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A Simple Proximal Stochastic Gradient Method for Nonsmooth Nonconvex Optimization
Zhize Li · Jian Li

Thu Dec 06 12:30 PM -- 12:35 PM (PST) @ Room 517 CD

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+. Our main contribution lies in the analysis of ProxSVRG+. It recovers several existing convergence results and improves/generalizes them (in terms of the number of stochastic gradient oracle calls and proximal oracle calls). 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 also 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., 2016]. Also, ProxSVRG+ uses much less proximal oracle calls than ProxSVRG [Reddi et al., 2016]. Moreover, for nonconvex functions satisfied Polyak-\L{}ojasiewicz condition, we prove that ProxSVRG+ achieves a global linear convergence rate without restart unlike ProxSVRG. Thus, it can \emph{automatically} switch to the faster linear convergence in some regions as long as the objective function satisfies the PL condition locally in these regions. Finally, we conduct several experiments and the experimental results are consistent with the theoretical results.

Author Information

Zhize Li (Tsinghua University)

Zhize Li is a PhD student at Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University (Advisor: Prof. Jian Li). His research interests lie in theoretical computer science and machine learning, in particular (non-)convex optimization algorithms, machine learning, algorithms and data structures.

Jian Li (Tsinghua University)

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