A direct proof of convergence of Davis-Yin splitting algorithm allowing larger stepsizes

TL;DR

This paper provides a direct convergence proof for the Davis-Yin splitting algorithm with larger stepsizes, extends the result to forward-backward splitting, introduces a new algorithm, and demonstrates the importance of parameter choices through experiments.

Contribution

It offers a direct convergence proof allowing larger stepsizes for Davis-Yin splitting, extends the result to forward-backward splitting, and proposes a new resolvent computation algorithm.

Findings

Convergence guaranteed for stepsizes less than four times the cocoercivity constant.

Doubling the allowable stepsize interval compared to previous results.

Numerical experiments highlight the impact of stepsize and relaxation parameters.

Abstract

This note is devoted to the splitting algorithm proposed by Davis and Yin in 2017 for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of "strengthening" of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.