Scaled, inexact and adaptive generalized FISTA for strongly convex optimization

TL;DR

This paper introduces SAGE-FISTA, an adaptive, inexact, variable metric algorithm for strongly convex optimization, with proven linear convergence and demonstrated effectiveness in image denoising and deblurring tasks involving TV regularization and Poisson noise.

Contribution

It presents a novel adaptive, inexact, variable metric FISTA variant with linear convergence analysis and practical validation on image processing problems.

Findings

Linear convergence of the proposed algorithm.

Effective in image denoising and deblurring with TV and Poisson noise.

Adaptive step-size improves convergence speed.

Abstract

We consider a variable metric and inexact version of the FISTA-type algorithm considered in (Chambolle, Pock, 2016, Calatroni, Chambolle, 2019) for the minimization of the sum of two (possibly strongly) convex functions. The proposed algorithm is combined with an adaptive (non-monotone) backtracking strategy, which allows for the adjustment of the algorithmic step-size along the iterations in order to improve the convergence speed. We prove a linear convergence result for the function values, which depends on both the strong convexity moduli of the two functions and the upper and lower bounds on the spectrum of the variable metric operators. We validate the proposed algorithm, named Scaled Adaptive GEneralized FISTA (SAGE-FISTA), on exemplar image denoising and deblurring problems where edge-preserving Total Variation (TV) regularization is combined with Kullback-Leibler-type fidelity terms, as it is common in applications where signal-dependent Poisson noise is assumed in the data.