We present a theoretical study of the robustness of parameterized neural networks to random input perturbations. Specifically, we analyze local robustness by quantifying the probability that a random L_2-perturbation of a given input results in a correct classification. For deep ReLU networks, we derive lower bounds on local robustness by combining tools from high-dimensional geometry, in particular concentration of measure, with a new characterization of the geometric structure induced by their input-output functions. We prove that each convex polyhedral region in the partition of the input space induced by a ReLU network has at most as many faces as there are network units, regardless of the network depth or architecture. This geometric property serves as the key ingredient in our robustness analysis. Finally, we analyze how local robustness scales with input dimension and characterize the sets of inputs whose neighborhoods are most likely to contain adversarial examples. We show that the width of decision-boundary neighborhoods containing vulnerable points shrinks rapidly as dimension increases and grows only logarithmically with the number of network units. We also discuss the volume of a set of vulnerable points in terms of approximately space-filling shapes of decision boundaries.

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