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A rate-distortion framework for MCMC algorithms: geometry and factorization of multivariate Markov chains (2404.12589v2)

Published 19 Apr 2024 in math.PR, cs.IT, math.IT, math.OC, and stat.CO

Abstract: We introduce a framework rooted in a rate distortion problem for Markov chains, and show how a suite of commonly used Markov Chain Monte Carlo (MCMC) algorithms are specific instances within it, where the target stationary distribution is controlled by the distortion function. Our approach offers a unified variational view on the optimality of algorithms such as Metropolis-Hastings, Glauber dynamics, the swapping algorithm and Feynman-Kac path models. Along the way, we analyze factorizability and geometry of multivariate Markov chains. Specifically, we demonstrate that induced chains on factors of a product space can be regarded as information projections with respect to a particular divergence. This perspective yields Han--Shearer type inequalities for Markov chains as well as applications in the context of large deviations and mixing time comparison. Finally, to demonstrate the significance of our framework, we propose a new projection sampler based on the swapping algorithm that provably accelerates the mixing time by multiplicative factors related to the number of temperatures and the dimension of the underlying state space.

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Summary

  • The paper establishes that optimal MCMC chains are solutions to specific rate-distortion problems, connecting information theory with sampling strategies.
  • The paper develops a unified variational approach by linking distortion cost functions with the geometry and factorization of multivariate Markov chains.
  • The paper demonstrates strong numerical results and universal applicability, which may lead to more effective adaptive MCMC designs.

A Rate-Distortion Framework for MCMC Algorithms: Geometry and Factorization of Multivariate Markov Chains

This paper introduces a novel perspective on Markov Chain Monte Carlo (MCMC) algorithms by framing them within the context of rate distortion theory, offering a unified variational approach to understanding their optimality. The framework posits that common MCMC algorithms are particular instances of a generalized rate distortion problem, where the target stationary distribution is modulated by the distortion cost function. The authors build upon this by exploring the geometry and factorizability of multivariate Markov chains and emphasizing the duality between product chains and the closest independent transition matrices.

The paper's core contribution is establishing the connection between MCMC algorithms and rate distortion optimization. Specifically, the authors show that the sought-after optimal chains in various MCMC algorithms, including Metropolis-Hastings, Glauber dynamics, and swapping algorithms, are solutions to specific rate distortion problems. The distortion from the source chain to the target can be controlled by adjusting the distortion cost, resulting in different MCMC behaviors.

The proposed framework offers a deeper insight into the informativeness and efficiency of MCMC methods. This is achieved by assessing their operation as achieving a balance between maintaining reasonable proximity to the source distribution while minimizing distortion cost.

Strong Numerical Results and Bold Claims

The paper introduces robust numerical frameworks, such as Han--Shearer type inequalities for Markov chains, and explores their implications in large deviations and mixing time comparison for induced chains. It also provides a detailed analysis of the geometry of Markov chains, showing that product chains form an exponential family whereas multivariate chains with prescribed marginals constitute a mixture family.

The authors make bold claims about the universal applicability of their framework across a wide range of MCMC methods, postulating that these methods can be understood as special cases arising from different source chains and cost functions. This unifying view not only provides intuitive geometrical insights but also grounds the algorithms within the established principles of information theory.

Implications and Future Directions

The theoretical implications expand the understanding of MCMC optimization, suggesting that these algorithms naturally emerge from a deeper informational principle minimizing divergence subject to a cost. The paper's results imply that improvements or variations in these algorithms can potentially be achieved through refined control of the source chain or sophisticated constructions of distortion functions.

From a practical standpoint, this framework could influence algorithm design by focusing on constructing distortion functions and the source chains to achieve better convergence rates and sampling efficiency. Moreover, by revealing the inherent structure of these algorithms as optimal solutions in the rate-distortion sense, this approach can facilitate the construction of more effective adaptive MCMC algorithms tailored for specific problems.

Given these findings, future research could further explore this framework's application in other areas of monte-carlo simulation and decision-making processes. Additionally, ongoing developments could consider extensions to continuous space models and distributed computation scenarios. Furthermore, exploring the framework in relation to other information-theoretic measures or aligning with risk-sensitive control theory may yield novel insights into adaptive algorithm design.

Overall, this paper provides a compelling synthesis of rate distortion theory and MCMC algorithms, offering a new lens for evaluating and developing these critical tools in statistical science and beyond.

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