On the Lipschitz Constant of Deep Networks and Double Descent

TL;DR

This paper investigates how the Lipschitz constant of deep networks relates to double descent phenomena, revealing non-monotonic trends and factors like loss landscape curvature and parameter distance that influence generalization.

Contribution

It provides an extensive experimental analysis linking empirical Lipschitz constants with double descent, and introduces a connection between parameter-space and input-space gradients.

Findings

Lipschitz constant exhibits non-monotonic behavior correlated with test error.

Loss landscape curvature impacts optimization dynamics.

Distance from initialization bounds model complexity.

Abstract

Existing bounds on the generalization error of deep networks assume some form of smooth or bounded dependence on the input variable, falling short of investigating the mechanisms controlling such factors in practice. In this work, we present an extensive experimental study of the empirical Lipschitz constant of deep networks undergoing double descent, and highlight non-monotonic trends strongly correlating with the test error. Building a connection between parameter-space and input-space gradients for SGD around a critical point, we isolate two important factors -- namely loss landscape curvature and distance of parameters from initialization -- respectively controlling optimization dynamics around a critical point and bounding model function complexity, even beyond the training data. Our study presents novels insights on implicit regularization via overparameterization, and effective model complexity for networks trained in practice.