Improved Rates for Stochastic Variance-Reduced Difference-of-Convex Algorithms

Abstract

In this work, we propose and analyze DCA-PAGE, a novel algorithm that integrates the difference-of-convex algorithm (DCA) with the ProbAbilistic Gradient Estimator (PAGE) to solve structured nonsmooth difference-of-convex programs. In the finite-sum setting, our method achieves a gradient computation complexity of $O(N+N^{1/2}{\epsilon }^{-2})$ with sample size $N$, surpassing the previous best-known complexity of $O(N+N^{2/3}{\epsilon }^{-2})$ for stochastic variance-reduced (SVR) DCA methods. Furthermore, DCA-PAGE readily extends to online settings with a similar optimal gradient computation complexity $O(b+b^{1/2}{\epsilon }^{-2})$ with batch size $b$, a significant advantage over existing SVR DCA approaches that only work for the finite-sum setting. We further refine our analysis with a gap function, which enables us to obtain comparable convergence guarantees under milder assumptions.

Accepted at IEEE Conference on Decision and Control (IEEE CDC 2025)