TY - JOUR
T1 - Variational Principles for Mirror Descent and Mirror Langevin Dynamics
AU - Tzen, Belinda
AU - Raj, Anant
AU - Raginsky, Maxim
AU - Bach, Francis
N1 - Publisher Copyright:
© 2017 IEEE.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Optimization
KW - optimal control
KW - stochastic optimal control
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U2 - 10.1109/LCSYS.2023.3274069
DO - 10.1109/LCSYS.2023.3274069
M3 - Article
AN - SCOPUS:85159836677
SN - 2475-1456
VL - 7
SP - 1542
EP - 1547
JO - IEEE Control Systems Letters
JF - IEEE Control Systems Letters
ER -