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Parameter-Free Algorithms for Performative Regret Minimization under Decision-Dependent Distributions

Sungwoo Park, Junyeop Kwon, Byeongnoh Kim, Suhyun Chae, Jeeyong Lee, Dabeen Lee

TL;DR

This work addresses performative risk minimization with decision-dependent distributions, a setting where standard stochastic optimization fails due to distribution shifts induced by decisions. It introduces parameter-free optimistic optimization algorithms DOOP and SOOP, along with two feedback regimes—full and data-driven—leveraging hierarchical partitioning and performative feedback to bound regret with respect to the performative optimum. Theoretical guarantees hinge on the near-optimality dimension and show favorable rates, including exponential decay when the dimension is zero, while remaining robust to unknown problem constants like \\varepsilon and \\alpha. Empirically, the methods outperform Lipschitz-bandit baselines and other black-box optimizers, demonstrating both numerical efficiency and practical viability for non-convex performative risks. The results advance practical, parameter-free strategies for decision-dependent optimization in stochastic settings.

Abstract

This paper studies performative risk minimization, a formulation of stochastic optimization under decision-dependent distributions. We consider the general case where the performative risk can be non-convex, for which we develop efficient parameter-free optimistic optimization-based methods. Our algorithms significantly improve upon the existing Lipschitz bandit-based method in many aspects. In particular, our framework does not require knowledge about the sensitivity parameter of the distribution map and the Lipshitz constant of the loss function. This makes our framework practically favorable, together with the efficient optimistic optimization-based tree-search mechanism. We provide experimental results that demonstrate the numerical superiority of our algorithms over the existing method and other black-box optimistic optimization methods.

Parameter-Free Algorithms for Performative Regret Minimization under Decision-Dependent Distributions

TL;DR

This work addresses performative risk minimization with decision-dependent distributions, a setting where standard stochastic optimization fails due to distribution shifts induced by decisions. It introduces parameter-free optimistic optimization algorithms DOOP and SOOP, along with two feedback regimes—full and data-driven—leveraging hierarchical partitioning and performative feedback to bound regret with respect to the performative optimum. Theoretical guarantees hinge on the near-optimality dimension and show favorable rates, including exponential decay when the dimension is zero, while remaining robust to unknown problem constants like \\varepsilon and \\alpha. Empirically, the methods outperform Lipschitz-bandit baselines and other black-box optimizers, demonstrating both numerical efficiency and practical viability for non-convex performative risks. The results advance practical, parameter-free strategies for decision-dependent optimization in stochastic settings.

Abstract

This paper studies performative risk minimization, a formulation of stochastic optimization under decision-dependent distributions. We consider the general case where the performative risk can be non-convex, for which we develop efficient parameter-free optimistic optimization-based methods. Our algorithms significantly improve upon the existing Lipschitz bandit-based method in many aspects. In particular, our framework does not require knowledge about the sensitivity parameter of the distribution map and the Lipshitz constant of the loss function. This makes our framework practically favorable, together with the efficient optimistic optimization-based tree-search mechanism. We provide experimental results that demonstrate the numerical superiority of our algorithms over the existing method and other black-box optimistic optimization methods.
Paper Structure (18 sections, 17 theorems, 99 equations, 3 figures, 4 algorithms)

This paper contains 18 sections, 17 theorems, 99 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Existing Methods for Performative Risk Minimization
  3. Our Contributions
  4. Related Work
  5. Performative Prediction and Performative Risk Minimization
  6. Optimistic Optimization
  7. Preliminaries: Optimization with Performative Feedback
  8. Optimistic Optimization-Based Parameter-Free Framework
  9. Data-Driven Setting
  10. Experiments
  11. Regret Analysis of DOOP
  12. Approximation Bounds under the Hierarchical Partitioning Scheme
  13. Proof of \ref{['thm:full-feedback']}
  14. Regret Analysis of SOOP
  15. Total Number of Solution Deployments
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that the distribution map satisfies the $\varepsilon$-sensitivity condition and the loss function $f(\theta,z)$ is $L_z$-Lipschitz continuous in $z$ for any $\theta$. Let $d$ denote the $L_z\varepsilon$-near-optimality dimension. For the full-feedback setting, algorithm0 after $T$ decision d

Figures (3)

  • Figure 1: Contour plots of the Ackley and Rastrigin functions on $[-5.12, 5.12]^2$
  • Figure 2: Cumulative regret comparison in the full-feedback case
  • Figure 3: Cumulative regret comparison in the data-driven setting

Theorems & Definitions (20)

  • Theorem 1: Full-Feedback Case, Informal
  • Theorem 2: Data-Driven Case, Informal
  • Lemma 3
  • Definition 4: near-optimality dimension
  • Lemma 5
  • Theorem 6
  • Definition 7: Rademacher complexity
  • Definition 8: Clean event
  • Lemma 9
  • Lemma 10
  • ...and 10 more