Deep Network with Approximation Error Being Reciprocal of Width to Power of Square Root of Depth

A new network with super approximation power is introduced. This network is built with Floor ($\lfloor x\rfloor$) or ReLU ($\max{0,x}$) activation function in each neuron and hence we call such networks Floor-ReLU networks. For any hyper-parameters $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, it is shown that Floor-ReLU networks with width $\max{d,\, 5N+13}$ and depth $64dL+3$ can uniformly approximate a H\"older function $f$ on $[0,1]^d$ with an approximation error $3\lambda d^{{\alpha/2}N{-\alpha\sqrt{L}}$,} where $\alpha \in(0,1]$ and $\lambda$ are the H\"older order and constant, respectively. More generally for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of continuity $\omegaf(\cdot)$, the constructive approximation rate is $\omegaf(\sqrt{d}\,N^{{-\sqrt{L}})+2\omega_f(\sqrt{d}){N{-\sqrt{L}}}$.} As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of $\omegaf(r)$ as $r\to 0$ is moderate (e.g., $\omegaf(r) \lesssim r^\alpha$ for H\"older continuous functions), since the major term to be considered in our approximation rate is essentially $\sqrt{d}$ times a function of $N$ and $L$ independent of $d$ within the modulus of continuity.