Convergence of the Iterates in Mirror Descent Methods

TL;DR

This paper proves the convergence of variables in mirror descent algorithms under certain step size conditions, extending understanding beyond function value convergence to variable convergence in both centralized and distributed settings.

Contribution

It establishes convergence of the iterates in mirror descent methods without requiring strong convexity, broadening applicability in distributed optimization and learning.

Findings

Numerical simulations compare entropic mirror descent and subgradient methods.

Convergence of variables is achieved with square summable but not summable step sizes.

Results apply to nonsmooth optimization problems in distributed settings.

Abstract

We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem.