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

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 $\mathcal{O}\big(\lambda\sqrt{d} (N^2L2\ln N)^{{-\alpha/d}\big)$,} 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 $\mathcal{O}\big(\,\sqrt{d}\,\omegaf\big( (N^2L2\ln N)^{{-1/d}\big)\,\big)$,} where $\omegaf(\cdot)$ 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.