Accelerated Mirror Descent Method through Variable and Operator Splitting

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.