Householder Activations for Provable Robustness against Adversarial Attacks

Training convolutional neural networks (CNNs) with a strict Lipschitz constraint under the l_{2} norm is useful for provable adversarial robustness, interpretable gradients and stable training. While 1-Lipschitz CNNs can be designed by enforcing a 1-Lipschitz constraint on each layer, training such networks requires each layer to have an orthogonal Jacobian matrix (for all inputs) to prevent gradients from vanishing during backpropagation. A layer with this property is said to be Gradient Norm Preserving (GNP). To construct expressive GNP activation functions, we first prove that the Jacobian of any GNP piecewise linear function is only allowed to change via Householder transformations for the function to be continuous. Building on this result, we introduce a class of nonlinear GNP activations with learnable Householder transformations called Householder activations. A householder activation parameterized by the vector $\mathbf{v}$ outputs $(\mathbf{I} - 2\mathbf{v}\mathbf{v}^{T})\mathbf{z}$ for its input $\mathbf{z}$ if $\mathbf{v}^{T}\mathbf{z} \leq 0$; otherwise it outputs $\mathbf{z}$. Existing GNP activations such as $\mathrm{MaxMin}$ can be viewed as special cases of $\mathrm{HH}$ activations for certain settings of these transformations. Thus, networks with $\mathrm{HH}$ activations have higher expressive power than those with $\mathrm{MaxMin}$ activations. Although networks with $\mathrm{HH}$ activations have nontrivial provable robustness against adversarial attacks, we further boost their robustness by (i) introducing a certificate regularization and (ii) relaxing orthogonalization of the last layer of the network. Our experiments on CIFAR-10 and CIFAR-100 show that our regularized networks with $\mathrm{HH}$ activations lead to significant improvements in both the standard and provable robust accuracy over the prior works (gain of 3.65\% and 4.46\% on CIFAR-100 respectively).