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arxiv: 2606.30018 · v1 · pith:NLKJEYOBnew · submitted 2026-06-29 · 🧮 math.ST · math.PR· stat.CO· stat.TH

Error bounds for simultaneous Wasserstein contractive adaptive increasingly rare MCMC

Julian Hofstadler , Daniel Rudolf This is my paper

Pith reviewed 2026-06-30 04:10 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.COstat.TH
keywords adaptive MCMCWasserstein contractionmean squared error boundsincreasingly rare MCMCtime-average estimatorMetropolis-Hastingsnormalizing flowsdoubly intractable problems
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The pith

Explicit mean squared error bounds are derived for the time-average estimator in adaptive increasingly rare MCMC under simultaneous Wasserstein contraction of the Markov kernels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines adaptive increasingly rare Markov chain Monte Carlo algorithms that use a time-average estimator to approximate expectations. It imposes a simultaneous Wasserstein contraction condition on the entire family of kernels and obtains explicit bounds on the mean squared error of the estimator. The resulting estimates are applied to adaptive stereographic algorithms, Metropolis-Hastings schemes that adapt via normalizing flows, and a generic procedure for doubly intractable problems, where a cost analysis for reaching a target precision is also supplied.

Core claim

Under a simultaneous Wasserstein contraction assumption on the underlying family of Markov kernels, explicit bounds are derived for the mean squared error of the time-average estimator in adaptive increasingly rare MCMC algorithms.

What carries the argument

The simultaneous Wasserstein contraction assumption on the family of Markov kernels, which supplies the contraction rate used to control the accumulated error in the time-average estimator.

If this is right

  • The bounds apply directly to adaptive stereographic algorithms and to Metropolis-Hastings kernels adapted by normalizing flows.
  • A cost analysis follows that tells how many steps are needed to reach a prescribed precision for doubly intractable target distributions.
  • The same contraction-based argument yields error control for any generic adaptive algorithm that respects the simultaneous contraction hypothesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contraction condition could be checked numerically on other families of kernels to certify the error bound before running the sampler.
  • The explicit bounds make it possible to compare the computational cost of different adaptation strategies on the same footing.
  • If the contraction rate can be made uniform over a larger class of targets, the same analysis would extend to non-adaptive rare-event sampling.

Load-bearing premise

The family of Markov kernels used by the adaptive algorithm satisfies a simultaneous Wasserstein contraction condition.

What would settle it

A concrete adaptive increasingly rare MCMC run in which the observed mean squared error grows faster than the explicit bound predicted by the contraction rate would falsify the claim.

read the original abstract

We investigate adaptive increasingly rare Markov chain Monte Carlo algorithms and the associated time-average estimator for approximating expectations. Under a simultaneous Wasserstein contraction assumption on the underlying family of Markov kernels we derive explicit bounds for the mean squared error. We illustrate the applicability of our estimate through adaptive stereographic algorithms and Metropolis-Hastings schemes that employ normalizing flows for adaptation. We also consider a generic adaptive algorithm for doubly intractable problems and provide a corresponding cost analysis to achieve a desired precision.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript derives explicit mean squared error bounds for the time-average estimator in adaptive increasingly rare Markov chain Monte Carlo algorithms, conditional on a simultaneous Wasserstein contraction assumption holding uniformly over the family of adaptive Markov kernels. The bounds are illustrated through applications to adaptive stereographic algorithms, Metropolis-Hastings schemes using normalizing flows, and a generic adaptive algorithm for doubly intractable problems, along with a cost analysis for achieving desired precision.

Significance. Conditional on the stated contraction assumption, the explicit (non-asymptotic) MSE bounds provide a useful theoretical contribution for error analysis in adaptive MCMC, where such quantitative guarantees are often lacking. The illustrations with stereographic projection, normalizing-flow adaptation, and doubly intractable targets, together with the accompanying cost analysis, strengthen the practical relevance of the result.

minor comments (3)
  1. The definition and verification of the simultaneous Wasserstein contraction (stated in the abstract as the key enabling hypothesis) should be given a dedicated subsection early in the paper, with explicit reference to how it is checked or assumed in each of the three illustrative examples.
  2. [§2] Notation for the family of kernels and the time-average estimator is introduced in the abstract but should be fixed with a single consistent set of symbols in §2 before the main theorem is stated.
  3. The cost analysis for the doubly intractable case would benefit from an explicit statement of the total computational complexity (in terms of the target precision) as a displayed equation or corollary.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from explicit assumption

full rationale

The paper states a simultaneous Wasserstein contraction assumption on the family of Markov kernels as the enabling hypothesis and derives explicit MSE bounds for the time-average estimator from it. No load-bearing step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the result is a standard conditional theoretical bound with independent mathematical content. The assumption is positioned as an input rather than something verified or redefined internally.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The load-bearing element is the simultaneous Wasserstein contraction assumption, which is a domain assumption imported from MCMC theory rather than derived inside the paper.

axioms (1)
  • domain assumption simultaneous Wasserstein contraction assumption on the family of Markov kernels
    Invoked in the abstract as the condition under which explicit MSE bounds are derived.

