Convergence Rates of Inertial Splitting Schemes for Nonconvex Composite Optimization

TL;DR

This paper analyzes the convergence rates of inertial splitting algorithms for nonconvex nonsmooth optimization, establishing broad applicability and different convergence behaviors based on the KL inequality.

Contribution

It introduces new convergence rate results for inertial algorithms under the KL inequality for nonconvex nonsmooth problems.

Findings

Convergence can be finite, linear, or sublinear depending on the KL exponent.

Results apply to various inertial algorithms in signal processing and machine learning.

Provides a unified framework for convergence analysis of inertial methods.

Abstract

We study the convergence properties of a general inertial first-order proximal splitting algorithm for solving nonconvex nonsmooth optimization problems. Using the Kurdyka--\L ojaziewicz (KL) inequality we establish new convergence rates which apply to several inertial algorithms in the literature. Our basic assumption is that the objective function is semialgebraic, which lends our results broad applicability in the fields of signal processing and machine learning. The convergence rates depend on the exponent of the "desingularizing function" arising in the KL inequality. Depending on this exponent, convergence may be finite, linear, or sublinear and of the form $O(k^{-p})$ for $p>1$.