Contraction of Markovian Operators in Orlicz Spaces and Error Bounds for Markov Chain Monte Carlo
Abstract: We introduce a novel concept of convergence for Markovian processes within Orlicz spaces, extending beyond the conventional approach associated with $L_p$ spaces. After showing that Markovian operators are contractive in Orlicz spaces, our key technical contribution is an upper bound on their contraction coefficient, which admits a closed-form expression. The bound is tight in some settings, and it recovers well-known results, such as the connection between contraction and ergodicity, ultra-mixing and Doeblin's minorisation. Specialising our approach to $L_p$ spaces leads to a significant improvement upon classical Riesz-Thorin's interpolation methods. Furthermore, by exploiting the flexibility offered by Orlicz spaces, we can tackle settings where the stationary distribution is heavy-tailed, a severely under-studied setup. As an application of the framework put forward in the paper, we introduce tighter bounds on the mixing time of Markovian processes, better exponential concentration bounds for MCMC methods, and better lower bounds on the burn-in period. To conclude, we show how our results can be used to prove the concentration of measure phenomenon for a sequence of Markovian random variables.
- G. O. Roberts and J. S. Rosenthal, “General state space Markov chains and MCMC algorithms,” Probability Surveys, 2004.
- D. Rudolf, “Explicit error bounds for Markov chain Monte Carlo,” Dissertationes Mathematicae, vol. 485, pp. 1–93, 2012.
- C. J. Geyer, “Practical Markov Chain Monte Carlo,” Statistical Science, vol. 7, no. 4, pp. 473 – 483, 1992.
- K. Łatuszyński, B. Miasojedow, and W. Niemiro, “Nonasymptotic bounds on the estimation error of mcmc algorithms,” Bernoulli, vol. 19, no. 5A, pp. 2033–2066, 2013.
- D. W. Gillman, “Hidden Markov chains: convergence rates and the complexity of inference,” Ph.D. dissertation, Massachusetts Institute of Technology, Boston, US, 1993.
- I. H. Dinwoodie, “A probability inequality for the occupation measure of a reversible Markov chain,” The Annals of Applied Probability, vol. 5, no. 1, pp. 37–43, 1995.
- C. A. León and F. Perron, “Optimal Hoeffding bounds for discrete reversible Markov chains,” The Annals of Applied Probability, vol. 14, no. 2, pp. 958–970, 2004.
- K.-M. Chung, H. Lam, Z. Liu, and M. Mitzenmacher, “Chernoff-hoeffding bounds for Markov chains: Generalized and simplified,” in Symposium on Theoretical Aspects of Computer Science, 2012.
- B. Miasojedow, “Hoeffding’s inequalities for geometrically ergodic Markov chains on general state space,” Statistics & Probability Letters, vol. 87, pp. 115–120, 2014.
- J. Fan, B. Jiang, and Q. Sun, “Hoeffding’s inequality for general markov chains and its applications to statistical learning,” Journal of Machine Learning Research, vol. 22, no. 139, pp. 1–35, 2021.
- A. R. Esposito and M. Mondelli, “Concentration without independence via information measures,” arXiv preprint arXiv:2303.07245, 2023.
- M. Raginsky, “Strong data processing inequalities and ϕitalic-ϕ\phiitalic_ϕ-sobolev inequalities for discrete channels,” IEEE Transactions on Information Theory, vol. 62, no. 6, pp. 3355–3389, 2016.
- S. Jarner and G. Roberts, “Polynomial convergence rates of markov chains,” Annals of Applied Probability, vol. 12, 2000.
- ——, “Convergence of heavy-tailed monte carlo markov chain algorithms,” Scandinavian Journal of Statistics, vol. 34, pp. 781–815, 2007.
- P. Del Moral, M. Ledoux, and L. Miclo, “On contraction properties of markov kernels,” Probability Theory and Related Fields, vol. 126, pp. 395–420, 2003.
- H. Hudzik and L. Maligranda, “Amemiya norm equals orlicz norm in general,” Indagationes Mathematicae, vol. 11, no. 4, pp. 573 – 585, 2000.
- C. Roberto and B. Zegarlinski, “Hypercontractivity for Markov semi-groups,” Journal of Functional Analysis, vol. 282, no. 12, p. 109439, 2022.
- M. Raginsky and I. Sason, “Concentration of measure inequalities in information theory, communications, and coding,” Foundations and Trends in Communications and Information Theory, vol. 10, no. 1-2, pp. 1–246, 2013.
- G. Roberts and J. Rosenthal, “Geometric Ergodicity and Hybrid Markov Chains,” Electronic Communications in Probability, vol. 2, no. none, pp. 13 – 25, 1997.
- S. Bubeck and N. Cesa-Bianchi, “Regret analysis of stochastic and nonstochastic multi-armed bandit problems,” Foundations and Trends® in Machine Learning, vol. 5, 04 2012.
- R. L. Dobrushin, “Central limit theorem for nonstationary Markov chains. i,” Theory of Probability & Its Applications, vol. 1, no. 1, pp. 65–80, 1956.
- A. R. Esposito, M. Gastpar, and I. Issa, “Generalization error bounds via Rényi-, f-divergences and maximal leakage,” IEEE Transactions on Information Theory, vol. 67, no. 8, pp. 4986–5004, 2021.
- I. Csiszár, “Eine informationstheoretische ungleichung und ihre anwendung auf den beweis der ergodizitat von markoffschen ketten,” Magyar. Tud. Akad. Mat. Kutató Int. Közl,, vol. 8, pp. 85–108, 1963.
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