Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes

Ohad Shamir, Tong Zhang
; Proceedings of the 30th International Conference on Machine Learning, PMLR 28(1):71-79, 2013.

Abstract

Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD \emphwithout such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the \emphlast SGD iterate scales as O(\log(T)/\sqrtT) for non-smooth convex objective functions, and O(\log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in \citetRakhShaSri12arxiv is not as simple to implement). Finally, we provide some experimental illustrations.

Cite this Paper


BibTeX
@InProceedings{pmlr-v28-shamir13, title = {Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes}, author = {Ohad Shamir and Tong Zhang}, booktitle = {Proceedings of the 30th International Conference on Machine Learning}, pages = {71--79}, year = {2013}, editor = {Sanjoy Dasgupta and David McAllester}, volume = {28}, number = {1}, series = {Proceedings of Machine Learning Research}, address = {Atlanta, Georgia, USA}, month = {17--19 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v28/shamir13.pdf}, url = {http://proceedings.mlr.press/v28/shamir13.html}, abstract = {Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD \emphwithout such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the \emphlast SGD iterate scales as O(\log(T)/\sqrtT) for non-smooth convex objective functions, and O(\log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in \citetRakhShaSri12arxiv is not as simple to implement). Finally, we provide some experimental illustrations.} }
Endnote
%0 Conference Paper %T Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes %A Ohad Shamir %A Tong Zhang %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-shamir13 %I PMLR %J Proceedings of Machine Learning Research %P 71--79 %U http://proceedings.mlr.press %V 28 %N 1 %W PMLR %X Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD \emphwithout such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the \emphlast SGD iterate scales as O(\log(T)/\sqrtT) for non-smooth convex objective functions, and O(\log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in \citetRakhShaSri12arxiv is not as simple to implement). Finally, we provide some experimental illustrations.
RIS
TY - CPAPER TI - Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes AU - Ohad Shamir AU - Tong Zhang BT - Proceedings of the 30th International Conference on Machine Learning PY - 2013/02/13 DA - 2013/02/13 ED - Sanjoy Dasgupta ED - David McAllester ID - pmlr-v28-shamir13 PB - PMLR SP - 71 DP - PMLR EP - 79 L1 - http://proceedings.mlr.press/v28/shamir13.pdf UR - http://proceedings.mlr.press/v28/shamir13.html AB - Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD \emphwithout such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the \emphlast SGD iterate scales as O(\log(T)/\sqrtT) for non-smooth convex objective functions, and O(\log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in \citetRakhShaSri12arxiv is not as simple to implement). Finally, we provide some experimental illustrations. ER -
APA
Shamir, O. & Zhang, T.. (2013). Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes. Proceedings of the 30th International Conference on Machine Learning, in PMLR 28(1):71-79

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