Query-Efficient Zeroth-Order Algorithms for Nonconvex Constrained Optimization

TL;DR

This paper introduces new zeroth-order algorithms for constrained nonconvex optimization that significantly improve query efficiency by estimating partial gradients, achieving optimal overall complexity and demonstrating strong empirical performance.

Contribution

The paper proposes two novel algorithms, ZOB-GDA and ZOB-SGDA, which reduce single-step gradient query complexity while maintaining optimal overall convergence rates.

Findings

Achieve the best-known overall query complexity of O(d/ε^4).

Reduce single-step gradient estimation queries from O(d) to O(1).

Numerical experiments show superior performance over existing methods.

Abstract

Zeroth-order optimization (ZO) has been a powerful framework for solving black-box problems, which estimates gradients using zeroth-order data to update variables iteratively. The practical applicability of ZO critically depends on the efficiency of single-step gradient estimation and the overall query complexities. However, existing constrained ZO algorithms cannot achieve efficiency on both simultaneously. In this work, we consider a general constrained optimization model with black-box objective and constraint functions. To solve it, we propose novel algorithms that can achieve the best-known overall query complexity bound of $\mathcal{O}(d/\epsilon^4)$ to find an $\epsilon$-stationary solution ($d$ is the dimension of variables), while reducing the queries for estimating a single-step gradient from $\mathcal{O}(d)$ to $\mathcal{O}(1)$. Specifically, we integrate block gradient estimators with gradient descent ascent, which leads to two algorithms, ZOB-GDA and ZOB-SGDA, respectively. Instead of constructing full gradients, they estimate only partial gradients along random blocks of dimensions, where the adjustable block sizes enable high single-step efficiency without sacrificing convergence guarantees. Our theoretical results establish the finite-sample convergence of the proposed algorithms for nonconvex optimization. Finally, numerical experiments demonstrate the superior performance of our algorithms compared to existing methods.