Variable Bregman Majorization-Minimization Algorithm and its Application to Dirichlet Maximum Likelihood Estimation

TL;DR

This paper introduces a Variable Bregman Majorization-Minimization algorithm that adaptively varies the Bregman divergence to accelerate convergence in convex optimization, with a novel application to Dirichlet maximum likelihood estimation.

Contribution

The paper develops a new adaptive Bregman descent algorithm that generalizes existing methods and demonstrates its effectiveness in Dirichlet parameter estimation.

Findings

VBMM converges to a minimizer under mild conditions.

VBMM outperforms existing methods in convergence speed.

Application to Dirichlet MLE shows practical efficiency.

Abstract

We propose a novel Bregman descent algorithm for minimizing a convex function that is expressed as the sum of a differentiable part (defined over an open set) and a possibly nonsmooth term. The approach, referred to as the Variable Bregman Majorization-Minimization (VBMM) algorithm, extends the Bregman Proximal Gradient method by allowing the Bregman function used in the divergence to adaptively vary at each iteration, provided it satisfies a majorizing condition on the objective function. This adaptive framework enables the algorithm to approximate the objective more precisely at each iteration, thereby allowing for accelerated convergence compared to the traditional Bregman Proximal Gradient descent. We establish the convergence of the VBMM algorithm to a minimizer under mild assumptions on the family of metrics used. Furthermore, we introduce a novel application of both the Bregman Proximal Gradient method and the VBMM algorithm to the estimation of the multidimensional parameters of a Dirichlet distribution through the maximization of its log-likelihood. Numerical experiments confirm that the VBMM algorithm outperforms existing approaches in terms of convergence speed.