  To achieve scalable and accurate inference for latent Gaussian processes, we
propose a variational approximation based on a family of Gaussian distributions
whose covariance matrices have sparse inverse Cholesky (SIC) factors. We
combine this variational approximation of the posterior with a similar and
efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior.
We then focus on a particular SIC ordering and nearest-neighbor-based sparsity
pattern resulting in highly accurate prior and posterior approximations. For
this setting, our variational approximation can be computed via stochastic
gradient descent in polylogarithmic time per iteration. We provide numerical
comparisons showing that the proposed double-Kullback-Leibler-optimal
Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate
for stationary kernels than alternative approaches such as inducing-point and
mean-field approximations at similar computational complexity.
