Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency

Jingfeng Wu, Peter L. Bartlett, Matus Telgarsky, Bin Yu
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:5019-5073, 2024.

Abstract

We consider \emph{gradient descent} (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $\eta$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly — in $O(\eta)$ steps, and subsequently achieves an $\tilde{O}(1 / (\eta t) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an \emph{accelerated} loss of $\tilde{O}(1/T^2)$ with an aggressive stepsize $\eta:= \Theta( T)$, without any use of momentum or variable stepsize schedulers. Our proof technique is versatile and also handles general classification loss functions (where exponential tails are needed for the $\tilde{O}(1/T^2)$ acceleration), nonlinear predictors in the \emph{neural tangent kernel} regime, and online \emph{stochastic gradient descent} (SGD) with a large stepsize, under suitable separability conditions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-wu24b, title = {Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency}, author = {Wu, Jingfeng and Bartlett, Peter L. and Telgarsky, Matus and Yu, Bin}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {5019--5073}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/wu24b/wu24b.pdf}, url = {https://proceedings.mlr.press/v247/wu24b.html}, abstract = {We consider \emph{gradient descent} (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $\eta$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly — in $O(\eta)$ steps, and subsequently achieves an $\tilde{O}(1 / (\eta t) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an \emph{accelerated} loss of $\tilde{O}(1/T^2)$ with an aggressive stepsize $\eta:= \Theta( T)$, without any use of momentum or variable stepsize schedulers. Our proof technique is versatile and also handles general classification loss functions (where exponential tails are needed for the $\tilde{O}(1/T^2)$ acceleration), nonlinear predictors in the \emph{neural tangent kernel} regime, and online \emph{stochastic gradient descent} (SGD) with a large stepsize, under suitable separability conditions.} }
Endnote
%0 Conference Paper %T Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency %A Jingfeng Wu %A Peter L. Bartlett %A Matus Telgarsky %A Bin Yu %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-wu24b %I PMLR %P 5019--5073 %U https://proceedings.mlr.press/v247/wu24b.html %V 247 %X We consider \emph{gradient descent} (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $\eta$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly — in $O(\eta)$ steps, and subsequently achieves an $\tilde{O}(1 / (\eta t) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an \emph{accelerated} loss of $\tilde{O}(1/T^2)$ with an aggressive stepsize $\eta:= \Theta( T)$, without any use of momentum or variable stepsize schedulers. Our proof technique is versatile and also handles general classification loss functions (where exponential tails are needed for the $\tilde{O}(1/T^2)$ acceleration), nonlinear predictors in the \emph{neural tangent kernel} regime, and online \emph{stochastic gradient descent} (SGD) with a large stepsize, under suitable separability conditions.
APA
Wu, J., Bartlett, P.L., Telgarsky, M. & Yu, B.. (2024). Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:5019-5073 Available from https://proceedings.mlr.press/v247/wu24b.html.

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