# Variational Principles for Mirror Descent and Mirror Langevin Dynamics

arXiv: https://arxiv.org/abs/2303.09532 · HTML: https://mlsanity.com/paper/2303.09532/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 16 Mar 2023.

## Builds on (most cited first, 4 of 4)

- [Mirrored Langevin Dynamics](https://mlsanity.com/paper/1802.10174/) (2018)
- [Exponential ergodicity of mirror-Langevin diffusions](https://mlsanity.com/paper/2005.09669/) (2020)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [Stochastic linear optimization never overfits with quadratically-bounded losses on general data](https://mlsanity.com/paper/2202.06915/) (2022)

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

- [Cautious Optimizers: Improving Training with One Line of Code](https://mlsanity.com/paper/2411.16085/) (2024)
- [Grams: Gradient Descent with Adaptive Momentum Scaling](https://mlsanity.com/paper/2412.17107/) (2024)
- [Is All Learning (Natural) Gradient Descent?](https://mlsanity.com/paper/2409.16422/) (2024)
- [Sinkhorn Flow: A Continuous-Time Framework for Understanding and Generalizing the Sinkhorn Algorithm](https://mlsanity.com/paper/2311.16706/) (2023)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)

## Abstract (arXiv)

Mirror descent, introduced by Nemirovski and Yudin in the 1970s, is a primal-dual convex optimization method that can be tailored to the geometry of the optimization problem at hand through the choice of a strongly convex potential function. It arises as a basic primitive in a variety of applications, including large-scale optimization, machine learning, and control. This paper proposes a variational formulation of mirror descent and of its stochastic variant, mirror Langevin dynamics. The main idea, inspired by the classic work of Brezis and Ekeland on variational principles for gradient flows, is to show that mirror descent emerges as a closed-loop solution for a certain optimal control problem, and the Bellman value function is given by the Bregman divergence between the initial condition and the global minimizer of the objective function.