Accelerated Mirror Descent Method through Variable and Operator Splitting
Optimization and Control
2026-01-28 v1
Abstract
Mirror descent uses the mirror function to encode geometry and constraints, improving convergence while preserving feasibility. Accelerated Mirror Descent Methods (Acc-MD) are derived from a discretization of an accelerated mirror ODE system using a variable--operator splitting framework. A geometric assumption, termed the Generalized Cauchy-Schwarz (GCS) condition, is introduced to quantify the compatibility between the objective and the mirror geometry, under which the first accelerated linear convergence for Acc-MD on a broad class of problems is established. Numerical experiments on smooth and composite optimization tasks demonstrate that Acc-MD consistently outperforms existing accelerated variants, both theoretically and empirically.
Cite
@article{arxiv.2601.19038,
title = {Accelerated Mirror Descent Method through Variable and Operator Splitting},
author = {Long Chen and Hao Luo and Jingrong Wei and Zeyi Xu and Yuan Yao},
journal= {arXiv preprint arXiv:2601.19038},
year = {2026}
}