IMPROVED ESTIMATION OF RELAXATION TIME IN NONREVERSIBLE MARKOV CHAINS

Geoffrey Wolfer; Aryeh Kontorovich

doi:10.1214/23-AAP1963

IMPROVED ESTIMATION OF RELAXATION TIME IN NONREVERSIBLE MARKOV CHAINS

Geoffrey Wolfer, Aryeh Kontorovich

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the minimax sample complexity for estimating the pseudo-spectral gap γ_ps of an ergodic Markov chain in constant multiplicative error is of the order of Θ̃ ⁽_γ_ps¹_π*⁾ , 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 language	English
Pages (from-to)	249-276
Number of pages	28
Journal	Annals of Applied Probability
Volume	34
Issue number	1 A
DOIs	https://doi.org/10.1214/23-AAP1963
State	Published - 1 Feb 2024

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/23-AAP1963

Cite this

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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 {\~Θ} (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.",

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N2 - 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.

AB - 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.

KW - Ergodic Markov chain

KW - empirical confidence interval

KW - mixing time

KW - pseudo-spectral gap

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IMPROVED ESTIMATION OF RELAXATION TIME IN NONREVERSIBLE MARKOV CHAINS

Abstract

Keywords

ASJC Scopus subject areas

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