Zeroth-Order Katyusha: An Accelerated Derivative-Free Method for Composite Convex Optimization

TL;DR

This paper introduces ZO-L-Katyusha, an accelerated zeroth-order algorithm that achieves linear convergence and optimal oracle complexity for constrained composite convex problems, bridging a key gap in zeroth-order optimization.

Contribution

The paper proposes ZO-L-Katyusha, combining variance reduction and acceleration to improve convergence and query complexity in zeroth-order composite convex optimization.

Findings

Achieves accelerated linear convergence for strongly convex problems.

Maintains the same oracle complexity as unconstrained methods.

Function query complexity can be reduced to O(1) on average.

Abstract

We investigate accelerated zeroth-order algorithms for smooth composite convex optimization problems. While for unconstrained optimization, existing methods that merge 2-point zeroth-order gradient estimators with first-order frameworks usually lead to satisfactory performance, for constrained/composite problems, there is still a gap in the complexity bound that is related to the non-vanishing variance of the 2-point gradient estimator near an optimal point. To bridge this gap, we propose the Zeroth-Order Loopless Katyusha (ZO-L-Katyusha) algorithm, leveraging the variance reduction as well as acceleration techniques from the first-order loopless Katyusha algorithm. We show that ZO-L-Katyusha is able to achieve accelerated linear convergence for compositve smooth and strongly convex problems, and has the same oracle complexity as the unconstrained case. Moreover, the number of function queries to construct a zeroth-order gradient estimator in ZO-L-Katyusha can be made to be O(1) on average. These results suggest that ZO-L-Katyusha provides a promising approach towards bridging the gap in the complexity bound for zeroth-order composite optimization.