Implicit bias of SGD in $L_2$-regularized linear DNNs: One-way jumps from high to low rank

Published: 16 Jan 2024, Last Modified: 14 Mar 2024ICLR 2024 spotlightEveryoneRevisionsBibTeX
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.
Keywords: implicit bias, SGD, low-rank, linear networks
Submission Guidelines: I certify that this submission complies with the submission instructions as described on
TL;DR: Low-rank bias of SGD in L2 reg. linear nets: SGD jumps from high to low rank minima, with no probability of jumping back.
Abstract: The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local minimum with the smallest rank that still fits the training data. While rank-underestimating minima can be avoided since they do not fit the data, GD might get stuck at rank-overestimating minima. We show that with SGD, there is always a probability to jump from a higher rank minimum to a lower rank one, but the probability of jumping back is zero. More precisely, we define a sequence of sets $B_{1}\subset B_{2}\subset\cdots\subset B_{R}$ so that $B_{r}$ contains all minima of rank $r$ or less (and not more) that are absorbing for small enough ridge parameters $\lambda$ and learning rates $\eta$: SGD has prob. 0 of leaving $B_{r}$, and from any starting point there is a non-zero prob. for SGD to go in $B_{r}$.
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.
Primary Area: learning theory
Submission Number: 3990