Deep Network Approximation Characterized by Number of Neurons

Abstract

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 $\mathcal{O}(\max \{d\lfloor N^{1/d}\rfloor ,N+1\})$ and depth $\mathcal{O}(L)$ can approximate an arbitrary Hölder continuous function of order $\alpha \in (0,1]$ on $[0,1{]}^d$ with a nearly tight approximation rate $\mathcal{O}\left(\sqrt{d}{N}^{-2\alpha /d}{L}^{-2\alpha /d}\right)$ 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 ${\omega }_f(\cdot )$, the constructive approximation rate is $\mathcal{O}(\sqrt{d}{\omega }_f(N^{-2/d}{L}^{-2/d}))$. We also extend our analysis to $f$ on irregular domains or those localized in an $\epsilon$-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 $\mathcal{O}({\omega }_f(\frac{\epsilon }{1-\delta }\sqrt{\frac{d}{d_{\delta }}}+\epsilon )+\sqrt{d}{\omega }_f(\frac{\sqrt{d}}{(1-\delta )\sqrt{d_{\delta }}}{N}^{-2/d_{\delta }}{L}^{-2/d_{\delta }}))$ for ReLU FNNs to approximate $f$ in the $\epsilon$-neighborhood, where $d_{\delta }=\mathcal{O}\left(d_{\mathcal{M}}\frac{\ln (d/\delta )}{{\delta }^2}\right)$ for any $\delta \in (0,1)$ as a relative error for a projection to approximate an isometry when projecting $\mathcal{M}$ to a $d_{\delta }$-dimensional domain.