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

• URL: http://arxiv.org/abs/2207.04173v3
• Date: Thu, 14 Mar 2024 01:41:34 GMT
• Title: Stochastic Approximation with Decision-Dependent Distributions: Asymptotic Normality and Optimality
• Authors: Joshua Cutler, Mateo Díaz, Dmitriy Drusvyatskiy,
• Abstract summary: We analyze an approximation for decision-dependent problems, wherein the data distribution used by the algorithm evolves along the iterate sequence. We show that under mild assumptions, the deviation between the iterate of the algorithm and its solution isally normal. We also show that the performance of the algorithm with averaging is locally minimax optimal.
• Score: 8.771678221101368
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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