A Coordinate Descent Primal-Dual Algorithm with Large Step Size and Possibly Non Separable Functions

TL;DR

This paper presents a coordinate descent primal-dual algorithm that handles large step sizes and non-separable functions, with proven convergence properties and applications to large-scale machine learning problems.

Contribution

It introduces a novel coordinate descent primal-dual algorithm with convergence guarantees for non-separable, non-differentiable problems, expanding the applicability of such methods.

Findings

Converges to saddle points under wider parameter ranges.

Achieves linear convergence under strong convexity.

Demonstrates effectiveness on large-scale SVM and total variation problems.

Abstract

This paper introduces a coordinate descent version of the V\~u-Condat algorithm. By coordinate descent, we mean that only a subset of the coordinates of the primal and dual iterates is updated at each iteration, the other coordinates being maintained to their past value. Our method allows us to solve optimization problems with a combination of differentiable functions, constraints as well as non-separable and non-differentiable regularizers. We show that the sequences generated by our algorithm converge to a saddle point of the problem at stake, for a wider range of parameter values than previous methods. In particular, the condition on the step-sizes depends on the coordinate-wise Lipschitz constant of the differentiable function's gradient, which is a major feature allowing classical coordinate descent to perform so well when it is applicable. We then prove a sublinear rate of convergence in general and a linear rate of convergence if the objective enjoys strong convexity properties. We illustrate the performances of the algorithm on a total-variation regularized least squares regression problem and on large scale support vector machine problems.