Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm (ADASAP) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. ADASAPoutperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, ADASAPscales to a dataset with > 3 108 samples, a feat which has not been accomplished in the literature.

We study last-iterate convergence of SGD with greedy step size over smooth quadratics in the interpolation regime, a setting which captures the classical Randomized Kaczmarz algorithm as well as other popular iterative linear system solvers. For these methods, we show that the $t$-th iterate attains an $O(1/t^{3/4})$ convergence rate, addressing a question posed by Attia, Schliserman, Sherman, and Koren, who gave an $O(1/t^{1/2})$ guarantee for this setting. In the proof, we introduce the family of stochastic contraction processes, whose behavior can be described by the evolution of a certain deterministic eigenvalue equation, which we analyze via a careful discrete-to-continuous reduction.

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Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which canbeimplemented efficiently andwhich perform wellinnumericalexperiments.

Finally, we extend LESS embeddings to include uniformly sparsified random sign matrices which canbeimplemented efficiently andwhich perform wellinnumericalexperiments.

Surprisingly we show that volume sampling can have poor behavior when we require a very accurate approximation - indeed worse than some i.i.d.

Matrix sketching is a powerful tool for reducing the size of large data matrices.