Inexact Proximal Newton methods in Hilbert spaces

TL;DR

This paper introduces inexact Proximal Newton methods tailored for Hilbert spaces, providing convergence analysis and demonstrating efficiency gains over exact methods through a model problem in function space.

Contribution

It develops new inexactness criteria suitable for Hilbert space settings and analyzes their impact on convergence and acceleration of Proximal Newton methods.

Findings

The inexact criteria ensure global convergence.

The method achieves local acceleration.

Efficiency gains over exact methods are demonstrated.

Abstract

We consider Proximal Newton methods with an inexact computation of update steps. To this end, we introduce two inexactness criteria which characterize sufficient accuracy of these update step and with the aid of these investigate global convergence and local acceleration of our method. The inexactness criteria are designed to be adequate for the Hilbert space framework we find ourselves in while traditional inexactness criteria from smooth Newton or finite dimensional Proximal Newton methods appear to be inefficient in this scenario. The performance of the method and its gain in effectiveness in contrast to the exact case are showcased considering a simple model problem in function space.