Bernoulli

  • Bernoulli
  • Volume 25, Number 4B (2019), 3912-3938.

Rademacher complexity for Markov chains: Applications to kernel smoothing and Metropolis–Hastings

Patrice Bertail and François Portier

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Abstract

The concept of Rademacher complexity for independent sequences of random variables is extended to Markov chains. The proposed notion of “regenerative block Rademacher complexity” (of a class of functions) follows from renewal theory and allows to control the expected values of suprema (over the class of functions) of empirical processes based on Harris Markov chains as well as the excess probability. For classes of Vapnik–Chervonenkis type, bounds on the “regenerative block Rademacher complexity” are established. These bounds depend essentially on the sample size and the probability tails of the regeneration times. The proposed approach is employed to obtain convergence rates for the kernel density estimator of the stationary measure and to derive concentration inequalities for the Metropolis–Hastings algorithm.

Article information

Source
Bernoulli, Volume 25, Number 4B (2019), 3912-3938.

Dates
Received: June 2018
Revised: December 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1569398789

Digital Object Identifier
doi:10.3150/19-BEJ1115

Mathematical Reviews number (MathSciNet)
MR4010977

Zentralblatt MATH identifier
07110160

Keywords
concentration inequalities Kernel smoothing Markov chains Metropolis Hastings Rademacher complexity

Citation

Bertail, Patrice; Portier, François. Rademacher complexity for Markov chains: Applications to kernel smoothing and Metropolis–Hastings. Bernoulli 25 (2019), no. 4B, 3912--3938. doi:10.3150/19-BEJ1115. https://projecteuclid.org/euclid.bj/1569398789


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