Randomized sketches for kernels: Fast and optimal nonparametric regression

Yun Yang, Mert Pilanci, Martin J. Wainwright

Research output: Contribution to journalArticle

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 O(n3) and O(n2), 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.

Original languageEnglish (US)
Pages (from-to)991-1023
Number of pages33
JournalAnnals of Statistics
Volume45
Issue number3
DOIs
StatePublished - Jun 2017

Keywords

  • Convex optimization
  • Dimensionality reduction
  • Kernel method
  • Nonparametric regression
  • Random projection

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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