Efficient randomized mirror descents in stochastic online convex optimization

TL;DR

This paper compares classical mirror descent, dual averaging, and the Grigoriadis-Khachiyan algorithm in stochastic online convex optimization, revealing the effectiveness of randomized mirror descent in solving sparse matrix games and related problems.

Contribution

It demonstrates that the Grigoriadis-Khachiyan algorithm is a form of randomized mirror descent with optimal randomization in KL-projection, advancing understanding of stochastic optimization methods.

Findings

Randomized mirror descent is effective for sparse matrix games.

The Grigoriadis-Khachiyan algorithm is a form of dual averaging with optimal randomization.

Randomization in KL-projection improves solutions in convex optimization and experts weighting.

Abstract

In the paper we consider an application of mirror descent (dual averaging) to the stochastic online convex optimization problems. We compare classical mirror descent (Nemirovski-Yudin, 1979) with dual averaging (Nesterov, 2005) and Grigoriadis-Khachiyan algorithm (1995). Grigoriadis-Khachiyan algorithm has just proved to be a randomized mirror descent (dual averaging) with randomization in KL-projection of (sub)gradient to a unit simplex. We found out that this randomization is an optimal way of solving sparse matrix games and some other problems arising in convex optimization and experts weighting.