Excellent variational approximations to Gaussian process posteriors have been developed which avoid the $\mathcal{O}\left(N^3\right)$ scaling with dataset size $N$. They reduce the computational cost to $\mathcal{O}\left(NM^2\right)$, with $M\ll N$ being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in $N$, the true complexity of the algorithm depends on how $M$ must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We show that with high probability the KL divergence can be made arbitrarily small by growing $M$ more slowly than $N$. A particular case of interest is that for regression with normally distributed inputs in D-dimensions with the popular Squared Exponential kernel, $M=\mathcal{O}(\log^D N)$ is sufficient. Our results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase $M$ in continual learning scenarios.

发展了出色的变分逼近高斯过程后验方法，可以避免数据集尺寸为N的时间复杂度O(N³)的问题，而将计算复杂度降低到O(NM²)的程度，M是总结进程的引出的变量数。结果表明，通过以比N更慢的速度增加M，可以使KL散度任意小。

稀疏变分高斯过程回归的收敛率