Vanishing Curvature and the Power of Adaptive Methods in Randomly Initialized Deep Networks

TL;DR

This paper analyzes the vanishing gradient problem in deep neural networks, showing it persists under certain conditions and explaining why adaptive methods and architectural features improve training.

Contribution

It provides a detailed theoretical analysis of vanishing gradients and Hessians, clarifying the role of adaptive methods and architectural components in training deep networks.

Findings

Vanishing gradients occur when network width scales less than O(depth).

Hessian eigenspectra vanish with increasing depth, creating flat plateaus.

Adaptive gradient methods help escape flat regions by adapting to curvature.

Abstract

This paper revisits the so-called vanishing gradient phenomenon, which commonly occurs in deep randomly initialized neural networks. Leveraging an in-depth analysis of neural chains, we first show that vanishing gradients cannot be circumvented when the network width scales with less than O(depth), even when initialized with the popular Xavier and He initializations. Second, we extend the analysis to second-order derivatives and show that random i.i.d. initialization also gives rise to Hessian matrices with eigenspectra that vanish as networks grow in depth. Whenever this happens, optimizers are initialized in a very flat, saddle point-like plateau, which is particularly hard to escape with stochastic gradient descent (SGD) as its escaping time is inversely related to curvature. We believe that this observation is crucial for fully understanding (a) historical difficulties of training deep nets with vanilla SGD, (b) the success of adaptive gradient methods (which naturally adapt to curvature and thus quickly escape flat plateaus) and (c) the effectiveness of modern architectural components like residual connections and normalization layers.