This paper establishes the optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N)$ and depth $\mathcal{O}(L)$ with an approximation error $\mathcal{O}(N^{-L})$. Through local Taylor expansions and their deep ReLU network approximations, we show that deep ReLU networks of width $\mathcal{O}(N\ln N)$ and depth $\mathcal{O}(L\ln L)$ can approximate $f\in C^s([0,1]^d)$ with a nearly optimal approximation error $\mathcal{O}(\|f\|_{C^s([0,1]^d)}N^{-2s/d}L^{-2s/d})$. Our estimate is non-asymptotic in the sense that it is valid for arbitrary width and depth specified by $N\in\mathbb{N}^+$ and $L\in\mathbb{N}^+$, respectively.

翻译：本文同时为宽度和深度平滑功能的深修正线性单位( ReLU) 网络设定最佳近似错误。 为此, 我们首先证明, 宽度为$\ mathcal{O} (N) $和深度为$\ mathcal{O} (L) $(N) $和深度为$\ mathcal{O} (L) $(美元) 和深度为$\ mathcal{(N)- L) 的深修正线性线性单位(ReLU) 网络。 通过本地泰勒扩展及其深为 ReLU 网络近似值, 我们显示, 宽度为$\ mathcal{O} (N\ lnN) $和深度的深深处 RELU 网络的深度为$\ 2\ d} (N\\\\\\\\\\\\\\\ ma} $( 美元) 美元, 以任意的深度和深度为准有效。