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

Abstract

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 $\omega_f(\cdot)$, the constructive approximation rate is $\omega_f(\sqrt{d}\,N^{-\sqrt{L}})+2\omega_f(\sqrt{d}){N^{-\sqrt{L}}}$. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of $\omega_f(r)$ as $r\to 0$ is moderate (e.g., $\omega_f(r) \lesssim r^\alpha$ 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.