Markov Chain Monte Carlo

Also known as MCMC

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Latest papers 145

27 papers in the last four weeks, up 50% on the four weeks before. 0.2% of all new papers.

Sep 23, 2026

Local Geometric Mixing via Dobrushin Contraction with Applications to Diffusion Path Monte Carlo and the Proximal Sampler

Local geometric mixing localizes geometric mixing by requiring geometric convergence to equilibrium in total variation only over finitely many transitions. It accommodates local convergence rates and captures rapid local equilibration, even when global mixing is much slower. We establish and discuss local geometric mixing bounds through Dobrushin contraction. We then apply this approach to Diffusion Path Monte Carlo, a recently proposed Markov chain Monte Carlo method, aimed at leveraging advances in score-based modeling, whose ideal transitions coincide with those of the Proximal Sampler. Our analysis covers both the ideal method and its implementable Metropolis-adjusted counterpart, providing mixing guarantees under minimal assumptions. For the ideal method, these guarantees complement recent spectral gap estimates, which we develop into mixing time bounds.

Stefan Oberdörster

Institute for Applied Mathematics, University of Bonn

Markov Chain Monte Carlo Posterior Sampling

Sep 21, 2026

Penalized Nonreversible Langevin for Constrained Sampling

We propose penalized nonreversible Langevin algorithms for sampling from

π(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)

, where

\mathcal C\subset\mathbb R^d

is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex

f

, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish

2

-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an

\mathcal{O}(\sqrtη)

neighborhood, where

η

is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.

Pervez Ali, Weihao Dong, Xiaoyu Wang

Department of Mathematics, Florida State University, Tallahassee, Florida, United States of America · Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong, People’s Republic of China · FinTech Thrust, Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong, People’s Republic of China

Markov Chain Monte Carlo Stochastic Gradient Langevin Dynamics Langevin Dynamics

Sep 19, 2026

Auditing Bayesian Graph Alignment: Diagnostic Comparisons and Reference Failure

Bayesian graph alignment estimates correspondence probabilities, but convergence of an alignment-score trace need not imply accurate correspondence marginals. We audit this gap on 240 new exact graph pairs from four source families, 240 larger pairs with 20-100 vertices, and a separate 60-case exact implementation check. Under an explicit edge-flip likelihood, we compare three samplers and score, marginal, indicator, categorical, and classifier-based diagnostics. Marginal disagreement improves error discrimination over score R-hat for the exact informed sampler, but its improvement for vanilla local sampling is uncertain. Assignment-based R* and short indicator panels are competitive; no diagnostic dominates across samplers and endpoints. At larger sizes, diagnostics predict subsequent marginal changes, not posterior error, and classification performance depends on the drift threshold. Disjoint-window and held-out-chain checks attenuate but preserve positive associations. Only 22 of 240 original reference sets pass an agreement screen. On forty failure-selected cases, eightfold SMC particle escalation does not resolve disagreement, whereas additional rejuvenation helps. Longer informed runs remain unstable. An elementary feasible-alignment bound demonstrates severely unrepresentative SMC and informed-chain scores in concentrated 100-vertex cases, independently of approximate reference consensus. We also exhibit common-start chains with near-zero disagreement despite exact marginal error near .967. These results support assignment-sensitive auditing while identifying limits of finite budgets, diagnostic rankings, and reference agreement as evidence of accuracy.

