Query-Efficient Zeroth-Order Algorithms for Nonconvex Optimization
Ruiyang Jin, Yuke Zhou, Yujie Tang, Jie Song, Siyang Gao
TL;DR
This work tackles black-box constrained optimization where both the objective and constraints are nonanalytic. It reformulates the problem as a deterministic min-max program f(x,y)=h(x)+y^T c(x) and develops two zeroth-order block-update methods, ZOB-GDA and ZOB-SGDA, to achieve strong query-efficiency while maintaining convergence guarantees in nonconvex-concave settings. Theoretical results show that ZOB-GDA attains a bound of $O(d/\epsilon^6)$ and ZOB-SGDA achieves the optimal $O(d/\epsilon^4)$ overall query complexity, with single-step costs controlled by block size $b$; empirical results on a large-scale energy-management problem demonstrate over 10x reductions in queries compared to prior methods. The proposed approach broadens the applicability of zeroth-order optimization to general constraints by leveraging block-gradient estimators within primal-dual min-max frameworks, offering practical benefits in high-dimensional, nonconvex settings.
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 complexity. However, existing 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 state-of-the-art overall query complexity bound of $\mathcal{O}(d/ε^4)$ to find an $ε$-stationary solution ($d$ is the dimension of variable space), while reducing the queries for estimating a single-step gradient from $\mathcal{O}(d)$ to $\mathcal{O}(1)$. Specifically, we integrate block updates with gradient descent ascent and a block gradient estimator, 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 on a practical problem demonstrate that our algorithms require over ten times fewer queries than existing methods.