Mirror Descent and Constrained Online Optimization Problems

TL;DR

This paper introduces an adaptive Mirror Descent method for constrained online optimization problems involving multiple convex, Lipschitz-continuous, non-smooth functionals, achieving optimal accuracy with minimal gradient evaluations.

Contribution

It proposes a novel adaptive Mirror Descent approach tailored for constrained online problems with non-smooth functionals, optimizing gradient usage and accuracy.

Findings

Achieves optimal accuracy estimates for the problem.

Handles non-smooth convex functionals efficiently.

Applicable to problems on the standard simplex.

Abstract

We consider the following class of online optimization problems with functional constraints. Assume, that a finite set of convex Lipschitz-continuous non-smooth functionals are given on a closed set of $n$-dimensional vector space. The problem is to minimize the arithmetic mean of functionals with a convex Lipschitz-continuous non-smooth constraint. In addition, it is allowed to calculate the (sub)gradient of each functional only once. Using some recently proposed adaptive methods of Mirror Descent the method is suggested to solve the mentioned constrained online optimization problem with optimal estimate of accuracy. For the corresponding non-Euclidean prox-structure the case of a set of $n$-dimensional vectors lying on the standard $n$-dimensional simplex is considered.