Melika Gorgi, Kourosh Mirsohi

Center for Complex Biological Systems University of California, Irvine Irvine, CA 92697, USA · Department of Computer Science University of California, Irvine Irvine, CA 92697, USA

Markov Chain Monte Carlo Benchmark Auditing Posterior Sampling Graph Matching

Sep 17, 2026

FlowSGS: Improving Flow Priors for Inverse Imaging with Stochastic Interpolants

Flow matching has emerged as the state-of-the-art generative model and has been used for plug-and-play (PnP) priors to solve inverse problems in computational imaging. However, existing flow-based inverse solvers make simplifying approximations in posterior sampling and/or assume linear forward models. In contrast, principled diffusion-based methods exist but are computationally expensive. To circumvent these problems, we introduce FlowSGS, a flow-based posterior sampling method using Split Gibbs Sampling (SGS) with annealed coupling, and make flow-specific interventions to the prior sampling and annealing steps, theoretically grounded in the Stochastic Interpolants (SI) framework. Using a novel timestep correction and accelerated warm-up with one-step flow models, our method is faster and more accurate than the state of the art across a range of imaging problems. We demonstrate the first use of flow-based samplers for a nonlinear inverse problem (Fourier phase retrieval) and apply one-step flow models to principled Bayesian inverse imaging.

Tianao Li, Xinhui Qian, Emma Alexander

Department of Computer Science, Northwestern University · NSF-Simons AI Institute for the Sky (SkAI) · Department of Statistics and Data Science, Northwestern University

Flow Matching Markov Chain Monte Carlo Posterior Sampling Flow-Based Generative Modeling Image Inverse Problems

Sep 13, 2026

GRPO-QPS: Target-Preserving Reinforcement Learning for Quantum Posterior Sampling

Bayesian quantum tomography requires efficient inference while preserving a posterior fixed by the prior and Born likelihood. Learned transport provides fast amortized samples, but reward tuning can reshape the generated distribution rather than improve exploration of this fixed target. We introduce GRPO-QPS, a target-preserving framework in which GRPO learns proposal behavior and an exact Metropolis correction preserves the posterior after training. Across the evaluated reconstruction benchmarks, GRPO-QPS improves over BuresTomFlow and Flow-GRPO on thermal, cat, Dicke, and cluster families, and it closely matches an exact two-qubit reference posterior. Tuned conventional MCMC is slightly stronger on several original continuous benchmarks where the available fixed proposals already match the posterior geometry well. To test whether this reflects a fundamental limitation of learned exploration, we evaluate a more challenging multimodal thermal posterior. At six qubits and 800 shots, the learned proposal achieves a minimum effective sample size of 102 per

1{,}000

likelihood calls, compared with 28 for prior independence, 27 for a tuned fixed mixture, and 20 for Haario adaptive Metropolis. A record-conditioned policy also transfers to unseen 3,000-shot records, matching or exceeding the strongest conventional baseline in all nine held-out seed-record comparisons. These results show that GRPO-QPS combines target-preserving Bayesian inference with broad gains over learned transport baselines and a sampling advantage when efficient exploration requires proposal geometry beyond the evaluated conventional kernels.

Yufeng Wang, Parivesh Priye, Lu Wei +1

Stony Brook University · Georgia Institute of Technology · Westlake University

Markov Chain Monte Carlo Quantum State Tomography Group Relative Policy Optimization Posterior Sampling

Sep 13, 2026

Evaluation of optimisation and Bayesian inference methods for reaction rates in atmospheric chemical mechanisms

Constraining reaction rate coefficients is a central challenge in the development of explicit atmospheric chemical mechanisms, particularly for autoxidation systems where many reaction pathways are only indirectly observed through high-resolution mass spectrometry. In this study, we evaluate rate-coefficient optimisation methods for a toy-case autoxidation mechanism using synthetic data with known ground truth. Two complementary approaches are compared: ODE-constrained neural-network optimisation, which provides efficient point estimates of uncertain rate coefficients, and the Markov Chain Monte Carlo (MCMC) approach, which samples the posterior distribution of rate coefficients and quantifies parameter uncertainty. The methods are tested using direct concentration observations and mass-spectral observations under different noise levels. For unperturbed and low-noise synthetic observations, both methods converged towards the known rate coefficients, with the neural-network optimiser providing faster point estimates. Under high-noise conditions (with the signal-to-noise ratio approximately S / N = 1), however, MCMC was substantially more robust in recovering the rate coefficients. The posterior analysis shows that mass-spectral aggregation broadens credible intervals even at low noise, and that high-noise mass spectra can leave many individual reaction rates weakly identifiable. Posterior predictive validation nevertheless shows how broad parameter uncertainty constrained by MCMC remains consistent with accurate reproduction of the observable mass spectrum. These results demonstrate that point-estimation and Bayesian sampling methods provide complementary information: neural-network optimisation is effective for informative data, whereas MCMC is essential for diagnosing uncertainty, non-uniqueness, and identifiability in noisy or aggregated inverse problems.

