A Bregman ADMM for Bethe variational problem

TL;DR

This paper introduces a novel Bregman ADMM algorithm with nonlinear dual updates for solving the Bethe variational problem, offering convergence guarantees and efficiency in non-convex, non-Lipschitz scenarios, with extensions to quantum problems.

Contribution

The paper presents a new Bregman ADMM method with convergence guarantees for non-convex BVP, addressing non-smoothness and enabling parallel computation, also extending to quantum cases.

Findings

Proves strict positivity of local minima entries in BVP.

Demonstrates high efficiency and robustness through numerical experiments.

Provides an open-source implementation of the algorithm.

Abstract

In this work, we propose a novel Bregman ADMM with nonlinear dual update to solve the Bethe variational problem (BVP), a key optimization formulation in graphical models and statistical physics. Our algorithm provides rigorous convergence guarantees, even if the objective function of BVP is non-convex and non-Lipschitz continuous on the boundary. A central result of our analysis is proving that the entries in local minima of BVP are strictly positive, effectively resolving non-smoothness issues caused by zero entries. Beyond theoretical guarantees, the algorithm possesses high level of separability and parallelizability to achieve highly efficient subproblem computation. Our Bregman ADMM can be easily extended to solve the quantum Bethe variational problem. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed method. Based on this research, we have released an open-source package of the proposed method at https://github.com/TTYmath/BADMM-BVP.