Inertial Bregman Proximal Gradient under Partial Smoothness

TL;DR

This paper introduces an inertial Bregman proximal gradient algorithm that relaxes traditional smoothness assumptions, guarantees convergence under certain geometric conditions, and demonstrates local linear convergence in nonconvex optimization problems.

Contribution

It develops an inertial Bregman proximal gradient method with partial smoothness and the triangle scaling property, extending convergence guarantees to nonconvex settings.

Findings

Global convergence under KL property with strongly convex entropy

Finite activity identification in partly smooth nonconvex problems

Local linear convergence rate estimate

Abstract

This work considers an Inertial version of Bregman Proximal Gradient algorithm (IBPG) for minimizing the sum of two single-valued functions in finite dimension. We suppose that one of the functions is proper, closed, and convex but non-necessarily smooth whilst the second is a smooth enough function but not necessarily convex. For the latter, we ask the smooth adaptable property (smad) with respect to some kernel or entropy which allows to remove the very popular global Lipschitz continuity requirement on the gradient of the smooth part. We consider the IBPG under the framework of the triangle scaling property (TSP) which is a geometrical property for which one can provably ensure acceleration for a certain subset of kernel/entropy functions in the convex setting. Based on this property, we provide global convergence guarantees when the entropy is strongly convex under the framework of the Kurdyka-\L{}ojasiewicz (KL) property. Turning to the local convergence properties, we show that when the nonsmooth part is partly smooth relative to a smooth submanifold, IBPG has a finite activity identification property before entering a local linear convergence regime for which we establish a sharp estimate of the convergence rate. We report numerical simulations to illustrate our theoretical results on low complexity regularized phase retrieval.