The Expressivity and Training of Deep Neural Networks: toward the Edge of Chaos?

TL;DR

This paper analyzes the expressivity of deep neural networks using a dynamic model and Hilbert space, revealing their evolution toward the edge of chaos with depth, and proposes a new activation function for improved spatial representation.

Contribution

It introduces a quantitative framework for neural network expressivity, analyzes the impact of activation functions and input perturbations, and proposes a Hermite polynomial-based activation for better information transfer.

Findings

DNNs tend to evolve toward the edge of chaos as depth increases.

The proposed Hermite polynomial-based activation improves spatial representation.

Empirical results confirm the theoretical analysis on time series prediction and image classification.

Abstract

Expressivity is one of the most significant issues in assessing neural networks. In this paper, we provide a quantitative analysis of the expressivity for the deep neural network (DNN) from its dynamic model, where the Hilbert space is employed to analyze the convergence and criticality. We study the feature mapping of several widely used activation functions obtained by Hermite polynomials, and find sharp declines or even saddle points in the feature space, which stagnate the information transfer in DNNs. We then present a new activation function design based on the Hermite polynomials for better utilization of spatial representation. Moreover, we analyze the information transfer of DNNs, emphasizing the convergence problem caused by the mismatch between input and topological structure. We also study the effects of input perturbations and regularization operators on critical expressivity. Our theoretical analysis reveals that DNNs use spatial domains for information representation and evolve to the edge of chaos as depth increases. In actual training, whether a particular network can ultimately arrive the edge of chaos depends on its ability to overcome convergence and pass information to the required network depth. Finally, we demonstrate the empirical performance of the proposed hypothesis via multivariate time series prediction and image classification examples.