Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks
This work addresses the problem of understanding generalization in non-convex deep learning for researchers, providing a novel theoretical framework that is incremental but offers new insights into capacity metrics.
The paper tackles the challenge of characterizing generalization in deep learning by proving generalization bounds for SGD trajectories approximated by Feller processes, showing that generalization error is controlled by the Hausdorff dimension linked to tail behavior, with experiments indicating that heavier-tailed processes achieve better generalization and the tail-index serves as an effective capacity metric.
Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a \emph{Feller process}, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the \emph{Hausdorff dimension} of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of "capacity metric". We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.