A telescoping Bregmanian proximal gradient method without the global Lipschitz continuity assumption

TL;DR

This paper introduces a novel proximal gradient method that operates without requiring the gradient of the smooth part to be globally Lipschitz continuous, broadening its applicability in various spaces.

Contribution

It proposes a telescoping Bregmanian proximal gradient method that relaxes the global Lipschitz continuity assumption, using a telescopic decomposition and Bregman divergence.

Findings

Establishes a non-asymptotic convergence rate in function values.

Proves weak convergence of the entire sequence to a minimizer.

Applicable to finite and infinite-dimensional spaces.

Abstract

The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. In this paper, we discuss, in a wide class of finite and infinite-dimensional spaces, a new variant of the proximal gradient method which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain telescopic decomposition of the constraint set into subsets. Moreover, we use a Bregman divergence in the proximal forward-backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest.