Zeroth-order gradient estimators for stochastic problems with decision-dependent distributions

TL;DR

This paper analyzes zeroth-order gradient estimators for stochastic problems with decision-dependent distributions, providing a unified sample complexity analysis and demonstrating the superiority of estimators averaging over multiple directions.

Contribution

It offers a comprehensive analysis of different search directions for zeroth-order gradient estimators and identifies the most sample-efficient strategies, improving existing methods.

Findings

Gradient estimators averaging over multiple directions are most sample-efficient.

The proposed methods outperform existing zeroth-order algorithms in nonconvex, unbounded settings.

Simulation results validate the practical effectiveness of the proposed estimators.

Abstract

Stochastic optimization problems with unknown decision-dependent distributions have attracted increasing attention in recent years due to its importance in applications. Since the gradient of the objective function is inaccessible as a result of the unknown distribution, various zeroth-order methods have been developed to solve the problem. However, it remains unclear which search direction to construct a gradient estimator is more appropriate and how to set the algorithmic parameters. In this paper, we conduct a unified sample complexity analysis of zeroth-order methods across gradient estimators with different search directions. As a result, we show that gradient estimators that average over multiple directions, either uniformly from the unit sphere or from a Gaussian distribution, achieve the lowest sample complexity. The attained sample complexities improve those of existing zeroth-order methods in the problem setting that allows nonconvexity and unboundedness of the objective function. Moreover, by simulation experiments on multiple products pricing and strategic classification applications, we show practical performance of zeroth-order methods with various gradient estimators.