A Depth Hierarchy for Computing the Maximum in ReLU Networks via Extremal Graph Theory
This provides foundational insights into the inherent complexity of a fundamental operator in neural networks, with implications for understanding depth-width trade-offs in deep learning theory.
The paper tackled the problem of exactly computing the maximum function over d real inputs using ReLU neural networks, proving that width Ω(d^(1+1/(2^(k-2)-1))) is necessary for depths 3 ≤ k ≤ log₂(log₂(d)), establishing the first unconditional super-linear lower bound for this operator at such depths.
We consider the problem of exact computation of the maximum function over $d$ real inputs using ReLU neural networks. We prove a depth hierarchy, wherein width $Ω\big(d^{1+\frac{1}{2^{k-2}-1}}\big)$ is necessary to represent the maximum for any depth $3\le k\le \log_2(\log_2(d))$. This is the first unconditional super-linear lower bound for this fundamental operator at depths $k\ge3$, and it holds even if the depth scales with $d$. Our proof technique is based on a combinatorial argument and associates the non-differentiable ridges of the maximum with cliques in a graph induced by the first hidden layer of the computing network, utilizing Turán's theorem from extremal graph theory to show that a sufficiently narrow network cannot capture the non-linearities of the maximum. This suggests that despite its simple nature, the maximum function possesses an inherent complexity that stems from the geometric structure of its non-differentiable hyperplanes, and provides a novel approach for proving lower bounds for deep neural networks.