would be helpful to construct a model with guarantees in finding performatively robust optimal points\n>  * The intuition for the usefulness of exponential smoothing in this setting is not well explained.\n> >  * *Minor Weakness* The results in appendix F are useful and might have more merits in the main body.  \n> >* *Minor Weakness* The notion of model in the paper is vague and it is unclear whether it refers to a Lemma or a set of algorithmic choices.", "was_revised": true, "final_justification": "This paper extends the distributionally robust framework to study distributions perturbed by Wasserstein balls. While the paper is well written, my initial concerns centered on its technical novelty, as it does not substantially distinguish itself from [1] aside from changes in the setting and the limited generality of the model itself, namely only dealing with quadratic loss functions. \n\nIn their response, the authors addressed these concerns by providing detailed insight into how the use of Wasserstein balls increases the proof’s complexity. They also explained how, beyond the usual contraction arguments in this literature, their proof leverages Danzig’s theorem and the exponential smoothing algorithm to handle the specific challenges of this work, and explained in detail during the rebuttal how and why these elements are used to derive the desired results. \n\nMoreover, although the results are constrained to compositions of Lipschitz and quadratic functions, this is a notable generalization from previous work which only considers *bounded* loss functions. \n\nFor these reasons, I raise my score to 4 (Borderline Accept).\n\n[1] Xue, S., & Sun, Y. (2024). Distributionally robust performative prediction. Advances in Neural Information Processing Systems, 37, 55030-55052.", "rebuttal": "We are grateful to the reviewer for their detailed and insightful comments. We have addressed each point carefully below and will reflect these improvements in the revised manuscript. We kindly invite the reviewer to reassess the paper in light of our responses.\n\n$\\textbf{Weekness}$\n\n1. Thank you for your comment. We would like to clarify that our model encompasses a broad class of problems studied in the existing performative optimization literature, including strategic classification [Perdomo et al. 2020] and strategic regression [Harris et al. 2021].  In fact, it represents one of the most general settings studied in the distributionally robust performative optimization framework. The dependence of the random parameters $Z$ is assumed to be affine in the loss function to ensure a tractable reformulation. Moreover, our Model 3 allows for a convex-concave loss function, which encompasses a wide range of applications. In light of this and related comments, we will revise the paper to highlight Model 3 more prominently in the main body.\n2. Thank you for your comment. We agree that, like most of the existing performative optimization literature, our convergence analysis builds on the classical fixed-point iteration techniques. However, extending these tools to the distributionally robust setting introduces additional challenges. Specifically, our framework requires additional approximation techniques, such as exponential smoothing in Model 2, to handle non-smooth worst-case objectives. For Model 3, our analysis relies on Danskin’s theorem, which necessitates concavity of the loss function in the uncertainty to obtain a well-defined derivative of the worst-case objective. Adapting these tools to reformulate the problem into a tractable form while ensuring convergence involves careful technical treatment. We will clarify these aspects in the revised version.\n3. Thank you for the suggestion. We agree that additional discussion on the decision-dependent nature of the experiments would be helpful. In all of our experimental settings, the data distribution changes as a function of the decision variable, reflecting the performative nature of the problem. For instance, in the strategic classification task, individuals respond strategically to the deployed classifier, thereby inducing a distribution shift dependent on the decisions. Similar decision-dependent dynamics are present in the revenue management and portfolio optimization settings, where decisions affect customer behavior and market conditions, respectively. We will revise the experimental section to make this connection more explicit.\n4. Thank you for your suggestion. In this paper, we focus on convergence to distributionally performatively stable points and provide a comparison lemma that relates these stable points to performatively optimal solutions under distributional robustness. Constructing a model with guarantees for directly finding performatively robust optimal points would require stronger assumptions on the distribution map, as explored in prior work under non-robust settings [Izzo et al. 2022, Kim and Perdomo 2022]. However, such assumptions may not hold in many real-world scenarios. We view this as an important direction for future research and will include a discussion of it in the revised manuscript.\n5. We apologize for the confusion. The piecewise quadratic structure of the loss function in Model 2 leads to non-smoothness, which hinders the convergence of our proposed algorithm. To address this challenge, we apply a log-sum-exp smoothing approximation with a smoothing parameter, which yields a differentiable surrogate loss function. We will revise the manuscript to better explain this intuition and the role of smoothing in our framework.