Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes

Description

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 ⌈log₂(n)⌉ 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.

Period	May 2023
Event title	11th International Conference on Learning Representations, ICLR 2023
Event type	Conference
Conference number	11
Sponsors	Baidu, DeepMind, et al., Google Research, Huawei, Meta AI
Location	Kigali, RwandaShow on map
Degree of Recognition	International