Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles (2211.08959v2)

Abstract: We derive the first explicit bounds for the spectral gap of a random walk Metropolis algorithm on $R^d$ for any value of the proposal variance, which when scaled appropriately recovers the correct $d^{-1}$ dependence on dimension for suitably regular invariant distributions. We also obtain explicit bounds on the ${\rm L}^2$-mixing time for a broad class of models. In obtaining these results, we refine the use of isoperimetric profile inequalities to obtain conductance profile bounds, which also enable the derivation of explicit bounds in a much broader class of models. We also obtain similar results for the preconditioned Crank--Nicolson Markov chain, obtaining dimension-independent bounds under suitable assumptions.

• The paper presents explicit bounds for the spectral gap and mixing times of RWM and pCN algorithms, highlighting their dimensional dependencies.
• It links isoperimetric profiles to conductance, enhancing the quantitative assessment of convergence behavior in Markov chains.
• The findings offer practical guidelines for tuning proposal variances in high-dimensional Bayesian inference, leading to optimized MCMC performance.

Convergence Bounds for Metropolis Markov Chains: A Comprehensive Analysis

The paper "Explicit convergence bounds for Metropolis Markov chains: isoperimetry, spectral gaps and profiles" by Andrieu et al. investigates the quantitative performance of the Random Walk Metropolis (RWM) algorithm and the preconditioned Crank–Nicolson (pCN) method in high-dimensional settings. The authors develop explicit bounds for the spectral gap, conductance, and mixing time, offering critical insights into the convergence behavior of these widely-used Markov Chain Monte Carlo (MCMC) techniques.

Key Results and Theoretical Contributions

1. Spectral Gap and Dimensional Dependence: The authors present the first explicit bounds for the spectral gap of the RWM on $\mathbb{R}^{d}$, demonstrating that appropriate scaling of proposal variance results in a spectral gap with tight $d^{-1}$ dependence on dimension for appropriately regular invariant distributions. This result is of considerable interest, as it highlights the expected deterioration of performance of RWM in high-dimensional spaces and provides a framework for analyzing convergence rates with explicit bounds.
2. Isoperimetric Profile and Conductance: By employing isoperimetric profile inequalities, the paper refines the use of conductance profile bounds, thereby extending the applicability of their results over a broader class of models. The authors establish a concrete link between the isoperimetric properties of the target distribution and the conductance of the corresponding Markov chain, providing a foundation to derive valid spectral gap bounds.
3. Mixing Time Estimates: The paper offers detailed, non-asymptotic estimates of the mixing times using spectral profiles. The authors note that these estimates account for initial rapid convergence, reflecting real-world scenarios where Markov chains achieve equilibrium faster than standard spectral gap analyses suggest.
4. Application to pCN Algorithm: Explicit bounds for the spectral gap of the pCN Markov chain are derived under certain assumptions, elucidating the chain's dimension-independent convergence behavior under specific conditions. The dimension-independence is achieved by employing a covariance operator with controlled trace, which provides insights for sampling in high-dimensional Bayesian inverse problems.

Implications and Future Directions

The results have both practical and theoretical implications for the design and application of MCMC algorithms, particularly in high-dimensional and complex sampling scenarios. Practically, these bounds provide guidance for tuning the proposal distribution's variance, suggesting a scaling approach that optimizes convergence rates with respect to the dimensionality of the state space. This is critical for statistical and computational efficiency in practice, especially when dealing with large-scale models prevalent in modern applications.

Theoretically, the connections established between isoperimetric and spectral profiles open avenues for further investigations into the robustness of MCMC methods under varying conditions of log-concavity and distributional tail behavior. The findings suggest that while certain strong assumptions (e.g., smoothness, log-concavity) facilitate explicit bound derivations, relaxing these assumptions could lead to the exploration of broader model classes, potentially making the methods adaptable to less structured target distributions.

Future research could focus on extending these techniques to non-Euclidean spaces, exploring alternative metrics for measuring chain convergence, and integrating more sophisticated proposal mechanisms that might better capture the dependency structures within complex datasets. Additionally, cross-examination of these approaches with deterministic optimization strategies could reveal synergistic methods that enhance convergence properties in challenging computational settings.

Overall, Andrieu et al. make significant strides in elucidating the interplay between proposal strategies and convergence characteristics of MCMC algorithms, offering tools to practitioners and fostering a deeper understanding of these stochastic processes.