Convergence rates for an inertial algorithm of gradient type associated to a smooth nonconvex minimization

TL;DR

This paper analyzes an inertial gradient algorithm for nonconvex optimization, establishing convergence to critical points and providing convergence rates based on the Kurdyka-Łojasiewicz property.

Contribution

It extends inertial gradient methods to nonconvex functions and derives convergence rates using the Kurdyka-Łojasiewicz property.

Findings

Sequences converge to critical points under certain regularity conditions.

Convergence rates are characterized in terms of the Łojasiewicz exponent.

The method generalizes Nesterov's accelerated gradient to nonconvex settings.

Abstract

We investigate an inertial algorithm of gradient type in connection with the minimization of a nonconvex differentiable function. The algorithm is formulated in the spirit of Nesterov's accelerated convex gradient method. We show that the generated sequences converge to a critical point of the objective function, if a regularization of the objective function satisfies the Kurdyka-{\L}ojasiewicz property. Further, we provide convergence rates for the generated sequences and the function values formulated in terms of the {\L}ojasiewicz exponent.