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Performative Scenario Optimization

Quanyan Zhu, Zhengye Han

Abstract

This paper introduces a performative scenario optimization framework for decision-dependent chance-constrained problems. Unlike classical stochastic optimization, we account for the feedback loop where decisions actively shape the underlying data-generating process. We define performative solutions as self-consistent equilibria and establish their existence using Kakutani's fixed-point theorem. To ensure computational tractability without requiring an explicit model of the environment, we propose a model-free, scenario-based approximation that alternates between data generation and optimization. Under mild regularity conditions, we prove that a stochastic fixed-point iteration, equipped with a logarithmic sample size schedule, converges almost surely to the unique performative solution. The effectiveness of the proposed framework is demonstrated through an emerging AI safety application: deploying performative guardrails against Large Language Model (LLM) jailbreaks. Numerical results confirm the co-evolution and convergence of the guardrail classifier and the induced adversarial prompt distribution to a stable equilibrium.

Performative Scenario Optimization

Abstract

This paper introduces a performative scenario optimization framework for decision-dependent chance-constrained problems. Unlike classical stochastic optimization, we account for the feedback loop where decisions actively shape the underlying data-generating process. We define performative solutions as self-consistent equilibria and establish their existence using Kakutani's fixed-point theorem. To ensure computational tractability without requiring an explicit model of the environment, we propose a model-free, scenario-based approximation that alternates between data generation and optimization. Under mild regularity conditions, we prove that a stochastic fixed-point iteration, equipped with a logarithmic sample size schedule, converges almost surely to the unique performative solution. The effectiveness of the proposed framework is demonstrated through an emerging AI safety application: deploying performative guardrails against Large Language Model (LLM) jailbreaks. Numerical results confirm the co-evolution and convergence of the guardrail classifier and the induced adversarial prompt distribution to a stable equilibrium.

Paper Structure

This paper contains 21 sections, 8 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. PRELIMINARIES AND PROBLEM STATEMENT
  3. Decision-Dependent Chance-Constrained Problem
  4. Performative Solution and Best-Response Mapping
  5. Scenario Approach
  6. Fixed-Point Iteration & Computational Schemes
  7. Deterministic Best-Response Iteration
  8. Scenario-Based (Stochastic) Fixed-Point Iteration
  9. Equivalent Game-Theoretic Model
  10. Agent's Problem with Baseline and Endogenous Change of Measure
  11. Agent utility relative to baseline
  12. KL-regularized utility as a canonical example
  13. Best-response correspondence
  14. Endogenous change of measure
  15. Induced distribution
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose that Assumptions am:Regularity and am:Boundary regularity hold and $\mathcal{X}_{\varepsilon}^{\gamma}(x) \neq \varnothing$. Then $\mathcal{X}_{\varepsilon}^{\gamma}(x)$ is compact and $\Phi(x)$ is nonempty and compact.

Figures (2)

  • Figure 3: Evolution of the performative guardrail and the induced semantic shift in the embedding space. The dashed line denotes the decision boundary $w_t$, and the data points represent benign (blue) and mutated malicious (orange) prompts.
  • Figure 4: (Left) Convergence of the stochastic iteration to the performative fixed point in the embedding space. (Right) Logarithmic growth of the sample size $N_{t}$ used in the scenario optimization.

Theorems & Definitions (22)

  • Definition 1: Performative chance-constrained problem
  • Definition 2: Best-response operator
  • Definition 3: Performative solution
  • Proposition 1: Compactness of feasible sets and best responses
  • proof
  • Proposition 2: Closed graph and upper hemicontinuity
  • proof
  • Theorem 1: Existence of performative solution
  • proof
  • Definition 4: Scenario approximation of the best-response problem
  • ...and 12 more