Kernel ridge regression (KRR) is a fundamental computational tool, appearing in problems that range from computational chemistry to health analytics, with a particular interest due to its starring role in Gaussian process regression. However, it is challenging to scale KRR solvers to large datasets: with $n$ training points, a direct solver (i.e., Cholesky decomposition) uses $O(n^2)$ storage and $O(n^3)$ flops. Iterative methods for KRR, such as preconditioned conjugate gradient (PCG), avoid the cubic scaling of direct solvers and often use low-rank preconditioners; a rank $r$ preconditioner uses $O(rn)$ storage and each iteration requires $O(n^2)$ flops. To reduce the storage and iteration complexity of iterative solvers for KRR, we propose ASkotch ($\textbf{A}$ccelerated $\textbf{s}$calable $\textbf{k}$ernel $\textbf{o}$p$\textbf{t}$imization using block $\textbf{c}$oordinate descent with $\textbf{H}$essian preconditioning). For a given block size $|b| << n$, each iteration of ASkotch uses $O(r|b| + n)$ storage and $O(n|b|)$ flops, so ASkotch scales better than Cholesky decomposition and PCG. We prove that ASkotch obtains linear convergence to the optimum, with the convergence rate depending on the square roots of the $\textit{preconditioned}$ block condition numbers. Furthermore, we solve KRR problems that were considered to be impossibly large while using limited computational resources: we show that ASkotch outperforms PCG methods with respect to generalization error on large-scale KRR (up to $n = 10^8$) and KRR classification tasks (up to $n = 10^7$) while running each of our experiments on $\textit{a single 12 GB Titan V GPU}$. Our work opens up the possibility of as-yet-unimagined applications of KRR across a wide range of disciplines.

This paper introduces PROMISE (Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates), a suite of sketching-based preconditioned stochastic gradient algorithms for solving large-scale convex optimization problems arising in machine learning. PROMISE includes preconditioned versions of SVRG, SAGA, and Katyusha; each algorithm comes with a strong theoretical analysis and effective default hyperparameter values. In contrast, traditional stochastic gradient methods require careful hyperparameter tuning to succeed, and degrade in the presence of ill-conditioning, a ubiquitous phenomenon in machine learning. Empirically, we verify the superiority of the proposed algorithms by showing that, using default hyperparameter values, they outperform or match popular tuned stochastic gradient optimizers on a test bed of 51 ridge and logistic regression problems assembled from benchmark machine learning repositories. On the theoretical side, this paper introduces the notion of quadratic regularity in order to establish linear convergence of all proposed methods even when the preconditioner is updated infrequently. The speed of linear convergence is determined by the quadratic regularity ratio, which often provides a tighter bound on the convergence rate compared to the condition number, both in theory and in practice, and explains the fast global linear convergence of the proposed methods.