In particular, ProxSVRG+ generalizes the best results given by the SCSG algorithm, recently proposed by [Lei et al.,

In particular, ProxSVRG+ generalizes the best results given by the SCSG algorithm, recently proposed by [Lei et al.,

This paper focuses on the optimization problem min f(x) h(x), where f is of a finite sum structure (with n functions in the sum), with nonconvex but smooth components, and h is a convex but possibly nonsmooth function. So, this is a nonconvex finite sum problem with a convex regularizer. Function h is treated using a prox step. The authors propose a small modification to ProxSVRG (called ProxSVRG), and prove that this small modification has surprisingly interesting consequences. The modification consists in replacing the full gradient computation in the outer loop of ProxSVRG by an approximation thereof through subsampling/minibatch (batch size B).

Abstract:: We propose and analyze several stochastic gradient algorithms for finding stationary points or local minimum in nonconvex, possibly with nonsmooth regularizer, finite-sum and online optimization problems. First, we propose a simple proximal stochastic gradient algorithm based on variance reduction called ProxSVRG . We provide a clean and tight analysis of ProxSVRG, which shows that it outperforms the deterministic proximal gradient descent (ProxGD) for a wide range of minibatch sizes, hence solves an open problem proposed in Reddi et al. (2016b). Also, ProxSVRG uses much less proximal oracle calls than ProxSVRG (Reddi et al., 2016b) and extends to the online setting by avoiding full gradient computations. Then, we further propose an optimal algorithm, called SSRGD, based on SARAH (Nguyen et al., 2017) and show that SSRGD further improves the gradient complexity of ProxSVRG and achieves the optimal upper bound, matching the known lower bound of (Fang et al., 2018; Li et al., 2021).

We study the idea of variance reduction applied to adaptive stochastic mirror descent algorithms in nonsmooth nonconvex finite-sum optimization problems. We propose a simple yet generalized adaptive mirror descent algorithm with variance reduction named SVRAMD and provide its convergence analysis in different settings. We prove that variance reduction reduces the gradient complexity of most adaptive mirror descent algorithms and boost their convergence. In particular, our general theory implies variance reduction can be applied to algorithms using time-varying step sizes and self-adaptive algorithms such as AdaGrad and RMSProp. Moreover, our convergence rates recover the best existing rates of non-adaptive algorithms. We check the validity of our claims using experiments in deep learning.

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., NIPS'17] for the smooth nonconvex case.

In this paper, we propose a new stochastic algorithmic framework to solve stochastic composite nonconvex optimization problems that covers both finite-sum and expectation settings. Our algorithms rely on the SARAH estimator introduced in (Nguyen et al., 2017a) and consist of two steps: a proximal gradient step and an averaging step that are different from existing nonconvex proximal-type algorithms. The algorithms only require a smoothness assumption of the nonconvex objective term. In the finite-sum case, we show that our algorithm achieves optimal convergence rate by matching the lower-bound worst-case complexity, while in the expectation case, it attains the best-known convergence rate under only standard smoothness and bounded variance assumptions. One key step of our algorithms is a new constant step-size that helps to achieve desired convergence rate. Our step-size is much larger than existing methods including proximal SVRG schemes in the single sample case. We generalize our algorithm to mini-batches for both inner and outer loops, and adaptive step-sizes. We also specify the algorithm to the non-composite case that covers and dominates existing state-of-the-arts in terms of convergence rate. We test the proposed algorithms on two composite nonconvex optimization problems and feedforward neural networks using several well-known datasets.

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., NIPS'17] 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., NIPS'16]. Also, ProxSVRG+ uses much less proximal oracle calls than ProxSVRG [Reddi et al., NIPS'16]. 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.

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., NIPS'17] 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., NIPS'16]. Also, ProxSVRG+ uses much less proximal oracle calls than ProxSVRG [Reddi et al., NIPS'16]. 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.

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-{\L}ojasiewicz 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.