# Algorithms of Inertial Mirror Descent in Convex Problems of Stochastic Optimization

## TL;DR

This paper introduces an inertial mirror descent method for convex stochastic optimization problems, extending classical mirror descent with a new approach inspired by the heavy ball method, and provides theoretical error bounds.

## Contribution

It proposes a novel inertial mirror descent algorithm that does not require averaging, applicable to convex problems, with proven error bounds and a discrete implementation.

## Key findings

- Inertial MD generalizes classical mirror descent.
- The method achieves a proven upper bound on objective function error.
- Discrete algorithm implementation is provided.

## Abstract

The goal is to modify the known method of mirror descent (MD), proposed by A.S. Nemirovsky and D.B. Yudin in 1979. The paper shows the idea of a new, so-called inertial MD method with the example of a deterministic optimization problem in continuous time. In particular, in the Euclidean case, the heavy ball method by B.T. Polyak is realized. It is noted that the new method does not use additional averaging. A discrete algorithm of inertial MD is described. The theorem on the upper bound on the error in the objective function is proved.