Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal statistical accuracy is achieved requiring essentially $O(n)$ memory and $O(n\sqrt{n})$ time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures.

Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal statistical accuracy is achieved requiring essentially $O(n)$ memory and $O(n\sqrt{n})$ time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures.

Neural Information Processing Systems http://nips.cc/

In this second case we can compare the accuracy of the two methods. The final version of the paper will compare the two methods on all considered datasets. Indeed by "time" we mean total training time To conclude, with "blocks" we mean that

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After further thought, however, I have updated my review to a 5. While this works seems to have very high potential, the current draft has several important shortcomings: (1) It is not yet polished enough for broad consumption. Significant revisions would be necessary to increase clarity. Summary: This paper presents a novel algorithm for estimating leverage scores for kernel matrices. The running time of this algorithm improves upon state of the art.

Kernel methods provide a principled way to perform non linear, nonparametric learning. They rely on solid functional analytic foundations and enjoy optimal statistical properties. However, at least in their basic form, they have limited applicability in large scale scenarios because of stringent computational requirements in terms of time and especially memory. In this paper, we take a substantial step in scaling up kernel methods, proposing FALKON, a novel algorithm that allows to efficiently process millions of points. FALKON is derived combining several algorithmic principles, namely stochastic subsampling, iterative solvers and preconditioning. Our theoretical analysis shows that optimal statistical accuracy is achieved requiring essentially O(n) memory and O(n n) time. An extensive experimental analysis on large scale datasets shows that, even with a single machine, FALKON outperforms previous state of the art solutions, which exploit parallel/distributed architectures.

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.

Kernel ridge regression (KRR) is a popular scheme for non-linear non-parametric learning. However, existing implementations of KRR require that all the data is stored in the main memory, which severely limits the use of KRR in contexts where data size far exceeds the memory size. Such applications are increasingly common in data mining, bioinformatics, and control. A powerful paradigm for computing on data sets that are too large for memory is the streaming model of computation, where we process one data sample at a time, discarding each sample before moving on to the next one. In this paper, we propose StreaMRAK - a streaming version of KRR. StreaMRAK improves on existing KRR schemes by dividing the problem into several levels of resolution, which allows continual refinement to the predictions. The algorithm reduces the memory requirement by continuously and efficiently integrating new samples into the training model. With a novel sub-sampling scheme, StreaMRAK reduces memory and computational complexities by creating a sketch of the original data, where the sub-sampling density is adapted to the bandwidth of the kernel and the local dimensionality of the data. We present a showcase study on two synthetic problems and the prediction of the trajectory of a double pendulum. The results show that the proposed algorithm is fast and accurate.