Limiting fluctuation and trajectorial stability of multilayer neural networks with mean field training

TL;DR

This paper develops a theoretical framework to analyze the fluctuations and stability of multilayer neural networks during training, extending mean field theory to capture complex inter-layer stochastic dependencies and demonstrating that training biases solutions towards minimal fluctuation in large-width regimes.

Contribution

It introduces the second-order mean field limit for multilayer networks, capturing the distribution of fluctuations and their interactions across layers, a significant extension beyond shallow network analysis.

Findings

Fluctuations in multilayer networks can be systematically characterized by the second-order MF limit.

Gradient descent training biases solutions towards minimal fluctuation in the large-width limit.

The framework applies to general loss functions beyond convex cases.

Abstract

The mean field (MF) theory of multilayer neural networks centers around a particular infinite-width scaling, where the learning dynamics is closely tracked by the MF limit. A random fluctuation around this infinite-width limit is expected from a large-width expansion to the next order. This fluctuation has been studied only in shallow networks, where previous works employ heavily technical notions or additional formulation ideas amenable only to that case. Treatment of the multilayer case has been missing, with the chief difficulty in finding a formulation that captures the stochastic dependency across not only time but also depth. In this work, we initiate the study of the fluctuation in the case of multilayer networks, at any network depth. Leveraging on the neuronal embedding framework recently introduced by Nguyen and Pham, we systematically derive a system of dynamical equations, called the second-order MF limit, that captures the limiting fluctuation distribution. We demonstrate through the framework the complex interaction among neurons in this second-order MF limit, the stochasticity with cross-layer dependency and the nonlinear time evolution inherent in the limiting fluctuation. A limit theorem is proven to relate quantitatively this limit to the fluctuation of large-width networks. We apply the result to show a stability property of gradient descent MF training: in the large-width regime, along the training trajectory, it progressively biases towards a solution with "minimal fluctuation" (in fact, vanishing fluctuation) in the learned output function, even after the network has been initialized at or has converged (sufficiently fast) to a global optimum. This extends a similar phenomenon previously shown only for shallow networks with a squared loss in the ERM setting, to multilayer networks with a loss function that is not necessarily convex in a more general setting.