Output Distribution over the Entire Input Space: A Novel Perspective to Understand Neural Networks

Understanding the input-output mapping relationship in the \emph{entire input space} contributes a novel perspective to a comprehensive understanding of deep neural networks. In this paper, we focus on binary neural classifiers and propose to first uncover the histogram about the number of inputs that are mapped to certain output values and then scrutinize the representative inputs from a certain output range of interest, such as the positive-logit region that corresponds to one of the classes. A straightforward solution is uniform sampling (or exhaustive enumeration) in the entire input space but when the inputs are high dimensional, it can take almost forever to converge. We connect the output histogram to the \emph{density of states} in physics by making an analogy between the energy of a system and the neural network output. Inspired by the Wang-Landau algorithm designed for sampling the density of states, we propose an efficient sampler that is driven to explore the under-explored output values through a gradient-based proposal. Compared with the random proposal in Wang-Landau algorithm, our gradient-based proposal converges faster as it can propose the inputs corresponding to the under-explored output values. Extensive experiments have verified the accuracy of the histogram generated by our sampler and also demonstrated interesting findings. For example, the models map many human unrecognizable images to very negative logit values. These properties of a neural model are revealed for the first time through our sampled statistics. We believe that our approach opens a new gate for neural model evaluation and shall be further explored in future works.