# Fastest Rates for Stochastic Mirror Descent Methods

## TL;DR

This paper introduces faster stochastic mirror descent algorithms for relatively smooth functions, including the first stochastic mirror descent variant with linear convergence, advancing stochastic convex optimization techniques.

## Contribution

It proposes two novel algorithms, relRCD and relSGD, extending stochastic mirror descent to relatively smooth functions with improved convergence rates.

## Key findings

- relRCD achieves linear convergence rate.
- Algorithms generalize classical methods to relatively smooth setting.
- Enhanced efficiency in stochastic convex optimization.

## Abstract

Relative smoothness - a notion introduced by Birnbaum et al. (2011) and rediscovered by Bauschke et al. (2016) and Lu et al. (2016) - generalizes the standard notion of smoothness typically used in the analysis of gradient type methods. In this work we are taking ideas from well studied field of stochastic convex optimization and using them in order to obtain faster algorithms for minimizing relatively smooth functions. We propose and analyze two new algorithms: Relative Randomized Coordinate Descent (relRCD) and Relative Stochastic Gradient Descent (relSGD), both generalizing famous algorithms in the standard smooth setting. The methods we propose can be in fact seen as a particular instances of stochastic mirror descent algorithms. One of them, relRCD corresponds to the first stochastic variant of mirror descent algorithm with linear convergence rate.