Contraction of Markovian Operators in Orlicz Spaces and Error Bounds for Markov Chain Monte Carlo

2024-02-17

Annual Conference Computational Learning Theory

AI Summary

This paper addresses the failure of conventional $L^p$-space methods for analyzing Markov chains whose stationary distributions exhibit heavy tails. To resolve this, we develop a contraction theory for Markov operators within Orlicz spaces. We introduce, for the first time, the notion of operator contraction under Orlicz norms and derive tight, closed-form upper bounds on the contraction coefficient. Our framework unifies characterizations of ergodicity, hypercontractivity, and the Doeblin condition, while refining the Riesz–Thorin interpolation theorem. Leveraging tools from Orlicz-space analysis, spectral theory, functional inequalities, and exponential concentration techniques, we establish sharper upper bounds on mixing times, stronger exponential concentration inequalities, and optimal lower bounds on burn-in periods. Moreover, we prove, for the first time, measure concentration phenomena for heavy-tailed Markov sequences—providing a novel theoretical foundation for analyzing heavy-tailed MCMC methods.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Other Foundations of Machine Learning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Extends Markov process convergence analysis to Orlicz spaces beyond L_p spaces
• Provides contraction coefficient bounds for Markov operators in Orlicz spaces
• Improves mixing time bounds and concentration results for MCMC methods

Innovation

• Markovian operators contract in Orlicz spaces
• Closed-form upper bound on contraction coefficient
• Improved bounds for heavy-tailed stationary distributions