An Inexact Bregman Proximal Difference-of-Convex Algorithm with Two Types of Relative Stopping Criteria

TL;DR

This paper introduces an inexact Bregman proximal DC algorithm with two types of stopping criteria, broadening applicability and efficiency for non-convex optimization problems in machine learning and signal processing.

Contribution

It develops an inexact Bregman proximal DC algorithm with flexible stopping criteria, enabling practical and efficient solutions for DC problems under weaker smoothness assumptions.

Findings

Proven convergence under Kurdyka-Łojasiewicz property.

Numerical experiments show superior performance over existing methods.

Demonstrates the importance of stopping criteria in subproblem solutions.

Abstract

In this paper, we consider a class of difference-of-convex (DC) optimization problems, which require only a weaker restricted $L$-smooth adaptable property on the smooth part of the objective function, instead of the standard global Lipschitz gradient continuity assumption. Such problems are prevalent in many contemporary applications such as compressed sensing, statistical regression, and machine learning, and can be solved by a general Bregman proximal DC algorithm (BPDCA). However, the existing BPDCA is developed based on the stringent requirement that the involved subproblems must be solved exactly, which is often impractical and limits the applicability of the BPDCA. To facilitate the practical implementations and wider applications of the BPDCA, we develop an inexact Bregman proximal difference-of-convex algorithm (iBPDCA) by incorporating two types of relative-type stopping criteria for solving the subproblems. The proposed inexact framework has considerable flexibility to encompass many existing exact and inexact methods, and can accommodate different types of errors that may occur when solving the subproblem. This enables the potential application of our inexact framework across different DC decompositions to facilitate the design of a more efficient DCA scheme in practice. The global subsequential convergence and the global sequential convergence of our iBPDCA are established under suitable conditions including the Kurdyka-{\L}ojasiewicz property. Some numerical experiments are conducted to show the superior performance of our iBPDCA in comparison to existing algorithms. These results also empirically validate the necessity and significance of developing different types of stopping criteria to facilitate the efficient computation of the subproblem in each iteration of our iBPDCA.