Deep Network Approximation Characterized by Number of Neurons

Understand

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 H\"older continuous function of order $\alpha\in (0,1]$ on $[0,1]^d$ with a nearly tight approximation rate $\mathcal{O}\big(\sqrt{d} N^{-2\alpha/d}L^{-2\alpha/d}\big)$ 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 $\omega_f(\cdot)$, the constructive approximation rate is $\mathcal{O}\big(\sqrt{d}\,\omega_f( N^{-2/d}L^{-2/d})\big)$.
• 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 $d_{\mathcal{M}}\ll d$.