Local Linearity of ReLU Neural Networks

Abstract

Despite their impressive practical results and broad usage, ReLU neural networks are still widely considered to be black boxes. Modern networks are complex, high-dimensional, nonlinear functions frequently applied to problems where other methods perform poorly. Building an understanding of their behavior is, therefore, both difficult and necessary. Significant progress has been made towards this, but results remain limited in comparison to empirical success. This research proposes two primary methodologies to add to current understanding: formal analysis of network behavior on simple problems and investigation of the piecewise linear behavior induced by the ReLU activation function. Network behavior on simple problems, such as approximation of Boolean functions or minimal n-dimensional classification, has reduced complexity that makes answering questions about optimal or minimal network size feasible. The behavior of networks on these restricted domains provides an interpretable method for determining the effects of network size on representational capacity and training success rate in a way that is still applicable to more complex problems of interest. The piecewise linear nature of the ReLU function also allows for simplified local evaluation of network behavior. ReLU neural networks form convex polytopes in the input space with the network behaving as a simple linear mapping from input to output within each of these polytopes. By exploiting this structure, it is possible to examine locally linear behavior in aggregate to construct metrics for network complexity and similarity. This analysis is able to avoid traditional difficulties stemming from symmetries and architectural differences by largely maintaining the interior of the network as a black box.