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Unified Inference Framework for Single and Multi-Player Performative Prediction: Method and Asymptotic Optimality

Zhixian Zhang, Xiaotian Hou, Linjun Zhang

TL;DR

This work develops a unified inference framework for performative prediction that covers both single- and multi-player settings, unifying stability and Nash-optimality analysis. It introduces ERR (Empirical Repeated Retraining) to estimate stable equilibria with a CLT and semiparametric efficiency, and a two-stage plug-in strategy combining RePPI-calibrated estimation of distributional parameters with Importance Sampling to obtain efficient plug-in Nash equilibria. The theory provides asymptotic normality results and lower-bound efficiency guarantees, along with robustness to mild distributional misspecification through the use of distribution atlases. Numerical simulations corroborate the finite-sample validity, showing accurate coverage and narrower confidence intervals for the recalibrated approach compared to empirical risk baselines. Overall, the paper delivers a principled, efficient toolkit for reliable estimation and decision-making in dynamic, feedback-driven environments.

Abstract

Performative prediction characterizes environments where predictive models alter the very data distributions they aim to forecast, triggering complex feedback loops. While prior research treats single-agent and multi-agent performativity as distinct phenomena, this paper introduces a unified statistical inference framework that bridges these contexts, treating the former as a special case of the latter. Our contribution is two-fold. First, we put forward the Repeated Risk Minimization (RRM) procedure for estimating the performative stability, and establish a rigorous inferential theory for admitting its asymptotic normality and confirming its asymptotic efficiency. Second, for the performative optimality, we introduce a novel two-step plug-in estimator that integrates the idea of Recalibrated Prediction Powered Inference (RePPI) with Importance Sampling, and further provide formal derivations for the Central Limit Theorems of both the underlying distributional parameters and the plug-in results. The theoretical analysis demonstrates that our estimator achieves the semiparametric efficiency bound and maintains robustness under mild distributional misspecification. This work provides a principled toolkit for reliable estimation and decision-making in dynamic, performative environments.

Unified Inference Framework for Single and Multi-Player Performative Prediction: Method and Asymptotic Optimality

TL;DR

This work develops a unified inference framework for performative prediction that covers both single- and multi-player settings, unifying stability and Nash-optimality analysis. It introduces ERR (Empirical Repeated Retraining) to estimate stable equilibria with a CLT and semiparametric efficiency, and a two-stage plug-in strategy combining RePPI-calibrated estimation of distributional parameters with Importance Sampling to obtain efficient plug-in Nash equilibria. The theory provides asymptotic normality results and lower-bound efficiency guarantees, along with robustness to mild distributional misspecification through the use of distribution atlases. Numerical simulations corroborate the finite-sample validity, showing accurate coverage and narrower confidence intervals for the recalibrated approach compared to empirical risk baselines. Overall, the paper delivers a principled, efficient toolkit for reliable estimation and decision-making in dynamic, feedback-driven environments.

Abstract

Performative prediction characterizes environments where predictive models alter the very data distributions they aim to forecast, triggering complex feedback loops. While prior research treats single-agent and multi-agent performativity as distinct phenomena, this paper introduces a unified statistical inference framework that bridges these contexts, treating the former as a special case of the latter. Our contribution is two-fold. First, we put forward the Repeated Risk Minimization (RRM) procedure for estimating the performative stability, and establish a rigorous inferential theory for admitting its asymptotic normality and confirming its asymptotic efficiency. Second, for the performative optimality, we introduce a novel two-step plug-in estimator that integrates the idea of Recalibrated Prediction Powered Inference (RePPI) with Importance Sampling, and further provide formal derivations for the Central Limit Theorems of both the underlying distributional parameters and the plug-in results. The theoretical analysis demonstrates that our estimator achieves the semiparametric efficiency bound and maintains robustness under mild distributional misspecification. This work provides a principled toolkit for reliable estimation and decision-making in dynamic, performative environments.
Paper Structure (64 sections, 26 theorems, 288 equations, 4 figures, 3 algorithms)

This paper contains 64 sections, 26 theorems, 288 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Overview of our results
  3. Performative Stability
  4. Performative Optimality
  5. Related Work
  6. Performative prediction
  7. Recalibrated Prediction-Powered Inference
  8. Notation and Definitions
  9. Preliminaries
  10. Problem Setup
  11. Multi-player Performative Prediction
  12. Strong Monotonicity
  13. Probability Measures
  14. Performatively Stable Equilibria and Repeated Retraining
  15. Nash Equilibria and Plug-in Optimization
  16. ...and 49 more sections

Key Result

Theorem 1

Suppose that for each player $i$, the distribution map is $\epsilon_i$-Lipschitz in Wasserstein-1 distance, the loss function is $\beta_i$-jointly smooth, and the gradient function is $\alpha_i$-strongly monotone on $\theta^i$, and locally Lipschitz on $\theta^i$. Suppose $\sum_{k=1}^m (\frac{\beta_ where the covariance $\Sigma_t$ is related to the covariance at all previous iterations. Suppose $\

Figures (4)

  • Figure 1: Mahalanobis qq-plot under different sensitivities
  • Figure 2: Coverage Rate for $\theta_t$ vs. sensitivity
  • Figure 3: Inferential results for $\theta_{PO}^{\beta^*}$ under different misspecifications
  • Figure 4: Coverage Rate for two entries of $\theta_{PS}$ vs. Misspecification

Theorems & Definitions (55)

  • Theorem 1: Stability, informal
  • Theorem 2: Optimality, informal
  • Proposition 1: Existence and convergence narang2023multiplayer
  • Remark 1
  • Proposition 2: lin2023plug
  • Remark 2
  • Theorem 3: Consistency and Asymptotic Normality
  • Theorem 4
  • Definition 1
  • Theorem 5: Convolution Theorem
  • ...and 45 more