Bregman proximal gradient method for linear optimization under entropic constraints

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.

💡 Research Summary

This paper introduces an advanced and efficient algorithmic framework designed to solve linear optimization problems subject to entropic constraints, a fundamental class of problems prevalent in both information theory and game theory. The core challenge addressed is the optimization of linear objectives while maintaining specific entropy-related constraints, which often involve navigating the geometry of a probability simplex.

The authors propose a Bregman proximal gradient method that leverages entropic Legendre functions. Unlike standard proximal methods that rely on Euclidean distances, the use of Bregman divergence is specifically tailored to the intrinsic geometry of entropic constraints, allowing for more natural and efficient convergence within the probability space. A significant technical contribution of this work is the bifurcated approach to handling constraints: the researchers distinguish between active and inactive constraint scenarios. For active constraints, where the constraint is satisfied as an equality, the authors implement a bisection procedure to accurately approximate the strictly positive Lagrange multiplier. This approach not only ensures precision but also provides a rigorous proof for the uniqueness of the associated Lagrange multiplier.

From a theoretical standpoint, the paper establishes a convergence rate of $O(1/n)$ for the objective values, providing a robust guarantee of the algorithm’s performance as the number of iterations increases. Furthermore, the paper achieves a significant theoretical milestone by providing a formal justification for the well-known Blahut-Arimoto algorithm. By demonstrating that the Blahut-Arimoto algorithm emerges as a specific instance of their proposed framework under certain cost structures, the authors bridge the gap between classical information-theoretic methods and modern proximal optimization techniques.

The practical utility of the proposed method is validated through rigorous comparative experiments against standard optimization solvers. Using representative examples from game theory, the authors demonstrate that their method maintains high efficiency even when extended to higher-dimensional settings. The results indicate that the proposed Bregman proximal gradient method outperforms conventional solvers in terms of computational scalability and accuracy in complex, high-dimensional environments. Ultimately, this research provides both the mathematical foundation and the practical tools necessary for tackling complex optimization tasks in modern computational science.