Convergence of Stochastic Gradient Descent for PCA

Ohad Shamir
Proceedings of The 33rd International Conference on Machine Learning, PMLR 48:257-265, 2016.

Abstract

We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in R^d. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in (Hardt & Price, 2014). Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap.

Cite this Paper

BibTeX
@InProceedings{pmlr-v48-shamirb16, title = {Convergence of Stochastic Gradient Descent for PCA}, author = {Shamir, Ohad}, booktitle = {Proceedings of The 33rd International Conference on Machine Learning}, pages = {257--265}, year = {2016}, editor = {Balcan, Maria Florina and Weinberger, Kilian Q.}, volume = {48}, series = {Proceedings of Machine Learning Research}, address = {New York, New York, USA}, month = {20--22 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v48/shamirb16.pdf}, url = {https://proceedings.mlr.press/v48/shamirb16.html}, abstract = {We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in R^d. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in (Hardt & Price, 2014). Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap.} }
Endnote
%0 Conference Paper %T Convergence of Stochastic Gradient Descent for PCA %A Ohad Shamir %B Proceedings of The 33rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2016 %E Maria Florina Balcan %E Kilian Q. Weinberger %F pmlr-v48-shamirb16 %I PMLR %P 257--265 %U https://proceedings.mlr.press/v48/shamirb16.html %V 48 %X We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in R^d. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in (Hardt & Price, 2014). Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap.
RIS
TY - CPAPER TI - Convergence of Stochastic Gradient Descent for PCA AU - Ohad Shamir BT - Proceedings of The 33rd International Conference on Machine Learning DA - 2016/06/11 ED - Maria Florina Balcan ED - Kilian Q. Weinberger ID - pmlr-v48-shamirb16 PB - PMLR DP - Proceedings of Machine Learning Research VL - 48 SP - 257 EP - 265 L1 - http://proceedings.mlr.press/v48/shamirb16.pdf UR - https://proceedings.mlr.press/v48/shamirb16.html AB - We consider the problem of principal component analysis (PCA) in a streaming stochastic setting, where our goal is to find a direction of approximate maximal variance, based on a stream of i.i.d. data points in R^d. A simple and computationally cheap algorithm for this is stochastic gradient descent (SGD), which incrementally updates its estimate based on each new data point. However, due to the non-convex nature of the problem, analyzing its performance has been a challenge. In particular, existing guarantees rely on a non-trivial eigengap assumption on the covariance matrix, which is intuitively unnecessary. In this paper, we provide (to the best of our knowledge) the first eigengap-free convergence guarantees for SGD in the context of PCA. This also partially resolves an open problem posed in (Hardt & Price, 2014). Moreover, under an eigengap assumption, we show that the same techniques lead to new SGD convergence guarantees with better dependence on the eigengap. ER -
APA
Shamir, O.. (2016). Convergence of Stochastic Gradient Descent for PCA. Proceedings of The 33rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 48:257-265 Available from https://proceedings.mlr.press/v48/shamirb16.html.

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