Adversarial Examples in Random Neural Networks with General Activations
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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( · ;θ) with random weights θ, and feature vector x, we show that an adversarial example x' can be found with high probability along the direction of the gradient ∇_ xf( x;θ). Our proof is based on a Gaussian conditioning technique. Instead of proving that f is approximately linear in a neighborhood of x, we characterize the joint distribution of f( x;θ) and f( x';θ) for x' = x-s( x)∇_ xf( x;θ).
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