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discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 6 canonical work pages · 3 internal anchors

  1. [1]

    Atchad´ e and Gersende Fort,Limit theorems for some adaptive MCMC algorithms with subgeometric kernels, Bernoulli16(2010), no

    [AF10] Yves F. Atchad´ e and Gersende Fort,Limit theorems for some adaptive MCMC algorithms with subgeometric kernels, Bernoulli16(2010), no. 1, 116–154. [AF12] ,Limit theorems for some adaptive MCMC algorithms with subgeometric kernels: Part II, Bernoulli18(2012), no. 3, 975–1001. [AFEB16] P. Alquier, N. Friel, R. Everitt, and A. Boland,Noisy Monte Carlo...

    2010

  2. [2]

    Brofos, M

    [BGBL22] J. Brofos, M. Gabri´ e, M. A. Brubaker, and R. R. Leder- man,Adaptation of the independent Metropolis-Hastings sam- pler with normalizing flow proposals, International Conference on Artificial Intelligence and Statistics, PMLR, 2022, pp. 5949–

    2022

  3. [3]

    Brown and G

    [BJ24] A. Brown and G. L. Jones,Exact convergence analysis for Metropolis–Hastings independence samplers in Wasserstein distances, Journal of Applied Probability61(2024), no. 1, 33–

    2024

  4. [4]

    [B LR24] C. Bell, K. Latuszy´ nski, and G. O. Roberts,Adaptive stereo- graphic MCMC, arXiv preprint arXiv:2408.11780,

    work page arXiv

  5. [5]

    Air Markov Chain Monte Carlo

    [C LR18] C. Chimisov, K. Latuszy´ nski, and G. O. Roberts,Air Markov Chain Monte Carlo, arXiv preprint arXiv:1801.09309,

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Dobrushin,Lectures on probability theory and statistics: Ecole d’et´ e de probabilit´ es de saint-flour xxiv—1994, ch

    [Dob96] R. Dobrushin,Lectures on probability theory and statistics: Ecole d’et´ e de probabilit´ es de saint-flour xxiv—1994, ch. Per- turbation methods of the theory of Gibbsian fields, pp. 1–66, Springer Berlin Heidelberg, Berlin, Heidelberg,

    1994

  7. [7]

    A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC

    [FMP11] G. Fort, ´E. Moulines, and P. Priouret,Convergence of adaptive and interacting Markov chain Monte Carlo algorithms, The Annals of Statistics39(2011), no. 6, 3262–3289. [GODMG23] L. Grenioux, A. Oliviero Durmus, ´E. Moulines, and M. Gabri´ e, On Sampling with Approximate Transport Maps, Proceedings of the 40th International Conference on Machine Le...

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  8. [8]

    Hofstadler,Optimal convergence rates of MCMC integration for functions with unbounded second moment, Journal of Ap- plied Probability62(2025), no

    [Hof25] J. Hofstadler,Optimal convergence rates of MCMC integration for functions with unbounded second moment, Journal of Ap- plied Probability62(2025), no. 3, 1069–1075. [Hof26] J. Hofstadler,Solving Poisson’s equation for Wasserstein con- tractive Markov chains, arXiv preprint arXiv:2602.19119,

    work page arXiv 2025

  9. [9]

    Habeck, D

    [HRS20] M. Habeck, D. Rudolf, and B. Sprungk,Stability of doubly- intractable distributions, Electronic Communications in Prob- ability25(2020), 1–13. 30 [HST01] H. Haario, E. Saksman, and J. Tamminen,An adaptive Metropolis algorithm, Bernoulli7(2001), no. 2, 223–242. [JO10] A. Joulin and Y. Ollivier,Curvature, concentration and error estimates for Markov...

    2020

  10. [10]

    MCMC for doubly-intractable distributions

    [LJSL16] F. Liang, I. H. Jin, Q. Song, and J. S. Liu,An adaptive exchange algorithm for sampling from distributions with in- tractable normalizing constants, Journal of the American Sta- tistical Association111(2016), no. 513, 377–393. [ LMN13] K. Latuszy´ nski, B. Miasojedow, and W. Niemiro,Nonasymp- totic bounds on the estimation error of MCMC algorithm...

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  11. [11]

    Rudolf,Explicit error bounds for Markov chain Monte Carlo, Dissertationes Math.485(2012), 93 pp

    [Rud12] D. Rudolf,Explicit error bounds for Markov chain Monte Carlo, Dissertationes Math.485(2012), 93 pp. [SK21] Y. Song and D. P. Kingma,How to train your energy-based models, arXiv preprint arXiv:2101.03288 (2021). [SV10] E. Saksman and M. Vihola,On the ergodicity of the adaptive Metropolis algorithm on unbounded domains, The Annals of Applied Probabi...

    work page arXiv 2012

  12. [12]

    Wang,Exact convergence analysis of the independent Metropolis-Hastings algorithms, Bernoulli28(2022), no

    32 [Wan22] G. Wang,Exact convergence analysis of the independent Metropolis-Hastings algorithms, Bernoulli28(2022), no. 3, 2012–2033. [Y LR24] J. Yang, K. Latuszy´ nski, and G. O. Roberts,Stereographic Markov chain Monte Carlo, The Annals of Statistics52(2024), no. 6, 2692 –

    2022

  13. [13]

    Zhang,Existence and application of optimal Markovian cou- pling with respect to non-negative lower semi-continuous func- tions, Acta Math

    [Zha00] S. Zhang,Existence and application of optimal Markovian cou- pling with respect to non-negative lower semi-continuous func- tions, Acta Math. Sin.16(2000), no. 2, 261–270. 33

    2000