Analysis Accelerated Mirror Descent via High-resolution ODEs

Abstract: Mirror descent plays a crucial role in constrained optimization and acceleration schemes, along with its corresponding low-resolution ordinary differential equations (ODEs) framework have been proposed. However, the low-resolution ODEs are unable to distinguish between Polyak's heavy-ball method and Nesterov's accelerated gradient method. This problem also arises with accelerated mirror descent. To address this issue, we derive high-resolution ODEs for accelerated mirror descent and propose a general Lyapunov function framework to analyze its convergence rate in both continuous and discrete time. Furthermore, we demonstrate that accelerated mirror descent can minimize the squared gradient norm at an inverse cubic rate.