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 $f(,cdot,;{boldsymbol theta})$
with random weights ${boldsymbol theta}$, and feature vector ${boldsymbol
x}$, we show that an adversarial example ${boldsymbol x}’$ can be found with
high probability along the direction of the gradient $nabla_{{boldsymbol
x}}f({boldsymbol x};{boldsymbol theta})$. Our proof is based on a Gaussian
conditioning technique. Instead of proving that $f$ is approximately linear in
a neighborhood of ${boldsymbol x}$, we characterize the joint distribution of
$f({boldsymbol x};{boldsymbol theta})$ and $f({boldsymbol x}’;{boldsymbol
theta})$ for ${boldsymbol x}’ = {boldsymbol x}-s({boldsymbol
x})nabla_{{boldsymbol x}}f({boldsymbol x};{boldsymbol theta})$.
