Bregman Alternating Direction Method of Multipliers
This work provides a unified framework for ADMM variants, potentially improving optimization efficiency for domain-specific problems like linear programming.
The paper introduces Bregman ADMM (BADMM), a generalization of ADMM using Bregman divergences to better exploit problem structures, achieving global convergence with O(1/T) iteration complexity and demonstrating speedups, such as being several times faster than Gurobi in solving mass transportation problems.
The mirror descent algorithm (MDA) generalizes gradient descent by using a Bregman divergence to replace squared Euclidean distance. In this paper, we similarly generalize the alternating direction method of multipliers (ADMM) to Bregman ADMM (BADMM), which allows the choice of different Bregman divergences to exploit the structure of problems. BADMM provides a unified framework for ADMM and its variants, including generalized ADMM, inexact ADMM and Bethe ADMM. We establish the global convergence and the $O(1/T)$ iteration complexity for BADMM. In some cases, BADMM can be faster than ADMM by a factor of $O(n/\log(n))$. In solving the linear program of mass transportation problem, BADMM leads to massive parallelism and can easily run on GPU. BADMM is several times faster than highly optimized commercial software Gurobi.