Empirical Bounds on Linear Regions of Deep Rectifier Networks

One form of comparing the expressiveness of rectifier networks is by the number of linear regions, or pieces, of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive in practice and the known analytical bounds on their numbers are identical for networks having the same dimensions. In this work, we approximate the number of linear regions of rectifier networks through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is an algorithm for probabilistic lower bounds of mixed-integer linear sets, which is several orders of magnitude faster than exact counting and obtain values reaching similar orders of magnitude. Our second contribution is a tighter activation-based bound for the maximum number of linear regions, which is particularly stronger in networks with narrow layers. Combined, these bounds yield a reasonable proxy for the number of linear regions and the accuracy of the networks.