# Convergence of the Iterates in Mirror Descent Methods

arXiv: https://arxiv.org/abs/1805.01526 · HTML: https://mlsanity.com/paper/1805.01526/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 May 2018.

## Built on by (most cited first, 8 of 8)

- [Differentiable Bilevel Programming for Stackelberg Congestion Games](https://mlsanity.com/paper/2209.07618/) (2022)
- [On Centralized and Distributed Mirror Descent: Convergence Analysis Using Quadratic Constraints](https://mlsanity.com/paper/2105.14385/) (2021)
- [Second-Order Mirror Descent: Convergence in Games Beyond Averaging and Discounting](https://mlsanity.com/paper/2111.09982/) (2021)
- [Distributed Bayesian Estimation in Sensor Networks: Consensus on Marginal Densities](https://mlsanity.com/paper/2312.01227/) (2023)
- [Convergence Analysis of Stochastic Saddle Point Mirror Descent Algorithm -- A Projected Dynamical View Point](https://mlsanity.com/paper/2404.04907/) (2024)
- [Non-KKT Accumulation in Entropic Mirror Descent](https://mlsanity.com/paper/2608.01658/) (2026)
- [Mirror Descent-Type Algorithms for the Variational Inequality Problem with Functional Constraints](https://mlsanity.com/paper/2605.16262/) (2026)
- [Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization](https://mlsanity.com/paper/2608.07248/) (2026)

## Abstract (arXiv)

We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem.