Exact Upper and Lower Bounds for the Output Distribution of Neural Networks with Random Inputs

We derive exact upper and lower bounds for the cumulative distribution function (cdf) of the output of a neural network over its entire support subject to noisy (stochastic) inputs. The upper and lower bounds converge to the true cdf over its domain as the resolution increases. Our method applies to any feedforward NN using continuous monotonic piecewise differentiable activation functions (e.g., ReLU, tanh and softmax) and convolutional NNs, which were beyond the scope of competing approaches. The novelty and an instrumental tool of our approach is to bound general NNs with ReLU NNs. The ReLU NN based bounds are then used to derive upper and lower bounds of the cdf of the NN output. Experiments demonstrate that our method delivers guaranteed bounds of the predictive output distribution over its support, thus providing exact error guarantees, in contrast to competing approaches.