Asynchronous Parallel Nonconvex Optimization Under the Polyak-Łojasiewicz Condition

Communication delays and synchronization are major bottlenecks for parallel computing, and tolerating asynchrony is therefore crucial for accelerating parallel computation. Motivated by optimization problems that do not satisfy convexity assumptions, we present an asynchronous block coordinate descent algorithm for nonconvex optimization problems whose objective functions satisfy the Polyak-Łojasiewicz condition. This condition is a generalization of strong convexity to nonconvex problems and requires neither convexity nor uniqueness of minimizers. Under only assumptions of mild smoothness of objective functions and bounded delays, we prove that a linear convergence rate is obtained. Numerical experiments for logistic regression problems are presented to illustrate the impact of asynchrony upon convergence.