Gradient-Free Method for Heavily Constrained Nonconvex Optimization

TL;DR

This paper introduces a doubly stochastic zeroth-order gradient method with momentum and adaptive step size, effectively handling heavily constrained nonconvex optimization problems with many black/white-box constraints, improving efficiency and accuracy.

Contribution

It proposes a novel DSZOG method that converges to an $ ext{epsilon}$-stationary point for heavily constrained nonconvex problems, addressing efficiency issues of existing zeroth-order methods.

Findings

Demonstrates superior training time over existing methods.

Achieves higher accuracy in constrained optimization tasks.

Proves convergence to an $ ext{epsilon}$-stationary point.

Abstract

Zeroth-order (ZO) method has been shown to be a powerful method for solving the optimization problem where explicit expression of the gradients is difficult or infeasible to obtain. Recently, due to the practical value of the constrained problems, a lot of ZO Frank-Wolfe or projected ZO methods have been proposed. However, in many applications, we may have a very large number of nonconvex white/black-box constraints, which makes the existing zeroth-order methods extremely inefficient (or even not working) since they need to inquire function value of all the constraints and project the solution to the complicated feasible set. In this paper, to solve the nonconvex problem with a large number of white/black-box constraints, we proposed a doubly stochastic zeroth-order gradient method (DSZOG) with momentum method and adaptive step size. Theoretically, we prove DSZOG can converge to the $\epsilon$-stationary point of the constrained problem. Experimental results in two applications demonstrate the superiority of our method in terms of training time and accuracy compared with other ZO methods for the constrained problem.