Inexact Bregman Proximal Gradient Method and its Inertial Variant with Absolute and Partial Relative Stopping Criteria

TL;DR

The paper addresses convex composite optimization of the form $F(\bm{x}) = P(\bm{x}) + f(\bm{x})$ over a closed convex set, where the proximal step is defined via a Bregman distance. It introduces inexact versions of the Bregman proximal gradient method (iBPGM) with verifiable absolute-type (AbSC) and partial relative-type (ReSC) stopping criteria for subproblem solves, establishing $O(1/k)$ convergence and sequence convergence under suitable conditions. To accelerate, the paper develops an inertial variant (v-iBPGM) that achieves $O(1/k^{\gamma})$ convergence with $\gamma \ge 1$ under a restricted relative smoothness assumption, attaining $\gamma=2$ in the Lipschitz-gradient and strongly convex kernel scenario. Preliminary numerical experiments on discrete quadratic regularized optimal transport (QROT) problems, using Sinkhorn-type subproblem solvers, illustrate the impact of different inexactness settings and demonstrate practical efficiency of the proposed methods.

Abstract

The Bregman proximal gradient method (BPGM), which uses the Bregman distance as a proximity measure in the iterative scheme, has recently been re-developed for minimizing convex composite problems without the global Lipschitz gradient continuity assumption. This makes the BPGM appealing for a wide range of applications, and hence it has received growing attention in recent years. However, most existing convergence results are only obtained under the assumption that the involved subproblems are solved exactly, which is unrealistic in many applications and limits the applicability of the BPGM. To make the BPGM implementable and practical, in this paper, we develop inexact versions of the BPGM (denoted by iBPGM) by employing either an absolute-type stopping criterion or a partial relative-type stopping criterion for solving the subproblems. The $\mathcal{O}(1/k)$ convergence rate and the convergence of the sequence are also established for our iBPGM under some conditions. Moreover, we develop an inertial variant of our iBPGM (denoted by v-iBPGM) and establish the $\mathcal{O}(1/k^γ)$ convergence rate, where $γ\geq1$ is a restricted relative smoothness exponent depending on the smooth function in the objective and the kernel function. Specially, when the smooth function in the objective has a Lipschitz continuous gradient and the kernel function is strongly convex, we have $γ=2$ and thus the v-iBPGM improves the convergence rate of the iBPGM from $\mathcal{O}(1/k)$ to $\mathcal{O}(1/k^2)$, in accordance with the existing results on the exact accelerated BPGM. Finally, some preliminary numerical experiments for solving the discrete quadratic regularized optimal transport problem are conducted to illustrate the convergence behaviors of our iBPGM and v-iBPGM under different inexactness settings.

Theorems & Definitions (29)

• Definition 2.1: Restricted relative smoothness on $\mathcal{X}$
• Lemma 2.1: Four points identity
• Lemma 2.2: p1987introduction
• Lemma 2.3
• Lemma 2.4
• Remark 3.1: Comments on the underlying feasibility difficulty
• Lemma 3.1
• proof
• Theorem 3.1: Convergence rate of the iBPGM with (AbSC)
• proof