# Asymptotic convergence of iterative optimization algorithms

## TL;DR

This paper presents a comprehensive framework for iterative optimization algorithms, proving their asymptotic geometric convergence and providing exact rates, applicable to various algorithms including EM and mirror descent.

## Contribution

It introduces a unified framework for analyzing convergence rates of iterative algorithms, including constrained cases and variants like alpha-EM and Mirror Prox.

## Key findings

- Convergence is asymptotically geometric under general assumptions.
- Exact asymptotic convergence rates are established.
- Conditions for systematic convergence of Mirror Prox are provided.

## Abstract

This paper introduces a general framework for iterative optimization algorithms and establishes under general assumptions that their convergence is asymptotically geometric. We also prove that under appropriate assumptions, the rate of convergence can be lower bounded. The convergence is then only geometric, and we provide the exact asymptotic convergence rate. This framework allows to deal with constrained optimization and encompasses the Expectation Maximization algorithm and the mirror descent algorithm, as well as some variants such as the alpha-Expectation Maximization or the Mirror Prox algorithm.Furthermore, we establish sufficient conditions for the convergence of the Mirror Prox algorithm, under which the method converges systematically to the unique minimizer of a convex function on a convex compact set.