  Several recent works have shown that state-of-the-art classifiers are
vulnerable to worst-case (i.e., adversarial) perturbations of the datapoints.
On the other hand, it has been empirically observed that these same classifiers
are relatively robust to random noise. In this paper, we propose to study a
\textit{semi-random} noise regime that generalizes both the random and
worst-case noise regimes. We propose the first quantitative analysis of the
robustness of nonlinear classifiers in this general noise regime. We establish
precise theoretical bounds on the robustness of classifiers in this general
regime, which depend on the curvature of the classifier's decision boundary.
Our bounds confirm and quantify the empirical observations that classifiers
satisfying curvature constraints are robust to random noise. Moreover, we
quantify the robustness of classifiers in terms of the subspace dimension in
the semi-random noise regime, and show that our bounds remarkably interpolate
between the worst-case and random noise regimes. We perform experiments and
show that the derived bounds provide very accurate estimates when applied to
various state-of-the-art deep neural networks and datasets. This result
suggests bounds on the curvature of the classifiers' decision boundaries that
we support experimentally, and more generally offers important insights onto
the geometry of high dimensional classification problems.
