Towards Lower Bounds on the Depth of ReLU Neural Networks

TL;DR

This paper explores the limitations of ReLU neural networks' depth in representing functions, providing mathematical bounds and settling a conjecture about piecewise linear functions, thereby deepening understanding of neural network expressiveness.

Contribution

It introduces new lower bounds on the depth needed for ReLU networks to represent certain functions and confirms a longstanding conjecture about piecewise linear functions.

Findings

Established lower bounds on neural network depth for function representation.

Confirmed Wang and Sun's conjecture on piecewise linear functions.

Provided upper bounds on network size for logarithmic depth functions.

Abstract

We contribute to a better understanding of the class of functions that can be represented by a neural network with ReLU activations and a given architecture. Using techniques from mixed-integer optimization, polyhedral theory, and tropical geometry, we provide a mathematical counterbalance to the universal approximation theorems which suggest that a single hidden layer is sufficient for learning any function. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). As a by-product of our investigations, we settle an old conjecture about piecewise linear functions by Wang and Sun (2005) in the affirmative. We also present upper bounds on the sizes of neural networks required to represent functions with logarithmic depth.