Asynchronous Stochastic Proximal Methods for Nonconvex Nonsmooth Optimization.

We study stochastic algorithms for solving non-convex optimization problems with a convex yet possibly non-smooth regularizer, which find wide applications in many practical machine learning applications. However, compared to asynchronous parallel stochastic gradient descent (AsynSGD), an algorithm targeting smooth optimization, the understanding of the behavior of stochastic algorithms for the non-smooth regularized optimization problems is limited, especially when the objective function is non-convex. To fill this gap, in this paper, we propose and analyze asynchronous parallel stochastic proximal gradient (AsynSPG) methods, including a full update version and a block-wise version, for non-convex problems. We establish an ergodic convergence rate of $O(1/sqrt{K})$ for the proposed AsynSPG, $K$ being the number of updates made on the model, matching the convergence rate currently known for AsynSGD (for smooth problems). To our knowledge, this is the first work that provides convergence rates of asynchronous parallel SPG algorithms for non-convex problems. Furthermore, our results are also the first to prove convergence of any stochastic proximal methods without assuming an increasing batch size or the use of additional variance reduction techniques. We implement the proposed algorithms on Parameter Server and demonstrate its convergence behavior and near-linear speedup, as the number of workers increases, for sparse learning problems on a real-world dataset.