Ergodicity of an Adaptive MCMC Sampler under a Probability Bound

Abstract: This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to converge to the target distribution. These conditions aim to make more easily usable classical conditions formulated over the transition kernels, without needing, as was done in other works, to assume the compactness of both sample and parameter spaces. The condition of compactness is replaced here with a probability bound over the sequence of both samples and parameters.

• The paper demonstrates that adaptive MCMC achieves ergodicity by bounding state and kernel parameters probabilistically instead of using compactness.
• It introduces a framework that decouples adaptation control from principal kernel parameters, ensuring stability through probability bounds and topological continuity.
• The theoretical innovations validate adaptive Metropolis methods, proving ergodicity in high-dimensional, unbounded domains with significant practical implications.

Ergodicity of Adaptive MCMC under Probability Bounds

Motivation and Context

Adaptive Markov Chain Monte Carlo (MCMC) algorithms provide increased sampling efficiency by tuning parameters online, but guaranteeing ergodicity—the convergence of the chain to a target distribution—remains technically challenging, especially when moving beyond compact domains for state and parameter spaces. Classical ergodicity proofs for adaptive MCMC typically rely on compactness assumptions or manage adaptation with additional constraints, such as bounding jumps or deterministic kernel switching outside compacts. The paper "Ergodicity of an Adaptive MCMC Sampler under a Probability Bound" (2602.06568) addresses this gap by providing sufficient, practical topological and probabilistic conditions for ergodicity without requiring compactness, instead employing a probability bound on the trajectories of both state and parameter sequences.

Framework and Extended Conditions

The paper builds upon coupling-based ergodicity criteria (notably Roberts and Rosenthal, 2007) and translates the standard "containment" and "diminishing adaptation" requirements—typically stated in total variation and kernel continuity—into probabilistic bounds explicitly on the sequences $(\vxt)_t$ and $(\vparamkt)_t$, with adaptation variables separated out in $(\vparamut)_t$. This decoupling allows greater flexibility: adaptation control, for example, can potentially diverge without affecting ergodicity if the state and principal kernel parameters remain bounded in probability. Furthermore, sufficient ergodicity conditions are established as:

• Containment in Probability: For all $\delta>0$, there exists a compact $C_\delta$ in $X \times \Theta$ such that $\Pr((\vxt, \vparamkt) \in C_\delta) \geq 1-\delta$ uniformly for $t$.
• Diminishing Adaptation in Probability: The metric distance $d(\vparamkt, \Theta_\infty)$ converges to zero in probability (with $\Theta_\infty$ a limiting parameter value).
• Kernel Ergodicity: For each fixed $\paramk$, $P_{\paramk}$ is ergodic w.r.t. $\pi$.

Notably, these conditions are shown to suffice even on unbounded domains and parameter spaces, provided regularity via continuity and tightness is maintained.

Key Technical Innovations

Topological Probability Bound

The classical approach (compactness) is replaced by requiring that, for any $\delta>0$, the state-parameter process remains within a compact set with probability $\geq1-\delta$. This is more generally applicable and operationally verifiable, especially in numerical contexts where compacts may be arbitrarily large.

Separating Parameters

By decomposing the adaptation parameter space into kernel ($\paramkt$) and adaptation control ($\paramut$), the paper permits unconstrained evolution of adaptation variables so long as the process governing kernel parameters and state stays stable probabilistically.

Continuity and Tightness

Rather than requiring full kernel continuity (which is infeasible for MH kernels due to their singular Dirac components), continuity is imposed separately on the absolutely continuous and discrete parts, circumventing the difficulties in total variation convergence required in classical analysis.

Local Tightness

The paper leverages local tightness properties to extend dominated convergence arguments to non-compact spaces. For any compact subset of the state-parameter space, the family of measures generated by adaptive kernels is tight, enabling continuity results to be proven via Dini’s theorem and dominated convergence.

Diminishing Adaptation

A weakened version is shown to suffice: for any compact $C_X \subset X$, $$\sup_{x \in C_X} \|P_\theta(x, \cdot) - P_{\theta'}(x, \cdot)\|$$ converges in probability to zero as kernel parameters adapt.

Main Theoretical Results

Ergodicity without Compactness

The core theorem demonstrates that if the sequence $(\vxt, \vparamkt)_t$ is bounded in probability and the adaptation reduces (in the principal kernel parameters), then—provided regularity—the adaptive MCMC chain is ergodic w.r.t. $\pi$, under minor additional continuity and kernel ergodicity hypotheses.

Sufficient Conditions for Metropolis-Hastings

In the case of adaptive Metropolis-Hastings (MH), the paper fully characterizes when adaptive MH inherits ergodicity, provided the density of the target and proposal are continuous, and the proposal adaptation variables and state sequence are bounded in probability.

Applications

Two illustrative examples are provided:

• Adaptive Metropolis (AM-MCMC): Classical covariance matrix adaptation, proven ergodic if the state sequence is $L^2$ bounded.
• Rank-One MH Covariance Matrix Adaptation: Combining MH adaptation with CMA-ES inspired covariance adaptation, ergodicity is proven subject to probabilistic bounds on state and key adaptation variables.

Strong Claims

The paper asserts that ergodicity proofs with solely compactness-based conditions are misleading when the algorithm is purported to operate on an unbounded domain. The main result is a strong counterclaim to classical approaches: ergodicity can be achieved for adaptive MCMC without compactness assumptions, using probability bounds and topological continuity.

Numerical Implications and Contradictions

While explicit numerical results are not reported, the theoretical structure directly supports practical adaptive MCMC on unbounded domains, e.g., in Bayesian statistics and stochastic optimization. The direct translation of classical ergodicity constraints into probabilistic bounds enables robust implementation for high-dimensional (and unconstrained) settings typical in contemporary applications, without resorting to artificial compactifications.

Theoretical Implications and Future Directions

This framework is likely to inform subsequent rigorous analysis of sophisticated adaptive algorithms (e.g., Hamiltonian MC with adaptation), as well as facilitating verification protocols for ergodicity in highly automated, machine-learned adaptive schemes. It obviates the need to artificially restrict parameter evolution and lays a foundation for probabilistic stability analysis in stochastic process controlled adaptation. Extension to more general adaptation mechanisms, multi-kernel settings, or hierarchical adaptation (as in MCMC within variational inference or reinforcement learning) is anticipated.

Conclusion

The paper provides a rigorous generalization of adaptive MCMC ergodicity theory, moving beyond classical compactness assumptions and re-casting sufficiency criteria in terms of probability bounds and topological continuity. By decoupling adaptation control from principal kernel parameters, and providing technical tools for continuity and tightness, the results both broaden and refine the practical and theoretical basis for guaranteed convergence in adaptive MCMC algorithms. This framework is expected to serve as a reference point for future adaptive MCMC analysis and deployment in high-dimensional, unconstrained scenarios (2602.06568).