  The local Lipschitz constant of a neural network is a useful metric with
applications in robustness, generalization, and fairness evaluation. We provide
novel analytic results relating the local Lipschitz constant of nonsmooth
vector-valued functions to a maximization over the norm of the generalized
Jacobian. We present a sufficient condition for which backpropagation always
returns an element of the generalized Jacobian, and reframe the problem over
this broad class of functions. We show strong inapproximability results for
estimating Lipschitz constants of ReLU networks, and then formulate an
algorithm to compute these quantities exactly. We leverage this algorithm to
evaluate the tightness of competing Lipschitz estimators and the effects of
regularized training on the Lipschitz constant.
