Adversarial examples in random neural networks with general activations

Abstract

A substantial body of empirical work documents the lack of robustness in deep learning models to adversarial examples. Recent theoretical work proved that adversarial examples are ubiquitous in two-layers networks with sub-exponential width and ReLU or smooth activations, and multi-layer ReLU networks with sub-exponential width. We present a result of the same type, with no restriction on width and for general locally Lipschitz continuous activations.

More precisely, given a neural network $f(\cdot ;\theta )$ with random weights $\theta$, and feature vector $\mathbf{x}$, we show that an adversarial example ${\mathbf{x}}^{\prime }$ can be found with high probability along the direction of the gradient ${\nabla }_{\mathbf{x}}f(\mathbf{x};\theta )$. Our proof is based on a Gaussian conditioning technique. Instead of proving that $f$ is approximately linear in a neighborhood of $\mathbf{x}$, we characterize the joint distribution of $f(\mathbf{x};\theta )$ and $f({\mathbf{x}}^{\prime };\theta )$ for ${\mathbf{x}}^{\prime }=\mathbf{x}-s(\mathbf{x}){\nabla }_{\mathbf{x}}f(\mathbf{x};\theta )$, where $s(\mathbf{x})=\mathrm{sign}(f(\mathbf{x};\theta ))\cdot s_d$ for some positive step size $s_d$.