Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networks
This work addresses a key obstacle in designing provably robust and stable deep learning models, particularly for applications like adversarial defense, but it appears incremental as it extends prior fully connected network methods to convolutional architectures.
The authors tackled the problem of gradient attenuation in Lipschitz constrained convolutional networks by developing the Block Convolution Orthogonal Parameterization (BCOP), which enables training of scalable and expressive networks with provable Lipschitz bounds. They showed that BCOP is competitive with existing methods for provable adversarial robustness and Wasserstein distance estimation, though no specific numerical results are provided in the 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.