Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes
Abstract
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized volume of faces of such polytopes.
Cite
Text
Haase et al. "Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes." International Conference on Learning Representations, 2023.Markdown
[Haase et al. "Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes." International Conference on Learning Representations, 2023.](https://mlanthology.org/iclr/2023/haase2023iclr-lower/)BibTeX
@inproceedings{haase2023iclr-lower,
title = {{Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes}},
author = {Haase, Christian Alexander and Hertrich, Christoph and Loho, Georg},
booktitle = {International Conference on Learning Representations},
year = {2023},
url = {https://mlanthology.org/iclr/2023/haase2023iclr-lower/}
}