The Annals of Statistics

Randomized sketches for kernels: Fast and optimal nonparametric regression

Yun Yang, Mert Pilanci, and Martin J. Wainwright

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Abstract

Kernel ridge regression (KRR) is a standard method for performing nonparametric regression over reproducing kernel Hilbert spaces. Given $n$ samples, the time and space complexity of computing the KRR estimate scale as $\mathcal{O}(n^{3})$ and $\mathcal{O}(n^{2})$, respectively, and so is prohibitive in many cases. We propose approximations of KRR based on $m$-dimensional randomized sketches of the kernel matrix, and study how small the projection dimension $m$ can be chosen while still preserving minimax optimality of the approximate KRR estimate. For various classes of randomized sketches, including those based on Gaussian and randomized Hadamard matrices, we prove that it suffices to choose the sketch dimension $m$ proportional to the statistical dimension (modulo logarithmic factors). Thus, we obtain fast and minimax optimal approximations to the KRR estimate for nonparametric regression. In doing so, we prove a novel lower bound on the minimax risk of kernel regression in terms of the localized Rademacher complexity.

Article information

Source
Ann. Statist., Volume 45, Number 3 (2017), 991-1023.

Dates
Received: September 2015
Revised: April 2016
First available in Project Euclid: 13 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1497319686

Digital Object Identifier
doi:10.1214/16-AOS1472

Mathematical Reviews number (MathSciNet)
MR3662446

Zentralblatt MATH identifier
1371.62039

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 68W20: Randomized algorithms

Keywords
Nonparametric regression random projection kernel method dimensionality reduction convex optimization

Citation

Yang, Yun; Pilanci, Mert; Wainwright, Martin J. Randomized sketches for kernels: Fast and optimal nonparametric regression. Ann. Statist. 45 (2017), no. 3, 991--1023. doi:10.1214/16-AOS1472. https://projecteuclid.org/euclid.aos/1497319686


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