Deep Network Approximation Characterized by Number of Neurons
It provides theoretical bounds for neural network approximation, which is foundational for understanding model capacity in machine learning, though incremental relative to existing approximation theory.
This paper tackles the problem of quantifying the approximation power of deep feed-forward neural networks in terms of neuron count, showing that ReLU networks with specific width and depth can approximate Hölder continuous functions with a nearly tight rate of O(√d N^{-2α/d} L^{-2α/d}) in L^p-norm, and extends this to functions on irregular or low-dimensional domains.
This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+1\}\big)$ and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $α\in (0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}\big(\sqrt{d} N^{-2α/d}L^{-2α/d}\big)$ measured in $L^p$-norm for any $N,L\in \mathbb{N}^+$ and $p\in[1,\infty]$. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $ω_f(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,ω_f( N^{-2/d}L^{-2/d})\big)$. We also extend our analysis to $f$ on irregular domains or those localized in an $\varepsilon$-neighborhood of a $d_{\mathcal{M}}$-dimensional smooth manifold $\mathcal{M}\subseteq [0,1]^d$ with $d_{\mathcal{M}}\ll d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}\big(ω_f(\tfrac{\varepsilon}{1-δ}\sqrt{\tfrac{d}{d_δ}}+\varepsilon)+\sqrt{d}\,ω_f(\tfrac{\sqrt{d}}{(1-δ)\sqrt{d_δ}}N^{-2/d_δ}L^{-2/d_δ})\big)$ for ReLU FNNs to approximate $f$ in the $\varepsilon$-neighborhood, where $d_δ=\mathcal{O}\big(d_{\mathcal{M}}\tfrac{\ln (d/δ)}{δ^2}\big)$ for any $δ\in(0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_δ$-dimensional domain.