A Simple Proximal Stochastic Gradient Method for Nonsmooth Nonconvex Optimization
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We analyze stochastic gradient algorithms for optimizing nonconvex, nonsmooth finite-sum problems. In particular, the objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a possibly non-differentiable but convex component. We propose a proximal stochastic gradient algorithm based on variance reduction, called ProxSVRG+. The algorithm is a slight variant of the ProxSVRG algorithm [Reddi et al., 2016b]. Our main contribution lies in the analysis of ProxSVRG+. It recovers several existing convergence results (in terms of the number of stochastic gradient oracle calls and proximal operations), and improves/generalizes some others. In particular, ProxSVRG+ generalizes the best results given by the SCSG algorithm, recently proposed by [Lei et al., 2017] for the smooth nonconvex case. ProxSVRG+ is more straightforward than SCSG and yields simpler analysis. Moreover, ProxSVRG+ outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, which partially solves an open problem proposed in [Reddi et al., 2016b]. Finally, for nonconvex functions satisfied Polyak-Łojasiewicz condition, we show that ProxSVRG+ achieves global linear convergence rate without restart. ProxSVRG+ is always no worse than ProxGD and ProxSVRG/SAGA, and sometimes outperforms them (and generalizes the results of SCSG) in this case.
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