  Communication delays and synchronization are major bottlenecks for parallel
computing, and tolerating asynchrony is therefore crucial for accelerating
distributed computation. Motivated by modern interest in 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-Lojasiewicz 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
in this setting.
