Random coordinate descent methods for nonseparable composite optimization

TL;DR

This paper introduces two novel coordinate descent algorithms for large-scale composite optimization problems, providing theoretical convergence guarantees and demonstrating practical efficiency through numerical experiments.

Contribution

The paper develops and analyzes two new coordinate descent methods with adaptive stepsizes for nonseparable composite optimization, including convergence proofs and complexity analysis.

Findings

Both algorithms are proven to be descent methods under certain conditions.

The methods have established worst-case complexity bounds in convex and nonconvex cases.

Numerical results show the algorithms are effective on practical problems.

Abstract

In this paper we consider large-scale composite optimization problems having the objective function formed as a sum of two terms (possibly nonconvex), one has (block) coordinate-wise Lipschitz continuous gradient and the other is differentiable but nonseparable. Under these general settings we derive and analyze two new coordinate descent methods. The first algorithm, referred to as coordinate proximal gradient method, considers the composite form of the objective function, while the other algorithm disregards the composite form of the objective and uses the partial gradient of the full objective, yielding a coordinate gradient descent scheme with novel adaptive stepsize rules. We prove that these new stepsize rules make the coordinate gradient scheme a descent method, provided that additional assumptions hold for the second term in the objective function. We present a complete worst-case complexity analysis for these two new methods in both, convex and nonconvex settings, provided that the (block) coordinates are chosen random or cyclic. Preliminary numerical results also confirm the efficiency of our two algorithms on practical problems.