We show that common choices of kernel functions for a highly accurate and massively scalable nearest-neighbour based GP regression model (GPnn: \cite{GPnn}) exhibit gradual convergence to asymptotic behaviour as dataset-size $n$ increases. For isotropic kernels such as Mat\'{e}rn and squared-exponential, an upper bound on the predictive MSE can be obtained as $O(n^{-\frac{p}{d}})$ for input dimension $d$, $p$ dictated by the kernel (and $d>p$) and fixed number of nearest-neighbours $m$ with minimal assumptions on the input distribution. Similar bounds can be found under model misspecification and combined to give overall rates of convergence of both MSE and an important calibration metric. We show that lower bounds on $n$ can be given in terms of $m$, $l$, $p$, $d$, a tolerance $\varepsilon$ and a probability $\delta$. When $m$ is chosen to be $O(n^{\frac{p}{p+d}})$ minimax optimal rates of convergence are attained. Finally, we demonstrate empirical performance and show that in many cases convergence occurs faster than the upper bounds given here.

翻译：我们证明，基于高度精确且可大规模扩展的最近邻高斯过程回归模型（GPnn：\cite{GPnn}）中常用的核函数选择，会随着数据集规模 $n$ 的增加而逐渐收敛于渐近行为。对于各向同性核函数（如 Matérn 核和平方指数核），在输入维度 $d$、由核函数决定的参数 $p$（满足 $d>p$）、固定最近邻数量 $m$ 以及对输入分布的最小假设条件下，预测均方误差的上界可表示为 $O(n^{-\frac{p}{d}})$。在模型误设定情形下可得到类似界值，并将这些界值组合后可给出 MSE 及一项重要校准度量的整体收敛速率。我们证明，可以基于 $m$、$l$、$p$、$d$、容差 $\varepsilon$ 和概率 $\delta$ 给出 $n$ 的下界。当 $m$ 取为 $O(n^{\frac{p}{p+d}})$ 时，可达到极小极大最优收敛速率。最后，我们通过实证性能验证，在许多情况下实际收敛速度快于本文给出的上界。