Empirical Bounds on Linear Regions of Deep Rectifier Networks

TL;DR

This paper introduces empirical methods to estimate the number of linear regions in deep ReLU networks, providing faster bounds that improve understanding of network expressiveness.

Contribution

It presents a novel sampling method using hash functions and MILP for empirical bounds, along with a tighter activation-based bound for linear regions.

Findings

The MIPBound algorithm provides faster lower bounds on linear regions.

The tighter activation-based bound is especially effective for narrow layers.

Combined bounds serve as a quick proxy for network expressiveness.

Abstract

We can compare the expressiveness of neural networks that use rectified linear units (ReLUs) by the number of linear regions, which reflect the number of pieces of the piecewise linear functions modeled by such networks. However, enumerating these regions is prohibitive and the known analytical bounds are identical for networks with same dimensions. In this work, we approximate the number of linear regions through empirical bounds based on features of the trained network and probabilistic inference. Our first contribution is a method to sample the activation patterns defined by ReLUs using universal hash functions. This method is based on a Mixed-Integer Linear Programming (MILP) formulation of the network and an algorithm for probabilistic lower bounds of MILP solution sets that we call MIPBound, which is considerably faster than exact counting and reaches values in 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 fast proxy for the number of linear regions of a deep neural network.