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

Abstract

This paper introduces a randomized coordinate-descent version of the Vu͂--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 and constraints as well as nonseparable and nondifferentiable regularizers. We show that the sequences generated by our algorithm almost surely 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 coordinatewise 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.