Towards Training Without Depth Limits: Batch Normalization Without Gradient Explosion

TL;DR

This paper demonstrates that it is possible to design batch-normalized neural networks that maintain optimal signal propagation without suffering from exploding gradients, enabling training of arbitrarily deep models.

Contribution

The authors construct a specific MLP with batch normalization and linear activations that provably avoids gradient explosion at any depth, supported by a rigorous theoretical analysis.

Findings

Bounded gradients at arbitrary depth for the constructed MLP

Theoretical characterization of forward signal propagation

Empirical activation shaping scheme for non-linear activations

Abstract

Normalization layers are one of the key building blocks for deep neural networks. Several theoretical studies have shown that batch normalization improves the signal propagation, by avoiding the representations from becoming collinear across the layers. However, results on mean-field theory of batch normalization also conclude that this benefit comes at the expense of exploding gradients in depth. Motivated by these two aspects of batch normalization, in this study we pose the following question: "Can a batch-normalized network keep the optimal signal propagation properties, but avoid exploding gradients?" We answer this question in the affirmative by giving a particular construction of an Multi-Layer Perceptron (MLP) with linear activations and batch-normalization that provably has bounded gradients at any depth. Based on Weingarten calculus, we develop a rigorous and non-asymptotic theory for this constructed MLP that gives a precise characterization of forward signal propagation, while proving that gradients remain bounded for linearly independent input samples, which holds in most practical settings. Inspired by our theory, we also design an activation shaping scheme that empirically achieves the same properties for certain non-linear activations.