Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909220079665152 |
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| author | Wu, Jingfeng Bartlett, Peter L. Telgarsky, Matus Yu, Bin |
| author_facet | Wu, Jingfeng Bartlett, Peter L. Telgarsky, Matus Yu, Bin |
| contents | We consider gradient descent (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $η$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly -- in $\mathcal{O}(η)$ steps -- and subsequently achieves an $\tilde{\mathcal{O}}(1 / (ηt) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an accelerated loss of $\tilde{\mathcal{O}}(1/T^2)$ with an aggressive stepsize $η:= Θ( T)$, 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 $\tilde{\mathcal{O}}(1/T^2)$ acceleration), nonlinear predictors in the neural tangent kernel regime, and online stochastic gradient descent (SGD) with a large stepsize, under suitable separability conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15926 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency Wu, Jingfeng Bartlett, Peter L. Telgarsky, Matus Yu, Bin Machine Learning We consider gradient descent (GD) with a constant stepsize applied to logistic regression with linearly separable data, where the constant stepsize $η$ is so large that the loss initially oscillates. We show that GD exits this initial oscillatory phase rapidly -- in $\mathcal{O}(η)$ steps -- and subsequently achieves an $\tilde{\mathcal{O}}(1 / (ηt) )$ convergence rate after $t$ additional steps. Our results imply that, given a budget of $T$ steps, GD can achieve an accelerated loss of $\tilde{\mathcal{O}}(1/T^2)$ with an aggressive stepsize $η:= Θ( T)$, 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 $\tilde{\mathcal{O}}(1/T^2)$ acceleration), nonlinear predictors in the neural tangent kernel regime, and online stochastic gradient descent (SGD) with a large stepsize, under suitable separability conditions. |
| title | Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2402.15926 |