O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions

Lijun Zhang, Tianbao Yang, Rong Jin, Xiaofei He
Proceedings of the 30th International Conference on Machine Learning, PMLR 28(3):1121-1129, 2013.

Abstract

Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as the positive semidefinite cone, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batches, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal O(1/T) rate of convergence with only O(logT) projections. Our empirical study verifies the theoretical result.

Cite this Paper

BibTeX
@InProceedings{pmlr-v28-zhang13e, title = {O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions}, author = {Zhang, Lijun and Yang, Tianbao and Jin, Rong and He, Xiaofei}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {1121--1129}, year = {2013}, editor = {Dasgupta, Sanjoy and McAllester, David}, volume = {28}, number = {3}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/zhang13e.pdf}, url = {https://proceedings.mlr.press/v28/zhang13e.html}, abstract = {Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as the positive semidefinite cone, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batches, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal O(1/T) rate of convergence with only O(logT) projections. Our empirical study verifies the theoretical result.} }
Endnote
%0 Conference Paper %T O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions %A Lijun Zhang %A Tianbao Yang %A Rong Jin %A Xiaofei He %B Proceedings of the 30th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Sanjoy Dasgupta %E David McAllester %F pmlr-v28-zhang13e %I PMLR %P 1121--1129 %U https://proceedings.mlr.press/v28/zhang13e.html %V 28 %N 3 %X Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as the positive semidefinite cone, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batches, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal O(1/T) rate of convergence with only O(logT) projections. Our empirical study verifies the theoretical result.
RIS
TY - CPAPER TI - O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions AU - Lijun Zhang AU - Tianbao Yang AU - Rong Jin AU - Xiaofei He BT - Proceedings of the 30th International Conference on Machine Learning DA - 2013/05/26 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-zhang13e PB - PMLR DP - Proceedings of Machine Learning Research VL - 28 IS - 3 SP - 1121 EP - 1129 L1 - http://proceedings.mlr.press/v28/zhang13e.pdf UR - https://proceedings.mlr.press/v28/zhang13e.html AB - Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as the positive semidefinite cone, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batches, extra-gradient, and epoch gradient descent, in order to effectively explore the smoothness and strong convexity. We show, both in expectation and with a high probability, that when the objective function is both smooth and strongly convex, the proposed algorithm achieves the optimal O(1/T) rate of convergence with only O(logT) projections. Our empirical study verifies the theoretical result. ER -
APA
Zhang, L., Yang, T., Jin, R. & He, X.. (2013). O(logT) Projections for Stochastic Optimization of Smooth and Strongly Convex Functions. Proceedings of the 30th International Conference on Machine Learning, in Proceedings of Machine Learning Research 28(3):1121-1129 Available from https://proceedings.mlr.press/v28/zhang13e.html.

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