Parallelizing Spectrally Regularized Kernel Algorithms

Abstract

We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an reproducing kernel Hilbert space (RKHS) framework. The data set of size $n$ is partitioned into $m=O(n^\alpha)$, $\alpha < \frac{1}{2}$, disjoint subsamples. On each subsample, some spectral regularization method (belonging to a large class, including in particular Kernel Ridge Regression, $L^2$-boosting and spectral cut-off) is applied. The regression function $f$ is then estimated via simple averaging, leading to a substantial reduction in computation time. We show that minimax optimal rates of convergence are preserved if $m$ grows sufficiently slowly (corresponding to an upper bound for $\alpha$) as $n \to \infty$, depending on the smoothness assumptions on $f$ and the intrinsic dimensionality. In spirit, the analysis relies on a classical bias/stochastic error analysis.