Valery Ashu, Wenqing Peng, Zhi-Song Liu +7

Department of Computational Engineering, LUT University, 15210, Lahti, Finland · Atmospheric Modelling Center-Lahti, 15140, 15140, Lahti, Finland · Institute for Atmospheric and Earth System Research / Physics, Faculty of Science, University of Helsinki, 00014, Helsinki, Finland +2

Parameter Estimation Bayesian Inverse Problems Markov Chain Monte Carlo Neural Network Optimization

Sep 10, 2026

Particle GFlowNets: Rethinking Generative Marginalization Models

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.

Tiago da Silva, Diego Mesquita, Salem Lahlou

MBZUAI · School of Applied Mathematics, Getulio Vargas Foundation

Markov Chain Monte Carlo Generative Flow Networks Generative Modeling Flow-Based Generative Modeling

Sep 8, 2026

Optimal Slice-Adaptive Tuning of Hybrid Slice Sampling

Slice sampling is a Markov chain Monte Carlo algorithm that draws its next state uniformly from a "slice"---a super-level set of the target density function---at each iteration, thereby providing automatic local adaptivity to the scale of the target. In practice the exact slice is not known, so general-purpose implementations use an approximate slice that is grown from a starting interval of length

w>0

, with a computational cost that depends on

w

. This work presents an analysis of the average per-iteration number of target density evaluations, as a function of

w

, of hybrid slice sampling with various slice-finding schemes for targets with contiguous slices. The paper uses the results of the analysis to develop automated, slice-adaptive tuning schemes along with suboptimality bounds and asymptotic convergence guarantees. Simulations demonstrate that the tuning schemes reliably yield near-optimal slice-adaptive tuning with essentially no dependence on the initial setting of

w

.

Trevor Campbell

Department of Statistics, UBC

Markov Chain Monte Carlo

Sep 7, 2026

Thermodynamic Cyclic Processes with Markov Samplers in Bayesian Inference

The concept of Markov chain Monte Carlo (MCMC) cycles, an analogy to cyclic processes in heat engines, is presented in order to examine Bayesian inference problems. In this effort, we develop adaptive ensemble schedulers that allow the tuning of external parameters of a Bayesian canonical ensemble during an MCMC run, realising the MCMC cycles in practice. We run these cycles on different statistical models. As a fundamental insight, we find (both theoretically and in practice) that such systems can produce a non-zero net work output if and only if the considered model is non-Gaussian. As such, they may serve as a measure of non-Gaussianity in Bayesian inference, which we test on an example from supernova cosmology.

Heinrich von Campe, Bjoern Malte Schaefer

Interdisziplinäres Zentrum für wissenschaftliches Rechnen, Universität Heidelberg · Tübingen AI Center, University of Tübingen · ELLIS Institute Tübingen +2

Markov Chain Monte Carlo Bayesian Inference Adaptive Sampling

Sep 7, 2026

Microcanonical Hamiltonian Monte Carlo and the Helmholtz Theorem

The recently proposed Microcanonical Hamiltonian Monte Carlo algorithm has not yet been studied in detail from a thermodynamic point of view; this work aims to fill that gap. We demonstrate how thermodynamical state variables and potentials can be derived and thereby demonstrate that the construction of the algorithm formally represents a microcanonical thermodynamic ensemble. In particular, we demonstrate (analytically and numerically) that the algorithm fulfils the Helmholtz theorem, an alternative formulation of the first law of thermodynamics. Furthermore, we construct a new sampling algorithm that extends the original to lower-dimensional inference problems. Finally, we argue that canonical Markov Chain Monte Carlo algorithms are more natural than Microcanonical Hamiltonian Monte Carlo from the thermodynamic and information-theoretic point of view.

