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Stochastic gradient descent (SGD) and its variants have establishedthemselves as the go-to algorithms for large-scale machine learning problemswith independent samples due to their generalization performance and intrinsiccomputational advantage. However, the fact that the stochastic gradient is abiased estimator of the full gradient with correlated samples has led to thelack of theoretical understanding of how SGD behaves under correlated settingsand hindered its use in such cases. In this paper, we focus on hyperparameterestimation for the Gaussian process (GP) and take a step forward towardsbreaking the barrier by proving minibatch SGD converges to a critical point ofthe full log-likelihood loss function, and recovers model hyperparameters withrate $O(\frac{1}{K})$ for $K$ iterations, up to a statistical error termdepending on the minibatch size. Our theoretical guarantees hold provided thatthe kernel functions exhibit exponential or polynomial eigendecay which issatisfied by a wide range of kernels commonly used in GPs. Numerical studies onboth simulated and real datasets demonstrate that minibatch SGD has bettergeneralization over state-of-the-art GP methods while reducing thecomputational burden and opening a new, previously unexplored, data size regimefor GPs.

Gaussian Process Inference Using Mini-batch Stochastic Gradient Descent: Convergence Guarantees and Empirical Benefits