Towards a General Theory of Infinite-Width Limits of Neural Classifiers

TL;DR

This paper introduces a unified framework linking mean-field and NTK theories for infinite-width neural networks, revealing new limits and limitations of existing models in approximating finite neural networks.

Contribution

It proposes a general framework that connects mean-field and NTK theories, introduces a discrete-time MF limit, and explores new infinite-width limits for neural networks.

Findings

A discrete-time MF limit is introduced and shown to better approximate finite networks.

For networks with more than two layers, RMSProp has a non-trivial limit, GD does not.

Both MF and NTK limits have significant limitations in modeling finite neural networks.

Abstract

Obtaining theoretical guarantees for neural networks training appears to be a hard problem in a general case. Recent research has been focused on studying this problem in the limit of infinite width and two different theories have been developed: a mean-field (MF) and a constant kernel (NTK) limit theories. We propose a general framework that provides a link between these seemingly distinct theories. Our framework out of the box gives rise to a discrete-time MF limit which was not previously explored in the literature. We prove a convergence theorem for it and show that it provides a more reasonable approximation for finite-width nets compared to the NTK limit if learning rates are not very small. Also, our framework suggests a limit model that coincides neither with the MF limit nor with the NTK one. We show that for networks with more than two hidden layers RMSProp training has a non-trivial discrete-time MF limit but GD training does not have one. Overall, our framework demonstrates that both MF and NTK limits have considerable limitations in approximating finite-sized neural nets, indicating the need for designing more accurate infinite-width approximations for them.