Bregman proximal gradient method for linear optimization under entropic constraints

TL;DR

This paper introduces an efficient Bregman proximal gradient algorithm for linear optimization problems with entropic constraints, providing convergence guarantees and theoretical insights into related algorithms like Blahut-Arimoto.

Contribution

It develops a novel algorithm tailored for entropic constraints, analyzing active/inactive cases, and offers theoretical justification for existing methods.

Findings

Achieves an $O(1/n)$ convergence rate in objective values.

Provides a theoretical basis for Blahut-Arimoto algorithm.

Demonstrates efficiency through comparisons with standard solvers.

Abstract

In this paper, we present an efficient algorithm for solving a linear optimization problem with entropic constraints, a class of problems that arises in game theory and information theory. Our analysis distinguishes between the cases of active and inactive constraints, addressing each using a Bregman proximal gradient method with entropic Legendre functions, for which we establish a convergence rate of $O(1/n)$ in objective values. For a specific cost structure, our framework provides a theoretical justification for the well-known Blahut-Arimoto algorithm and the uniqueness of the Lagrange multiplier associated with the entropic constraint. In the active constraint setting, we include a bisection procedure to approximate the strictly positive Lagrange multiplier. The efficiency of the proposed method is illustrated through comparisons with standard optimization solvers on a representative example from game theory, including extensions to higher-dimensional settings.