Unifying Abstract Inexact Convergence Theorems and Block Coordinate Variable Metric iPiano

Abstract

An abstract convergence theorem for a class of generalized descent methods that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems. It is applicable to possibly nonsmooth and nonconvex lower semicontinuous functions that satisfy the Kurdyka--Łojasiewicz (KL) inequality, which comprises a huge class of problems. Many of the recent algorithms that explicitly prove convergence using the KL inequality can be cast in the abstract framework of this paper and, therefore, the generated sequence converges to a stationary point of the objective function. Additional flexibility compared to related approaches is gained by a descent property that is formulated with respect to a function that is allowed to change along the iterations, a generic distance measure, and an explicit/implicit relative error condition with respect to finite linear combinations of distance terms. As an application of the gained flexibility, the convergence of a block coordinate variable metric version of iPiano (an inertial forward-backward splitting algorithm) is proved, which performs favorably on an inpainting problem with a Mumford--Shah-like regularization from image processing.