The Annals of Statistics

“Local” vs. “global” parameters—breaking the Gaussian complexity barrier

Shahar Mendelson

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We show that if $F$ is a convex class of functions that is $L$-sub-Gaussian, the error rate of learning problems generated by independent noise is equivalent to a fixed point determined by “local” covering estimates of the class (i.e., the covering number at a specific level), rather than by the Gaussian average, which takes into account the structure of $F$ at an arbitrarily small scale. To that end, we establish new sharp upper and lower estimates on the error rate in such learning problems.

Article information

Ann. Statist., Volume 45, Number 5 (2017), 1835-1862.

Received: October 2015
Revised: August 2016
First available in Project Euclid: 31 October 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62C20: Minimax procedures 60G15: Gaussian processes

Error rates Gaussian averages covering numbers


Mendelson, Shahar. “Local” vs. “global” parameters—breaking the Gaussian complexity barrier. Ann. Statist. 45 (2017), no. 5, 1835--1862. doi:10.1214/16-AOS1510.

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Supplemental materials

  • Supplement to "`Local’ vs. ‘global’ parameters—breaking the Gaussian complexity barrier”. We prove two observations: the first shows that the setup of the Young–Barron theorem is different from the one we study here, and the other is that for $p>1$ there is a true gap between the “local” and “global” complexities of $B_{p}^{n}$.