Accelerated Bregman Proximal Gradient Methods for Relatively Smooth Convex Optimization

TL;DR

This paper introduces accelerated Bregman proximal gradient methods for convex optimization, achieving adaptive convergence rates that outperform traditional methods in practice, especially with non-Euclidean distances.

Contribution

The paper develops ABPG methods utilizing the triangle scaling property of Bregman distances, providing adaptive algorithms with empirical fast convergence rates.

Findings

Achieves empirical $O(k^{-2})$ convergence rates.

Develops adaptive methods that perform well with non-Euclidean Bregman distances.

Provides numerical certificates for fast convergence in applications.

Abstract

We consider the problem of minimizing the sum of two convex functions: one is differentiable and relatively smooth with respect to a reference convex function, and the other can be nondifferentiable but simple to optimize. We investigate a triangle scaling property of the Bregman distance generated by the reference convex function and present accelerated Bregman proximal gradient (ABPG) methods that attain an $O(k^{-\gamma})$ convergence rate, where $\gamma\in(0,2]$ is the triangle scaling exponent (TSE) of the Bregman distance. For the Euclidean distance, we have $\gamma=2$ and recover the convergence rate of Nesterov's accelerated gradient methods. For non-Euclidean Bregman distances, the TSE can be much smaller (say $\gamma\leq 1$), but we show that a relaxed definition of intrinsic TSE is always equal to 2. We exploit the intrinsic TSE to develop adaptive ABPG methods that converge much faster in practice. Although theoretical guarantees on a fast convergence rate seem to be out of reach in general, our methods obtain empirical $O(k^{-2})$ rates in numerical experiments on several applications and provide posterior numerical certificates for the fast rates.