This paper establishes the (nearly) 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的平滑功能设置深修正线性单位( RELU) 网络的最佳近似错误 。 为此, 我们首先证明, 宽度为$\ mathcal{O} (N) 的深修改线性单位( ReLU) 网络和深度为 $\ mathcal{O} (L) 的深修改线性线性单位( ReLU) 最佳近似错误 $\ mathcal{O} (L) 美元和深度为 $\ mathcal{( N) $( N) 和深度的深度为 $\ mathcal{O} (L\ l) 美元可以近似于$\ in c% ([ 0, 1\ d) 美元和深度为近似最佳的近似近似近似近误 $\ mathca{( O} ( ⁇ f ⁇ C} ([ 1\ d} N_ 2s/ d}L} \\\ 2s/ d} $/ d) $。 在任意和深度的深度中, 我们的估算中, $_\\\\\\ masmabbbbbbroc.