Title: Accelerated Mirror Descent Method through Variable and Operator Splitting
Primary area: optimization
Keywords: Accelerated Mirror Descent, Convex Optimization, Relative Smoothness, Lyapunov Analysis, Non-Euclidean Geometry, Mirror Map, Acceleration in Optimization

Abstract: Accelerated Mirror Descent (Acc-MD) is derived from a discretization of an accelerated mirror ODE system using a variable--operator splitting framework. A new Cauchy--Schwarz type inequality enables the first proof of linear accelerated convergence for mirror descent on a broad class of problems. Unlike prior methods based on the triangle scaling exponent (TSE), Acc-MD achieves acceleration in some cases where TSE fails. Experiments on smooth and composite optimization tasks show that Acc-MD consistently outperforms existing accelerated variants, both theoretically and empirically.

TL;DR: Acc-MD is a new accelerated mirror descent algorithm that attains optimal convergence rates via a rigorous Lyapunov analysis and outperforms existing methods both theoretically and empirically.