Distributionally Robust Performative Optimization

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.

$**Weekness**$

1. 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.

2. 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.

3. 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.

   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.

4. 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.

   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.

$**Questions**$

1. 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.

$**References**$

• D. Fonseca and M. Junca. Decision-dependent distributionally robust optimization. arXiv preprint arXiv:2303.03971, 2023
• 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
• 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
• 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
• 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.
• S. Xue and Y. Sun. Distributionally robust performative prediction. Advances in Neural Information Processing Systems, 37:55030–55052, 2024
• G. Byeon. Comparative analysis of two-stage distributionally robust optimization over 1-wasserstein and 2-Wasserstein balls. arXiv preprint arXiv:2501.05619, 2025

Dear Reviewer 1Jfc,

We deeply appreciate the time and effort you devoted to carefully reviewing our work and providing thoughtful, detailed feedback. Your insightful questions and constructive suggestions have been invaluable in enhancing both the technical depth and clarity of our paper.

Thank you once again for your thoughtful review and support.

Sincerely,

The Authors

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.

$**Weekness**$

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

• 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.
• 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.

$**Questions**$

1. 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.

2. 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.

3. 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.

   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].

4. 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.

   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.

$**References**$

• J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020
• K. Harris, H. Heidari, and S. Z. Wu. Stateful strategic regression. Advances in Neural Information Processing Systems, 34:28728–28741, 2021
• 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.
• M. P. Kim and J. C. Perdomo. Making decisions under outcome performativity. arXiv preprint arXiv:2210.01745, 2022

J_{\theta_t}^\tau(\overline \theta ) = E_{ \hat{P}(\theta_t) } [g( Z, \overline \theta )],

Dear Reviewer PzxX,

Continuing from our earlier response, we now provide a detailed explanation of the exponential smoothing algorithm, its mathematical role in our framework, and the theoretical challenges it helps address.

$**Background on Exponential Smoothing Algorithm**$

Let $\mathcal{Z} = \{ z_1, \dots, z_n\}$ denote a finite support set. Given a loss function $\ell: \Theta \times \mathcal{Z} \to \mathbb{R}$ and a distribution ${\hat p}_{\theta} \in \Delta_n$, where $\Delta_n$ denotes the probability simplex. We define the exponentially smoothed objective as:

$

f_\mu(\theta) := \mu \cdot \log \left( \sum_{i=1}^n \hat{p}_\theta(z_i) \cdot \exp\left( \frac{\ell(\theta, z_i)}{\mu} \right) \right),

$

where $\mu > 0$ is a smoothing parameter.

This function, also known as the log-sum-exp function, is a 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 a smooth approximation to $\max_{i} \ell(\theta, z_i)$ whenever $\hat{p}_\theta(z_i) > 0$ for all $i$.

$**Properties.**$ The function $f_\mu(\theta)$ satisfies the following:

• Approximation Bounds: The approximation error can be precisely quantified:

$

\max_{i} \ell(\theta, z_i) \leq f_\mu(\theta) \leq \max_{i} \ell(\theta, z_i) + \mu \cdot \log \left( \frac{1}{\min_i \hat{p}_\theta(z_i)} \right),

$

provided $\hat{p}\_\theta(z_i) > 0$ for all $i$. Thus, as $\mu \to 0$, the smoothed objective $f_\mu(\theta)$ approaches the exact maximum, with the error vanishing linearly in $\mu$ up to a logarithmic multiplicative factor.

• Convexity and Differentiability: If each function $\ell(\theta, z_i)$ is convex in $\theta$, then $f_\mu(\theta)$ is also convex, as it is a composition of convex functions closed under nonnegative weighted log-sum-exp operations. Moreover, $f_\mu(\theta)$ is continuously differentiable for all $\mu > 0$, with gradient:

$

\nabla_\theta f_\mu(\theta) = \sum_{i=1}^n \pi_\mu(z_i; \theta) \cdot \nabla_\theta \ell(\theta, z_i),

$

where the weight vector $\pi_\mu(\cdot; \theta) \in \Delta_n$ defines a softmax distribution:

$

\pi_\mu(z_i; \theta) := \frac{\hat{p}_\theta(z_i) \cdot \exp\left( \ell(\theta, z_i)/\mu \right)}{ \sum_{j=1}^n \hat{p}_\theta(z_j) \cdot \exp\left( \ell(\theta, z_j)/\mu \right)}.

$

This smoothing mechanism not only ensures differentiability but also facilitates efficient computation of gradients for robust optimization objectives involving maxima.

---

We appreciate your suggestion and have included the above discussion, along with additional references, in the revised appendix. Please let us know if further clarification would be helpful.

Thank you again for your careful review and constructive feedback.

Sincerely,

The Authors

• Bertsekas, Dimitri. Convex optimization algorithms. Athena Scientific, 2015.

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.

$**Weekness**$

1. 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.

   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.

   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].

   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.

2. 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,

$

    P^n[W(P, \hat P) \leq \rho_N(\beta)] \geq 1 - \beta.

$

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.

3. 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].

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. 

$**Questions**$

1. 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.

   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.

   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.

2. We apologize for the confusion. Please refer to our response to Weakness 2.

3. 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 $\frac{ W_1(P(\theta), P(\theta'))}{\|\theta - \theta' \|_2}$.

4. 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.

$**References**$

• J. H. Kim and W. B. Powell. Optimal energy commitments with storage and intermittent supply. Operations Research, 59(6):1347–1360, 2011.
• S. Shalev-Shwartz and S. Ben-David. Understanding machine learning: From theory to algorithms. Cambridge University Press, 2014
• 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
• 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
• 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
• J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020
• 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
• M. Jagadeesan, T. Zrnic, and C. Mendler-Dünner. Regret minimization with performative feedback. In International Conference on Machine Learning, pages 9760–9785. PMLR, 2022
• K. Harris, H. Heidari, and S. Z. Wu. Stateful strategic regression. Advances in Neural Information Processing Systems, 34:28728–28741, 2021
• 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

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:

$**Weekness**$

1. 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 $\frac{ W_1(P(\theta), P(\theta'))}{\|\theta - \theta' \|_2}$.

2. 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.

3. 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.

   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.

4. 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.

$**Questions**$

We 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.

$**Limitations**$

We 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.

$**References**$

• 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
• J. Perdomo, T. Zrnic, C. Mendler-Dünner, and M. Hardt. Performative prediction. In International Conference on Machine Learning, pages 7599–7609. PMLR, 2020
• 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
• M. Jagadeesan, T. Zrnic, and C. Mendler-Dünner. Regret minimization with performative feedback. In International Conference on Machine Learning, pages 9760–9785. PMLR, 2022

Dear Reviewer MXSY,

We sincerely thank you for taking the time to thoroughly evaluate our work and provide such detailed and constructive feedback. Your insightful comments, questions, and suggestions have significantly helped us improve both the technical content and the presentation of our paper.

Thank you again for your time and effort in reviewing our submission.

Sincerely,

The Authors