IMPROVED ESTIMATION OF RELAXATION TIME IN NONREVERSIBLE MARKOV CHAINS

Geoffrey Wolfer, Aryeh Kontorovich

Research output: Contribution to journalArticlepeer-review

Abstract

We show that the minimax sample complexity for estimating the pseudo-spectral gap γps of an ergodic Markov chain in constant multiplicative error is of the order of Θ̃ (γps1π*) , where π* is the minimum stationary probability, recovering the known bound in the reversible setting for estimating the absolute spectral gap (Hsu et al., Ann. Appl. Probab. 29 (2019) 2439–2480), and resolving an open problem of Wolfer and Kontorovich (In Proceedings of the Thirty-Second Conference on Learning Theory (2019) 3120–3159 PMLR). Furthermore, we strengthen the known empirical procedure by making it fully-adaptive to the data, thinning the confidence intervals and reducing the computational complexity. Along the way, we derive new properties of the pseudo-spectral gap and introduce the notion of a reversible dilation of a stochastic matrix.

Original languageEnglish
Pages (from-to)249-276
Number of pages28
JournalAnnals of Applied Probability
Volume34
Issue number1 A
DOIs
StatePublished - 1 Feb 2024

Keywords

  • empirical confidence interval
  • Ergodic Markov chain
  • mixing time
  • pseudo-spectral gap

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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