Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent

Yunwen Lei, Yiming Ying
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:5809-5819, 2020.

Abstract

Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introduction of a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting using stability approach. Secondly, the smoothness assumption is relaxed by considering loss functions with Holder continuous (sub)gradients for which we show that optimal bounds are still achieved by balancing computation and stability. To our best knowledge, this gives the first-ever-known stability and generalization bounds for SGD with non-smooth loss functions (e.g., hinge loss). Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-lei20c, title = {Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent}, author = {Lei, Yunwen and Ying, Yiming}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {5809--5819}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/lei20c/lei20c.pdf}, url = {https://proceedings.mlr.press/v119/lei20c.html}, abstract = {Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introduction of a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting using stability approach. Secondly, the smoothness assumption is relaxed by considering loss functions with Holder continuous (sub)gradients for which we show that optimal bounds are still achieved by balancing computation and stability. To our best knowledge, this gives the first-ever-known stability and generalization bounds for SGD with non-smooth loss functions (e.g., hinge loss). Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.} }
Endnote
%0 Conference Paper %T Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent %A Yunwen Lei %A Yiming Ying %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-lei20c %I PMLR %P 5809--5819 %U https://proceedings.mlr.press/v119/lei20c.html %V 119 %X Recently there are a considerable amount of work devoted to the study of the algorithmic stability and generalization for stochastic gradient descent (SGD). However, the existing stability analysis requires to impose restrictive assumptions on the boundedness of gradients, smoothness and convexity of loss functions. In this paper, we provide a fine-grained analysis of stability and generalization for SGD by substantially relaxing these assumptions. Firstly, we establish stability and generalization for SGD by removing the existing bounded gradient assumptions. The key idea is the introduction of a new stability measure called on-average model stability, for which we develop novel bounds controlled by the risks of SGD iterates. This yields generalization bounds depending on the behavior of the best model, and leads to the first-ever-known fast bounds in the low-noise setting using stability approach. Secondly, the smoothness assumption is relaxed by considering loss functions with Holder continuous (sub)gradients for which we show that optimal bounds are still achieved by balancing computation and stability. To our best knowledge, this gives the first-ever-known stability and generalization bounds for SGD with non-smooth loss functions (e.g., hinge loss). Finally, we study learning problems with (strongly) convex objectives but non-convex loss functions.
APA
Lei, Y. & Ying, Y.. (2020). Fine-Grained Analysis of Stability and Generalization for Stochastic Gradient Descent. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:5809-5819 Available from https://proceedings.mlr.press/v119/lei20c.html.

Related Material