Optimal Approximation Rate of ReLU Networks in terms of Width and Depth

TL;DR

This paper establishes the optimal approximation rates of deep ReLU neural networks in terms of width and depth for functions on [0,1]^d, improving existing bounds by including a logarithmic factor and extending results to arbitrary continuous functions.

Contribution

It proves that ReLU networks can achieve nearly optimal approximation rates with explicit constructions, including new bounds involving logarithmic factors and fixed-depth networks for Lipschitz functions.

Findings

ReLU networks with specified width and depth approximate Hölder functions at optimal rates.

The approximation rate for arbitrary continuous functions depends on the modulus of continuity.

Fixed-depth networks can approximate Lipschitz functions with a rate involving W ln W, a novel result.

Abstract

This paper concentrates on the approximation power of deep feed-forward neural networks in terms of width and depth. It is proved by construction that ReLU networks with width $\mathcal{O}\big(\max\{d\lfloor N^{1/d}\rfloor,\, N+2\}\big)$ and depth $\mathcal{O}(L)$ can approximate a H\"older continuous function on $[0,1]^d$ with an approximation rate $\mathcal{O}\big(\lambda\sqrt{d} (N^2L^2\ln N)^{-\alpha/d}\big)$, where $\alpha\in (0,1]$ and $\lambda>0$ are H\"older order and constant, respectively. Such a rate is optimal up to a constant in terms of width and depth separately, while existing results are only nearly optimal without the logarithmic factor in the approximation rate. More generally, for an arbitrary continuous function $f$ on $[0,1]^d$, the approximation rate becomes $\mathcal{O}\big(\,\sqrt{d}\,\omega_f\big( (N^2L^2\ln N)^{-1/d}\big)\,\big)$, where $\omega_f(\cdot)$ is the modulus of continuity. We also extend our analysis to any continuous function $f$ on a bounded set. Particularly, if ReLU networks with depth $31$ and width $\mathcal{O}(N)$ are used to approximate one-dimensional Lipschitz continuous functions on $[0,1]$ with a Lipschitz constant $\lambda>0$, the approximation rate in terms of the total number of parameters, $W=\mathcal{O}(N^2)$, becomes $\mathcal{O}(\tfrac{\lambda}{W\ln W})$, which has not been discovered in the literature for fixed-depth ReLU networks.