Composite Online-to-Nonconvex Conversion with Optimal Oracle Complexity

2026-10-08

AI Summary

This study addresses the absence of optimal first-order oracle complexity bounds for stochastic nonsmooth nonconvex composite optimization. Building upon an online-to-nonconvex conversion framework, the proposed method introduces a novel regularized loss function to handle composite objectives, integrating a variant of online mirror descent with Goldstein stationarity conditions and both stochastic gradient and zeroth-order access techniques. The primary contribution lies in establishing the first optimal oracle complexity for this setting, theoretically demonstrating that convex regularization incurs no additional complexity overhead and thereby extending online learning theory. The algorithm achieves optimal complexities of O(δ⁻¹ε⁻³) for first-order queries and O(dδ⁻¹ε⁻³) for zeroth-order queries, with numerical experiments validating its practical effectiveness.

Abstract

We consider stochastic nonsmooth nonconvex composite optimization, which includes several important problems such as constrained optimization and the regularized training of neural networks. The objective is the sum of a possibly nonsmooth nonconvex Lipschitz function and a convex regularizer, and the function is accessed through stochastic gradients or function values. The goal is to find a point that satisfies a Goldstein-type stationarity condition designed for composite objectives. To our knowledge, no oracle complexity bound for this setting is known under first-order access, and existing complexities under zeroth-order access are suboptimal. To handle this issue, we employ the framework of online-to-nonconvex conversion, which chooses update directions by an online learner and is known to achieve optimal rates for noncomposite problems. We extend the framework to our composite scenario by introducing new losses for the learner, which contain the regularizer itself rather than its linearization and for which a variant of online mirror descent achieves low regret. We show that the resulting algorithm finds such a point with $O(δ^{-1}\varepsilon^{-3})$ stochastic gradient queries or $O(dδ^{-1}\varepsilon^{-3})$ function-value queries, where $δ$ is the Goldstein radius, $\varepsilon$ is the stationarity tolerance, and $d$ is the dimension. These rates match the optimal ones for noncomposite nonsmooth nonconvex optimization, demonstrating that the additional convex regularizer does not worsen the oracle complexity. We also give rates for the smooth case and present numerical experiments.

Problem

• stochastic nonsmooth nonconvex composite optimization
• oracle complexity
• Goldstein stationarity
• online-to-nonconvex conversion

Innovation

• Composite Optimization
• Online-to-Nonconvex Conversion
• Oracle Complexity
• Goldstein Stationarity
• Online Mirror Descent