Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes
This provides theoretical depth lower bounds for integral ReLU networks, addressing a foundational problem in neural network expressivity, though it is incremental as it builds on existing tropical geometry methods.
The paper proves that ReLU neural networks with integer weights require at least ⌈log₂(n)⌉ hidden layers to compute the maximum of n numbers, matching known upper bounds, using a duality between networks and lattice polytopes from tropical geometry.
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.