Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems

Dan Garber, Atara Kaplan
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:286-294, 2019.

Abstract

Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipchitz parameter of the smoothed gradient vector and with $1/\epsilon$, where $\epsilon$ is the target accuracy. In this paper we develop efficient algorithms for \textit{stochastic} optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In particular, to the best of our knowledge, we present the first algorithm that enjoys all following critical properties for large-scale problems: i) (nearly) optimal sample complexity, ii) each iteration requires only a single \textit{low-rank} SVD computation, and iii) overall number of thin-SVD computations scales only with $\log{1/\epsilon}$ (as opposed to $\textrm{poly}(1/\epsilon)$ in previous methods). We also give an algorithm for the closely-related finite-sum setting. We empirically demonstrate our results on the problem of recovering a simultaneously low-rank and sparse matrix.

Cite this Paper

BibTeX
@InProceedings{pmlr-v89-garber19a, title = {Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems}, author = {Garber, Dan and Kaplan, Atara}, booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics}, pages = {286--294}, year = {2019}, editor = {Chaudhuri, Kamalika and Sugiyama, Masashi}, volume = {89}, series = {Proceedings of Machine Learning Research}, month = {16--18 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v89/garber19a/garber19a.pdf}, url = {https://proceedings.mlr.press/v89/garber19a.html}, abstract = {Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipchitz parameter of the smoothed gradient vector and with $1/\epsilon$, where $\epsilon$ is the target accuracy. In this paper we develop efficient algorithms for \textit{stochastic} optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In particular, to the best of our knowledge, we present the first algorithm that enjoys all following critical properties for large-scale problems: i) (nearly) optimal sample complexity, ii) each iteration requires only a single \textit{low-rank} SVD computation, and iii) overall number of thin-SVD computations scales only with $\log{1/\epsilon}$ (as opposed to $\textrm{poly}(1/\epsilon)$ in previous methods). We also give an algorithm for the closely-related finite-sum setting. We empirically demonstrate our results on the problem of recovering a simultaneously low-rank and sparse matrix.} }
Endnote
%0 Conference Paper %T Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems %A Dan Garber %A Atara Kaplan %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-garber19a %I PMLR %P 286--294 %U https://proceedings.mlr.press/v89/garber19a.html %V 89 %X Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipchitz parameter of the smoothed gradient vector and with $1/\epsilon$, where $\epsilon$ is the target accuracy. In this paper we develop efficient algorithms for \textit{stochastic} optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In particular, to the best of our knowledge, we present the first algorithm that enjoys all following critical properties for large-scale problems: i) (nearly) optimal sample complexity, ii) each iteration requires only a single \textit{low-rank} SVD computation, and iii) overall number of thin-SVD computations scales only with $\log{1/\epsilon}$ (as opposed to $\textrm{poly}(1/\epsilon)$ in previous methods). We also give an algorithm for the closely-related finite-sum setting. We empirically demonstrate our results on the problem of recovering a simultaneously low-rank and sparse matrix.
APA
Garber, D. & Kaplan, A.. (2019). Fast Stochastic Algorithms for Low-rank and Nonsmooth Matrix Problems. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:286-294 Available from https://proceedings.mlr.press/v89/garber19a.html.

Related Material