  In this paper, we determine analytical bounds on the local Lipschitz
constants of of affine functions composed with rectified linear units (ReLUs).
Affine-ReLU functions represent a widely used layer in deep neural networks,
due to the fact that convolution, fully-connected, and normalization functions
are all affine, and are often followed by a ReLU activation function. Using an
analytical approach, we mathematically determine upper bounds on the local
Lipschitz constant of an affine-ReLU function, show how these bounds can be
combined to determine a bound on an entire network, and discuss how the bounds
can be efficiently computed, even for larger layers and networks. We show
several examples by applying our results to AlexNet, as well as several smaller
networks based on the MNIST and CIFAR-10 datasets. The results show that our
method produces tighter bounds than the standard conservative bound (i.e. the
product of the spectral norms of the layers' linear matrices), especially for
small perturbations.
