  Variational approximations to Gaussian processes (GPs) typically use a small
set of inducing points to form a low-rank approximation to the covariance
matrix. In this work, we instead exploit a sparse approximation of the
precision matrix. We propose variational nearest neighbor Gaussian process
(VNNGP), which introduces a prior that only retains correlations within K
nearest-neighboring observations, thereby inducing sparse precision structure.
Using the variational framework, VNNGP's objective can be factorized over both
observations and inducing points, enabling stochastic optimization with a time
complexity of O($K^3$). Hence, we can arbitrarily scale the inducing point
size, even to the point of putting inducing points at every observed location.
We compare VNNGP to other scalable GPs through various experiments, and
demonstrate that VNNGP (1) can dramatically outperform low-rank methods, and
(2) is less prone to overfitting than other nearest neighbor methods.
