On the Lipschitz Constant of Deep Networks and Double Descent
This work provides insights into implicit regularization and effective model complexity for deep networks, addressing a gap in generalization error analysis for practitioners, though it is incremental in nature.
The authors investigated the empirical Lipschitz constant of deep networks during double descent, finding non-monotonic trends that strongly correlate with test error, and connected parameter-space and input-space gradients to isolate factors like loss landscape curvature and parameter distance from initialization that control optimization and model complexity.
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.