Empirical Bounds on Linear Regions of Deep Rectifier Networks
Authors: Thiago Serra, Srikumar Ramalingam
Keywords: linear regions, approximate model counting, mixed-integer linear programming
TL;DR: We provide improved upper bounds for the number of linear regions used in network expressivity, and an highly efficient algorithm (w.r.t. exact counting) to obtain probabilistic lower bounds on the actual number of linear regions.
Abstract: One form of characterizing the expressiveness of a piecewise linear neural network is by the number of linear regions, or pieces, of the function modeled. We have observed substantial progress in this topic through lower and upper bounds on the maximum number of linear regions and a counting procedure. However, these bounds only account for the dimensions of the network and the exact counting may take a prohibitive amount of time, therefore making it infeasible to benchmark the expressiveness of networks. In this work, we approximate the number of linear regions of specific rectifier networks with an algorithm for probabilistic lower bounds of mixed-integer linear sets. In addition, we present a tighter upper bound that leverages network coefficients. We test both on trained networks. The algorithm for probabilistic lower bounds is several orders of magnitude faster than exact counting and the values reach similar orders of magnitude, hence making our approach a viable method to compare the expressiveness of such networks. The refined upper bound is particularly stronger on networks with narrow layers.