Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks

TL;DR

This paper introduces BCOP, a novel parameterization for orthogonal convolutional layers that enables scalable, provably Lipschitz-constrained neural networks, improving adversarial robustness and Wasserstein distance estimation.

Contribution

The paper extends gradient norm preservation techniques to convolutional networks by proposing BCOP, a new orthogonal convolution parameterization that is scalable and expressive.

Findings

BCOP allows training large Lipschitz-constrained convolutional networks.

Empirical results show BCOP is competitive with existing methods for robustness.

BCOP effectively represents a wide range of orthogonal convolutions despite the disconnected space.

Abstract

Lipschitz constraints under L2 norm on deep neural networks are useful for provable adversarial robustness bounds, stable training, and Wasserstein distance estimation. While heuristic approaches such as the gradient penalty have seen much practical success, it is challenging to achieve similar practical performance while provably enforcing a Lipschitz constraint. In principle, one can design Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al. recently identified a key obstacle to this approach: gradient norm attenuation. They showed how to circumvent this problem in the case of fully connected networks by designing each layer to be gradient norm preserving. We extend their approach to train scalable, expressive, provably Lipschitz convolutional networks. In particular, we present the Block Convolution Orthogonal Parameterization (BCOP), an expressive parameterization of orthogonal convolution operations. We show that even though the space of orthogonal convolutions is disconnected, the largest connected component of BCOP with 2n channels can represent arbitrary BCOP convolutions over n channels. Our BCOP parameterization allows us to train large convolutional networks with provable Lipschitz bounds. Empirically, we find that it is competitive with existing approaches to provable adversarial robustness and Wasserstein distance estimation.