\n - Thank you for your suggestion. Due to page limits, we initially placed these results in the appendix. However, in response to your comment, we will move key results from Appendix F into the main body of the paper in the revised version to improve clarity and accessibility for readers.\n - We apologize for the confusion. In our paper, Model 1, Model 2, and Model 3 each refer to distinct problem formulations that differ in their assumptions on the loss function and structure of the distributional map, which in turn lead to different reformulations of the distributionally robust performative optimization problem. These are not lemmas or algorithms, but rather modeling frameworks that guide both the algorithmic design and the theoretical analysis. We will clarify this terminology in the revised manuscript to avoid ambiguity.\n\n$\\textbf{Questions}$\n\n1. We apologize for the confusion. To clarify,$\\tilde z$ is not a specific data point, but rather a random vector drawn from the unknown distribution $\\mathbb{P}$.We will revise the text to make this distinction clearer.\n2. We apologize for the confusion. In the performative optimization setting, the empirical reference distribution refers to the empirical distribution constructed from observed samples collected under a fixed decision $\\theta$. Specifically, at each iteration, we observe a finite set of samples drawn from the corresponding distribution $P(\\theta)$, and define the empirical reference distribution $\\hat P(\\theta)$ as the uniform distribution over these samples. Due to the limited sample size, $\\hat P(\\theta)$ may not represent the true distribution. We will revise the text to make this definition more precise.\n3. Thank you for your question. Broadening the assumption to include all strongly convex functions poses challenges, primarily because it may prevent a tractable reformulation of the distributionally robust problem. In our setting, we assume that the dependence of the random parameters $Z$ is affine in the loss function to ensure such tractability. Extending the analysis to more general strongly convex functions would likely require new machinery and is an interesting direction for future work.\n    \n    That said, we would like to emphasize that our model is already quite general and encompasses a broad class of problems studied in the performative optimization literature, including strategic classification [Perdomo et al. 2020]  and strategic regression [Harris et al. 2021].\n4. Thank you for your comment. The assumptions in our convergence analysis are chosen to enable a tractable and rigorous analysis using established tools such as Danskin’s theorem and fixed-point methods. While these assumptions are not necessarily tight, they are carefully selected to reflect the key structural properties that underpin the robustness and stability of our algorithm.\n\n    Relaxing any of them would likely require new proof techniques or a different algorithmic framework. At the same time, we believe our assumptions strike a reasonable balance between generality and tractability, as they already cover a broad class of performative optimization problems studied in prior work. We will clarify this point in the revised manuscript and discuss opportunities for generalization in future research.\n\n$\\textbf{References}$\n\n- J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020\n- K. Harris, H. Heidari, and S. Z. Wu. Stateful strategic regression. Advances in Neural Information Processing Systems, 34:28728–28741, 2021\n- Z. Izzo, L. Ying, and J. Zou. How to learn when data reacts to your model: performative gradient descent. In International Conference on Machine Learning, pages 4641–4650. PMLR, 2021.\n- M. P. Kim and J. C. Perdomo. Making decisions under outcome performativity. arXiv preprint arXiv:2210.01745, 2022"}]

Author rebuttal:

Meta-review: The paper studies the performative optimization under a distributionally robust setting. The paper is well-written, and it addresses technical challenges arising in this new formulation. All the reviewers are leaning positive.

On the positive side, I am in support of this paper based on the new setting studied, the efforts made by the authors to overcome the technical challenge, and the writing. While I support accepting the paper, I still want to point out my reservations:
- Studying a setting "different" or "hasn't been studied before by other papers", or "resolving some technical challenges that don't appear in the previous setting", all of these don't mean that it is a setting worth working on. In other words, a niche setting that hasn't been studied before doesn't justify its usefulness. In the past few years, we have seen tons of papers on the topic of distributionally robust optimization; hardly any have been applied to a real application context. Many papers motivate the considered setting or Wasserstein distance from ML applications, such as Wasserstein GANs or the usage of distributional robustness in improving the ML model performance. While I think it is inappropriate to say that these imaginary robustness motivations become largely obsolete by the new methods, such as CLIP (OpenAI 2021) and others, I sincerely hope the researchers and the community working on DRO/RO to look for new problems and learn new techniques, rather than spending further time filling the gaps in the existing scope.
Decision: Accept (poster)