Variational Principles for Mirror Descent and Mirror Langevin Dynamics

Belinda Tzen, Anant Raj, Maxim Raginsky, Francis Bach

Research output: Contribution to journalArticlepeer-review


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 letter 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.

Original languageEnglish (US)
Pages (from-to)1542-1547
Number of pages6
JournalIEEE Control Systems Letters
StatePublished - 2023


  • Optimization
  • optimal control
  • stochastic optimal control

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Control and Optimization


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