Optimizing the Performative Risk under Weak Convexity Assumptions

TL;DR

This paper investigates performative prediction where models influence data distribution, relaxing previous convexity assumptions to enable effective optimization under weaker conditions.

Contribution

It introduces weaker convexity assumptions for performative risk minimization, broadening the scope of models that can be optimized efficiently.

Findings

Weaker convexity conditions still allow for effective optimization.

Relaxed assumptions expand applicability to more models.

The approach maintains convergence properties under weaker conditions.

Abstract

In performative prediction, a predictive model impacts the distribution that generates future data, a phenomenon that is being ignored in classical supervised learning. In this closed-loop setting, the natural measure of performance named performative risk ($\mathrm{PR}$), captures the expected loss incurred by a predictive model \emph{after} deployment. The core difficulty of using the performative risk as an optimization objective is that the data distribution itself depends on the model parameters. This dependence is governed by the environment and not under the control of the learner. As a consequence, even the choice of a convex loss function can result in a highly non-convex $\mathrm{PR}$ minimization problem. Prior work has identified a pair of general conditions on the loss and the mapping from model parameters to distributions that implies the convexity of the performative risk. In this paper, we relax these assumptions and focus on obtaining weaker notions of convexity, without sacrificing the amenability of the $\mathrm{PR}$ minimization problem for iterative optimization methods.