Implicit Regularization in Deep Learning May Not Be Explainable by Norms

TL;DR

This paper challenges the idea that implicit regularization in deep learning can be explained solely by norm minimization, showing that in some cases it drives norms to infinity and suggesting rank minimization as a better explanation.

Contribution

It proves that norm-based explanations for implicit regularization in matrix factorization are insufficient, proposing rank minimization as a more accurate interpretation.

Findings

Norms can tend to infinity in certain matrix factorization problems.

Implicit regularization may be better explained by rank minimization.

Empirical evidence suggests rank minimization extends to some neural networks.

Abstract

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may apply, and a standard test-bed for studying this prospect is matrix factorization (matrix completion via linear neural networks). It is an open question whether norms can explain the implicit regularization in matrix factorization. The current paper resolves this open question in the negative, by proving that there exist natural matrix factorization problems on which the implicit regularization drives all norms (and quasi-norms) towards infinity. Our results suggest that, rather than perceiving the implicit regularization via norms, a potentially more useful interpretation is minimization of rank. We demonstrate empirically that this interpretation extends to a certain class of non-linear neural networks, and hypothesize that it may be key to explaining generalization in deep learning.