Shuffling the Stochastic Mirror Descent via Dual Lipschitz Continuity and Kernel Conditioning

TL;DR

This paper introduces dual kernel conditioning to extend convergence analysis of stochastic mirror descent algorithms to non-Lipschitz smooth settings, providing new complexity bounds and convergence guarantees.

Contribution

It proposes the dual kernel conditioning (DKC) condition, enabling analysis of stochastic mirror descent without primal Lipschitz smoothness, and establishes convergence results in this broader context.

Findings

DKC is widely satisfied by common kernels.

DKC is closed under affine and conic operations.

First complexity bounds for random reshuffling mirror descent in non-Lipschitz settings.

Abstract

The global Lipschitz smoothness condition underlies most convergence and complexity analyses via two key consequences: the descent lemma and the gradient Lipschitz continuity. How to study the performance of optimization algorithms in the absence of Lipschitz smoothness remains an active area. The relative smoothness framework from Bauschke-Bolte-Teboulle (2017) and Lu-Freund-Nesterov (2018) provides an extended descent lemma, ensuring convergence of Bregman-based proximal gradient methods and their vanilla stochastic counterparts. However, many widely used techniques (e.g., momentum schemes, random reshuffling, and variance reduction) additionally require the Lipschitz-type bound for gradient deviations, leaving their analysis under relative smoothness an open area. To resolve this issue, we introduce the dual kernel conditioning (DKC) regularity condition to regulate the local relative curvature of the kernel functions. Combined with the relative smoothness, DKC provides a dual Lipschitz continuity for gradients: even though the gradient mapping is not Lipschitz in the primal space, it preserves Lipschitz continuity in the dual space induced by a mirror map. We verify that DKC is widely satisfied by popular kernels and is closed under affine composition and conic combination. With these novel tools, we establish the first complexity bounds as well as the iterate convergence of random reshuffling mirror descent for constrained nonconvex relative smooth problems.