1 research outputs found
The Expressivity and Training of Deep Neural Networks: toward the Edge of Chaos?
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