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.

翻译：Nemirovski和Yudin于20世纪70年代提出的镜面对称下降，是一种可通过选取强凸势函数来适配优化问题几何结构的原始-对偶凸优化方法。该方法作为基础算法广泛应用于大规模优化、机器学习与控制等场景。本文提出了镜面对称下降及其随机变体——镜面朗之万动力学的变分形式。受Brezis与Ekeland关于梯度流变分原理的经典工作启发，核心思想在于证明镜面对称下降可表示为特定最优控制问题的闭环解，且Bellman值函数由初始条件与目标函数全局极小点之间的Bregman散度给出。