Tight Lower Bounds and Improved Convergence in Performative Prediction

TL;DR

This paper advances performative prediction by establishing tight bounds and demonstrating that leveraging historical data can significantly improve convergence speed to stable solutions in evolving environments.

Contribution

It introduces Affine Risk Minimizers, extends RRM with historical datasets, and provides the first lower bound analysis for this class, showing potential for faster convergence.

Findings

New upper bound for last-iteration methods

Empirical evidence of faster convergence with historical data

First lower bound analysis for Affine Risk Minimizers

Abstract

Performative prediction is a framework accounting for the shift in the data distribution induced by the prediction of a model deployed in the real world. Ensuring rapid convergence to a stable solution where the data distribution remains the same after the model deployment is crucial, especially in evolving environments. This paper extends the Repeated Risk Minimization (RRM) framework by utilizing historical datasets from previous retraining snapshots, yielding a class of algorithms that we call Affine Risk Minimizers and enabling convergence to a performatively stable point for a broader class of problems. We introduce a new upper bound for methods that use only the final iteration of the dataset and prove for the first time the tightness of both this new bound and the previous existing bounds within the same regime. We also prove that utilizing historical datasets can surpass the lower bound for last iterate RRM, and empirically observe faster convergence to the stable point on various performative prediction benchmarks. We offer at the same time the first lower bound analysis for RRM within the class of Affine Risk Minimizers, quantifying the potential improvements in convergence speed that could be achieved with other variants in our framework.