Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball

Othmane Sebbouh, Robert M Gower, Aaron Defazio
Proceedings of Thirty Fourth Conference on Learning Theory, PMLR 134:3935-3971, 2021.

Abstract

We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem. For SGD, in the convex and smooth setting, we provide the first \emph{almost sure} asymptotic convergence \emph{rates} for a weighted average of the iterates . More precisely, we show that the convergence rate of the function values is arbitrarily close to $o(1/\sqrt{k})$, and is exactly $o(1/k)$ in the so-called overparametrized case. We show that these results still hold when using a decreasing step size version of stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime. Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer \emph{almost surely}. Additionally, we prove that the function values of the deterministic HB converge at a $o(1/k)$ rate, which is faster than the previously known $O(1/k)$. Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.

Cite this Paper

BibTeX
@InProceedings{pmlr-v134-sebbouh21a, title = {Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball}, author = {Sebbouh, Othmane and Gower, Robert M and Defazio, Aaron}, booktitle = {Proceedings of Thirty Fourth Conference on Learning Theory}, pages = {3935--3971}, year = {2021}, editor = {Belkin, Mikhail and Kpotufe, Samory}, volume = {134}, series = {Proceedings of Machine Learning Research}, month = {15--19 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v134/sebbouh21a/sebbouh21a.pdf}, url = {https://proceedings.mlr.press/v134/sebbouh21a.html}, abstract = {We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem. For SGD, in the convex and smooth setting, we provide the first \emph{almost sure} asymptotic convergence \emph{rates} for a weighted average of the iterates . More precisely, we show that the convergence rate of the function values is arbitrarily close to $o(1/\sqrt{k})$, and is exactly $o(1/k)$ in the so-called overparametrized case. We show that these results still hold when using a decreasing step size version of stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime. Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer \emph{almost surely}. Additionally, we prove that the function values of the deterministic HB converge at a $o(1/k)$ rate, which is faster than the previously known $O(1/k)$. Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.} }
Endnote
%0 Conference Paper %T Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball %A Othmane Sebbouh %A Robert M Gower %A Aaron Defazio %B Proceedings of Thirty Fourth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2021 %E Mikhail Belkin %E Samory Kpotufe %F pmlr-v134-sebbouh21a %I PMLR %P 3935--3971 %U https://proceedings.mlr.press/v134/sebbouh21a.html %V 134 %X We study stochastic gradient descent (SGD) and the stochastic heavy ball method (SHB, otherwise known as the momentum method) for the general stochastic approximation problem. For SGD, in the convex and smooth setting, we provide the first \emph{almost sure} asymptotic convergence \emph{rates} for a weighted average of the iterates . More precisely, we show that the convergence rate of the function values is arbitrarily close to $o(1/\sqrt{k})$, and is exactly $o(1/k)$ in the so-called overparametrized case. We show that these results still hold when using a decreasing step size version of stochastic line search and stochastic Polyak stepsizes, thereby giving the first proof of convergence of these methods in the non-overparametrized regime. Using a substantially different analysis, we show that these rates hold for SHB as well, but at the last iterate. This distinction is important because it is the last iterate of SGD and SHB which is used in practice. We also show that the last iterate of SHB converges to a minimizer \emph{almost surely}. Additionally, we prove that the function values of the deterministic HB converge at a $o(1/k)$ rate, which is faster than the previously known $O(1/k)$. Finally, in the nonconvex setting, we prove similar rates on the lowest gradient norm along the trajectory of SGD.
APA
Sebbouh, O., Gower, R.M. & Defazio, A.. (2021). Almost sure convergence rates for Stochastic Gradient Descent and Stochastic Heavy Ball. Proceedings of Thirty Fourth Conference on Learning Theory, in Proceedings of Machine Learning Research 134:3935-3971 Available from https://proceedings.mlr.press/v134/sebbouh21a.html.

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