Theoretical and numerical comparison of first-order algorithms for cocoercive equations and smooth convex optimization

TL;DR

This paper compares classical first-order algorithms for smooth convex optimization and cocoercive equations, providing theoretical convergence rates and numerical validation in inverse problems, highlighting the advantages of proximal methods.

Contribution

It offers a comprehensive theoretical comparison of convergence rates for various first-order algorithms in convex optimization and cocoercive equations, with improved rates and practical insights.

Findings

Gradient descent and related algorithms have better convergence rates in optimization than in cocoercive equations.

Proximal-based strategies outperform gradient methods in inverse problems involving sparsity.

Numerical experiments confirm theoretical advantages of proximal methods over gradient-based approaches.

Abstract

This paper provides a theoretical and numerical comparison of classical first-order splitting methods for solving smooth convex optimization problems and cocoercive equations. From a theoretical point of view, we compare convergence rates of gradient descent, forward-backward, Peaceman-Rachford, and Douglas-Rachford algorithms for minimizing the sum of two smooth convex functions when one of them is strongly convex. A similar comparison is given in the more general cocoercive setting under the presence of strong monotonicity and we observe that the convergence rates in optimization are strictly better than the corresponding rates for cocoercive equations for some algorithms. We obtain improved rates with respect to the literature in several instances by exploiting the structure of our problems. Moreover, we indicate which algorithm has the lowest convergence rate depending on strong convexity and cocoercive parameters. From a numerical point of view, we verify our theoretical results by implementing and comparing previous algorithms in well-established signal and image inverse problems involving sparsity. We replace the widely used $\ell_1$ norm with the Huber loss and we observe that fully proximal-based strategies have numerical and theoretical advantages with respect to methods using gradient steps.