Understanding Deep Neural Networks with Rectified Linear Units

TL;DR

This paper analyzes the expressive power of ReLU deep neural networks, providing algorithms for training to global optimality, establishing new lower bounds on network size for approximation, and exploring the complexity of piecewise linear functions.

Contribution

It introduces an algorithm for globally optimal training of shallow ReLU networks, improves lower bounds on network size for approximation, and constructs explicit families of functions demonstrating these bounds.

Findings

Polynomial-time algorithm for training shallow ReLU networks to global optimality

Super exponential lower bounds on size for approximating deep ReLU networks with shallow ones

Explicit constructions of functions with many affine pieces using zonotopes

Abstract

In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give an algorithm to train a ReLU DNN with one hidden layer to *global optimality* with runtime polynomial in the data size albeit exponential in the input dimension. Further, we improve on the known lower bounds on size (from exponential to super exponential) for approximating a ReLU deep net function by a shallower ReLU net. Our gap theorems hold for smoothly parametrized families of "hard" functions, contrary to countable, discrete families known in the literature. An example consequence of our gap theorems is the following: for every natural number $k$ there exists a function representable by a ReLU DNN with $k^2$ hidden layers and total size $k^3$, such that any ReLU DNN with at most $k$ hidden layers will require at least $\frac{1}{2}k^{k+1}-1$ total nodes. Finally, for the family of $\mathbb{R}^n\to \mathbb{R}$ DNNs with ReLU activations, we show a new lowerbound on the number of affine pieces, which is larger than previous constructions in certain regimes of the network architecture and most distinctively our lowerbound is demonstrated by an explicit construction of a *smoothly parameterized* family of functions attaining this scaling. Our construction utilizes the theory of zonotopes from polyhedral theory.