# Stochastic mirror descent dynamics and their convergence in monotone variational inequalities

arXiv: https://arxiv.org/abs/1710.01551 · HTML: https://mlsanity.com/paper/1710.01551/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Oct 2017.

## Built on by (most cited first, 5 of 5)

- [Independent Policy Gradient Methods for Competitive Reinforcement Learning](https://mlsanity.com/paper/2101.04233/) (2021)
- [The Last-Iterate Convergence Rate of Optimistic Mirror Descent in Stochastic Variational Inequalities](https://mlsanity.com/paper/2107.01906/) (2021)
- [First-Order Methods for Convex Optimization](https://mlsanity.com/paper/2101.00935/) (2021)
- [The impact of uncertainty on regularized learning in games](https://mlsanity.com/paper/2506.13286/) (2025)
- [Multi-agent learning under uncertainty: Recurrence vs. concentration](https://mlsanity.com/paper/2512.08132/) (2025)

## Abstract (arXiv)

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system's controllable parameters are two variable weight sequences that respectively pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around this average.