Asynchronous Optimization over Graphs: Linear Convergence under Error Bound Conditions

TL;DR

This paper introduces a distributed asynchronous optimization algorithm for partially separable problems, achieving linear convergence under error bound conditions, with proven guarantees and practical effectiveness in matrix completion and LASSO tasks.

Contribution

It presents the first distributed asynchronous algorithm with convergence guarantees for this class of problems, including nonconvex cases under error bound conditions.

Findings

Linear convergence under error bound conditions

Effective in matrix completion and LASSO problems

Converges to stationary solutions in nonconvex settings

Abstract

We consider convex and nonconvex constrained optimization with a partially separable objective function: agents minimize the sum of local objective functions, each of which is known only by the associated agent and depends on the variables of that agent and those of a few others. This partitioned setting arises in several applications of practical interest. We propose what is, to the best of our knowledge, the first distributed, asynchronous algorithm with rate guarantees for this class of problems. When the objective function is nonconvex, the algorithm provably converges to a stationary solution at a sublinear rate whereas linear rate is achieved when the objective satisfies under the renowned Luo-Tseng error bound condition (which is less stringent than strong convexity). Numerical results on matrix completion and LASSO problems show the effectiveness of our method.