On the Depth of Monotone ReLU Neural Networks and ICNNs

We study two models of $\mathsf{ReLU}$ neural networks: monotone networks ($\mathsf{ReLU}^+$) and input convex neural networks ($\mathsf{ICNN}$). Our focus is on expressivity, mostly in terms of depth and our motivation is to gain better understanding of depth requirement needed to exactly represent functions in terms of neural networks with ReLU activations: a subject that has received a lot of attention lately. We prove several lower bounds on the depth required to represent functions by monotone ReLU networks and $\mathsf{ICNN}$. For the maximum function $\mathsf{MAX}_n$ computing the maximum of $n$ real numbers, we show that $\mathsf{ReLU}^+$ networks cannot compute $\mathsf{MAX}_n$, or even approximate it. We prove a sharp $n$ lower bound on the $\mathsf{ICNN}$ depth complexity of $\mathsf{MAX}_n$. We also prove depth separations between $\mathsf{ReLU}$ networks and $\mathsf{ICNN}$s; for every $k$, there is a depth-$2$ $\mathsf{ReLU}$ network of size $O(k^2)$ that cannot be simulated by a depth-$k$ $\mathsf{ICNN}$. The proofs combine ideas from some structural results for $\mathsf{ReLU}^+$ networks.