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