Implicit regularization in Heavy-ball momentum accelerated stochastic gradient descentDownload PDF

Published: 01 Feb 2023, Last Modified: 16 Feb 2023ICLR 2023 notable top 25%Readers: Everyone
Abstract: It is well known that the finite step-size ($h$) in Gradient descent (GD) implicitly regularizes solutions to flatter minimas. A natural question to ask is \textit{Does the momentum parameter $\beta$ (say) play a role in implicit regularization in Heavy-ball (H.B) momentum accelerated gradient descent (GD+M)?}. To answer this question, first, we show that the trajectory traced by discrete H.B momentum update (GD+M) is $O(h^2)$ close to a continuous trajectory induced by a modified loss, which consists of an original loss and an implicit regularizer. This implicit regularizer for (GD+M) is indeed stronger than that of (GD) by factor of $(\frac{1+\beta}{1-\beta})$, thus explaining why (GD+M) shows better generalization performance and higher test accuracy than (GD). Furthermore, we extend our analysis to stochastic version of gradient descent with momentum (SGD+M) and propose a deterministic continuous trajectory that is $O(h^2)$ close to the discrete update of (SGD+M) in a strong approximation sense. We explore the implicit regularization in (SGD+M) and (GD+M) through a series of experiments validating our theory.
Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.
No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.
Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics
Submission Guidelines: Yes
Please Choose The Closest Area That Your Submission Falls Into: Deep Learning and representational learning
Supplementary Material: zip
20 Replies