Proximal-Proximal-Gradient Method

TL;DR

The paper introduces the proximal-proximal-gradient (PPG) method, a new optimization algorithm that efficiently handles large sums of coupled convex functions, with proven convergence and a stochastic variant for enhanced scalability.

Contribution

It presents the PPG algorithm, generalizing existing methods like proximal-gradient and ADMM, and introduces a stochastic version with convergence guarantees for complex convex optimization problems.

Findings

PPG is simple to implement and parallelize.

PPG handles large sums of coupled non-differentiable functions with constant stepsize.

S-PPG extends Finito and MISO to coupled non-differentiable functions.

Abstract

In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to minimization problems written as a sum of many differentiable and many non-differentiable convex functions. The non-differentiable functions can be coupled. We furthermore present a related stochastic variation, which we call stochastic PPG (S-PPG). S-PPG can be interpreted as a generalization of Finito and MISO over to the sum of many coupled non-differentiable convex functions. We present many applications that can benefit from PPG and S-PPG and prove convergence for both methods. A key strength of PPG and S-PPG is, compared to existing methods, its ability to directly handle a large sum of non-differentiable non-separable functions with a constant stepsize independent of the number of functions. Such non-diminishing stepsizes allows them to be fast.