Skew Orthogonal Convolutions
This addresses the need for efficient and accurate provably robust neural networks, with incremental improvements over prior GNP methods.
The authors tackled the problem of training gradient norm preserving (GNP) convolutional layers for Lipschitz-constrained neural networks, which suffer from slow training and reduced accuracy. They proposed Skew Orthogonal Convolutions (SOC), achieving faster training and significant improvements in standard and certified robust accuracies on CIFAR-10 and CIFAR-100.
Training convolutional neural networks with a Lipschitz constraint under the $l_{2}$ norm is useful for provable adversarial robustness, interpretable gradients, stable training, etc. While 1-Lipschitz networks can be designed by imposing a 1-Lipschitz constraint on each layer, training such networks requires each layer to be gradient norm preserving (GNP) to prevent gradients from vanishing. However, existing GNP convolutions suffer from slow training, lead to significant reduction in accuracy and provide no guarantees on their approximations. In this work, we propose a GNP convolution layer called Skew Orthogonal Convolution (SOC) that uses the following mathematical property: when a matrix is {\it Skew-Symmetric}, its exponential function is an {\it orthogonal} matrix. To use this property, we first construct a convolution filter whose Jacobian is Skew-Symmetric. Then, we use the Taylor series expansion of the Jacobian exponential to construct the SOC layer that is orthogonal. To efficiently implement SOC, we keep a finite number of terms from the Taylor series and provide a provable guarantee on the approximation error. Our experiments on CIFAR-10 and CIFAR-100 show that SOC allows us to train provably Lipschitz, large convolutional neural networks significantly faster than prior works while achieving significant improvements for both standard and certified robust accuracies.