Every Layer Counts: An Exponential $L_2$ Depth Hierarchy for ReLU Networks

Abstract

We prove a depth hierarchy for ReLU neural networks in which every additional ReLU layer can save exponentially many neurons. For all $k\geq2$, we construct a globally $[0,1]$-valued, $1$-Lipschitz function realized by a depth-$(k+1)$ network of width $\mathcal{O}(d^4)$, whereas any depth-$k$ network with unrestricted weights and width at most $\frac{2^d}{2d(k-1)}$ has squared $L_2$ error at least $1/24$ under an absolutely continuous distribution supported at exponential distance from the origin. To the best of our knowledge, this is the first exponential hierarchy across all adjacent fixed depths, and the first exponential separation for ReLU networks between two fixed depths whose shallower network has depth at least $3$. The lower bound also immediately yields the corresponding hierarchy for exact computation. Moreover, the case $k=2$ gives a compactly supported separation between depths $3$ and $2$ with unrestricted shallow-network weights, answering a question raised by Safran, Eldan, and Shamir (2019). The distribution used in our construction nevertheless has all its mass at exponential radius, placing the hierarchy outside the regularity regime in which such a separation would imply major threshold-circuit lower bounds. We also prove an exact separation for a more regular target, which is globally $[0,1]$-valued and $\mathcal{O}(\sqrt d)$-Lipschitz and maps the unit hypercube onto $[0,1]$. It is computed by a polynomial-width depth-$4$ network, whereas any depth-$3$ network agreeing with it on the unit hypercube requires exponentially many first-layer neurons, even with unrestricted weights.