Publication Type
Journal Article
Version
acceptedVersion
Publication Date
8-2022
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). Moreover, we show that both ProxSVRG+ and SSRGD enjoy automatic adaptation with local structure of the objective function such as the Polyak-\L{}ojasiewicz (PL) condition for nonconvex functions in the finite-sum case, i.e., we prove that both of them can automatically switch to faster global linear convergence without any restart performed in prior work ProxSVRG (Reddi et al., 2016b). Finally, we focus on the more challenging problem of finding an $(\epsilon, \delta)$-local minimum instead of just finding an $\epsilon$-approximate (first-order) stationary point (which may be some bad unstable saddle points). We show that SSRGD can find an $(\epsilon, \delta)$-local minimum by simply adding some random perturbations. Our algorithm is almost as simple as its counterpart for finding stationary points, and achieves similar optimal rates.
Keywords
nonconvex optimization, optimal algorithm, proximal gradient descent, variance reduction, local minimum
Discipline
Databases and Information Systems
Research Areas
Data Science and Engineering; Intelligent Systems and Optimization
Publication
Journal of Machine Learning Research
Volume
23
Issue
1
First Page
10891
Last Page
10951
ISSN
1532-4435
Identifier
10.5555/3586589.3586828
Publisher
Journal of Machine Learning Research
Citation
LI, Zhize and LI, Jian.
Simple and optimal stochastic gradient methods for nonsmooth nonconvex optimization. (2022). Journal of Machine Learning Research. 23, (1), 10891-10951.
Available at: https://ink.library.smu.edu.sg/sis_research/8692
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Additional URL
https://doi.org/10.5555/3586589.3586828