Deep Network Approximation Characterized by Number of Neurons (1906.05497v5)

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 $\mathcal{O}\big(\max{d\lfloor N^{1/d}\rfloor,\, N+1}\big)$ 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 $d_{\mathcal{M}}\ll d$. Especially, in the case of an essentially low-dimensional domain, we show an approximation rate $\mathcal{O}\big(\omega_f(\tfrac{\varepsilon}{1-\delta}\sqrt{\tfrac{d}{d_\delta}}+\varepsilon)+\sqrt{d}\,\omega_f(\tfrac{\sqrt{d}}{(1-\delta)\sqrt{d_\delta}}N^{-2/d_\delta}L^{-2/d_\delta})\big)$ for ReLU FNNs to approximate $f$ in the $\varepsilon$-neighborhood, where $d_\delta=\mathcal{O}\big(d_{\mathcal{M}}\tfrac{\ln (d/\delta)}{\delta^2}\big)$ 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.

• The paper derives a nearly optimal quantitative approximation rate for deep ReLU networks approximating Hölder continuous functions on compact domains.
• It leverages the interplay between network width and depth, supported by VC-dimension theory and projection arguments, to rigorously analyze approximation capabilities.
• The findings facilitate efficient network design for high-dimensional and irregular domain scenarios, offering actionable insights for future architecture advancements.

Deep Network Approximation Characterized by the Number of Neurons

The paper entitled "Deep Network Approximation Characterized by Number of Neurons" by Zuowei Shen, Haizhao Yang, and Shijun Zhang provides a comprehensive analysis of the approximation capabilities of deep feed-forward neural networks (FNNs) using ReLU activation functions. This paper focuses on the ability of these networks to approximate continuous functions, quantified in terms of both network width and depth. The authors set out to establish an analytic framework, complemented by rigorous proofs, to characterize the approximation power of networks, particularly for continuous functions on compact domain spaces.

The principal contribution of the paper is the derivation of a quantitative and nearly optimal approximation rate for deep ReLU FNNs. The authors convincingly demonstrate that such networks, with width $$\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor, N+1\}\big)$$ and depth $\mathcal{O}(L)$, can approximate any arbitrary Hölder continuous function of order $\alpha$ on $[0,1]^d$ to an error of $$\mathcal{O}\big(\sqrt{d}N^{-2\alpha/d}L^{-2\alpha/d}\big)$$. This result is achieved by leveraging the interplay between width and depth, offering a framework that goes beyond previously, predominantly width-focused analyses. Notably, this rate is shown to be asymptotically almost tight, strengthening the paper’s contribution to foundational theory in neural network approximation.

The authors extend their results to domains with irregular structures. They present methods for approximating functions localized within an $\varepsilon$-neighborhood of a lower-dimensional manifold embedded in $[0,1]^d$. The approximation rate adjusted for the manifold's intrinsic dimensionality $d_{\mathcal{M}}$ can be expressed as $$\mathcal{O}\big(\omega_f(\frac{2\varepsilon}{1-\delta}\sqrt{\frac{d}{d_\delta}+\varepsilon})\big)$$.

The implications of these findings are manifold. Practically, they guide the architectural decisions in network design, particularly in applications prioritizing computational efficiency, such as high-dimensional data scenarios and irregular domain mappings. Theoretically, they shed light on dimensionality reduction and domain-specific neural architecture adaptations, paving the way for more efficient learning in both general and specific context settings.

A significant strength of this research lies in its mathematically grounded methodologies, which include deriving results from VC-dimension theory and empirical projection arguments. However, the paper abstains from over-claiming its practical immediacy, recognizing that the observed approximation rates assume optimal parameter selection and training, which remain contingent on empirical advancements.

Future research could explore generalizing these findings to other network architectures and non-ReLU activations or integrating identity mappings into the learning structure to potentially optimize network sizes. Moreover, exploring applications in more diverse function spaces or leveraging parallel computing frameworks to harness these findings for large-scale learning problems warrants serious investigation.

Overall, the insights from this paper significantly enhance the theoretical framework governing neural network approximation, providing robust analytical tools to inspire ongoing development in the field of machine learning and approximation theory.