Towards Lower Bounds on the Depth of ReLU Neural Networks

Christoph Hertrich, Amitabh Basu, Marco Di Summa, Martin Skutella

Keywords: ReLU Neural Networks, Expressivity, Depth Bounds, Polyhedral Methods, Mixed-Integer Optimization

TL;DR: Using polyhedral methods, we provide a better understanding of the class of functions represented by ReLU neural networks.

Abstract: We contribute to a better understanding of the class of functions that is 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 tasks. In particular, we investigate whether the class of exactly representable functions strictly increases by adding more layers (with no restrictions on size). This problem has potential impact on algorithmic and statistical aspects because of the insight it provides into the class of functions represented by neural hypothesis classes. However, to the best of our knowledge, this question has not been investigated in the neural network literature. We also present upper bounds on the sizes of neural networks required to represent functions in these neural hypothesis classes.

Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.

Code: https://github.com/ChristophHertrich/relu-mip-depth-bound

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2105.14835/code)