Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality

TL;DR

This paper studies a stochastic approximation method for problems where data distributions depend on decisions, demonstrating asymptotic normality and local minimax optimality of the algorithm under mild conditions.

Contribution

It introduces an analysis of stochastic approximation with decision-dependent distributions, establishing asymptotic normality and optimality results.

Findings

Asymptotic normality of the average iterate around the solution.

Decoupling of gradient noise and distributional shift effects.

Local minimax optimality of the averaged algorithm.

Abstract

We analyze a stochastic approximation algorithm for decision-dependent problems, wherein the data distribution used by the algorithm evolves along the iterate sequence. The primary examples of such problems appear in performative prediction and its multiplayer extensions. We show that under mild assumptions, the deviation between the average iterate of the algorithm and the solution is asymptotically normal, with a covariance that clearly decouples the effects of the gradient noise and the distributional shift. Moreover, building on the work of H\'ajek and Le Cam, we show that the asymptotic performance of the algorithm with averaging is locally minimax optimal.