Almost sure convergence rates of adaptive increasingly rare Markov chain Monte Carlo

TL;DR

This work analyzes adaptive increasingly rare MCMC (AIR MCMC) methods, where adaptation occurs less frequently over time, and proves almost-sure convergence rates for renormalized Monte Carlo sums under a Wasserstein-like contraction framework. The authors develop an augmented-state formulation and a martingale approximation based on Poisson's equation, yielding pathwise bounds of the form $\frac{1}{r(n)}\sum_{j=1}^n\left(f(X_j)-\nu(f)\right)\to 0$ a.s., with $r(n)$ matching near-LIL scalings such as $\sqrt{n}(\log n)^{1/2+\varepsilon}$ or $n^{1/2+\varepsilon}$. They treat two main regimes—uniformly bounded eccentricity and Lyapunov drift—showing that rates hold in both, and extend the results to uniformly, geometrically, and weak-Harris ergodic kernels. Importantly, the analysis does not require diminishing adaptation, and the AIR framework is shown to cover a range of practical MCMC settings via the three examples. The findings provide pathwise, almost-sure guarantees for AIR MCMC performance and offer a unified contraction-based method to assess convergence beyond traditional LLN/CLT results.

Abstract

We consider adaptive increasingly rare Markov chain Monte Carlo (MCMC) algorithms, which are adaptive MCMC methods, where the adaptation concerning the "past'' happens less and less frequently over time. Under a contraction assumption with respect to a Wasserstein-like function we deduce upper bounds of the convergence rate of Monte Carlo sums taking a renormalisation factor into account that is "almost'' the one that appears in a law of the iterated logarithm. We demonstrate the applicability of our results by considering different settings, among which are those of simultaneous geometric and uniform ergodicity. All proofs are carried out on an augmented state space, including the classical non-augmented setting as a special case. In contrast to other adaptive MCMC limit theory, some technical assumptions, like diminishing adaptation, are not needed.

Theorems & Definitions (39)

• Lemma 2.1
• proof
• Remark 3.2
• Proposition 3.3
• proof
• Theorem 3.5
• Theorem 3.6
• Lemma 3.7
• proof
• Lemma 3.8