Heinrich von Campe, Bjoern Malte Schaefer

Interdisziplinäres Zentrum für wissenschaftliches Rechnen, Universität Heidelberg · Tübingen AI Center, University of Tübingen · ELLIS Institute Tübingen +2

Markov Chain Monte Carlo

Sep 7, 2026

Smoothed Picard Hamiltonian Monte Carlo

We develop a new low-accuracy sampler, called smoothed Picard Hamiltonian Monte Carlo, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target

π\propto \exp(-V)

in dimension

d

satisfying

0 \prec αI \preceq \nabla^2 V \preceq βI

, with condition number

κ:= β/α

, smoothed Picard HMC returns a sample with

\sqrt α\,W_2(\cdot,π) \le \varepsilon

using

\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})

gradient queries. We also prove stronger

W_q

bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these

W_q

bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity

\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))

.

Fan Chen, Sinho Chewi, Jianfeng Lu +1

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. · Department of Statistics and Data Science, Yale University. · Department of Mathematics, Duke University. +1

Markov Chain Monte Carlo Log-Concave Sampling

Sep 7, 2026

Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

We study the problem of sampling from

μ(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x

on

\mathbb{R}^d

, where

V

is

α

-strongly convex and

β

-smooth, and write

κ:=β/α

. We design and analyze the Proximal Bouncy Particle Sampler (Proximal BPS), a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler. From a warm start initialization with

O(1)

Rényi divergence w.r.t.

μ

, Proximal BPS returns a sample whose law is

\varepsilon

-close to

μ

in total variation distance using

\widetilde O(\sqrtκ\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))

gradient queries in expectation.

Fan Chen, Sinho Chewi, Jianfeng Lu +1

Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. · Department of Statistics and Data Science, Yale University. · Department of Mathematics, Duke University. +1

Markov Chain Monte Carlo Log-Concave Sampling

Sep 2, 2026

HyperMC: Multi-Fidelity Hyperparameter Tuning for Stochastic Gradient MCMC

Stochastic gradient Markov chain Monte Carlo (SGMCMC) methods enable scalable Bayesian inference, but their performance depends strongly on hyperparameters such as the step size, mini-batch size, and number of leapfrog steps. Since most SGMCMC algorithms lack a Metropolis-Hastings acceptance rate, standard acceptance-based tuning methods are not directly applicable. We propose HyperMC, a multi-fidelity tuning framework that combines Hyperband-style resource allocation with kernel Stein discrepancy (KSD) evaluation. By running multiple successive-halving brackets, HyperMC balances broad exploration of a continuous hyperparameter space with increasingly accurate evaluation of promising configurations under a fixed computational budget. We further introduce Robust HyperMC, which uses global grid initialization followed by elite-guided local refinement to reduce sensitivity to random candidate generation and noisy finite-budget evaluations. Under suitable approximation and concentration conditions for the estimated KSD, we establish that the successive-halving component selects a near-optimal configuration among the sampled candidates with high probability and derive a sufficient computational budget for successful selection. Experiments on logistic regression, probabilistic matrix factorization, and Bayesian neural networks show that HyperMC improves posterior approximation or predictive calibration relative to MAMBA, grid search, and heuristic baselines, while Robust HyperMC yields more stable and reproducible tuning results.

Ming Tan, Xiyun Jiao

Department of Statistics and Data Science Southern University of Science and Technology Shenzhen, Guangdong, China · Department of Statistics and Data Science2026 Southern University of Science and Technology Shenzhen, Guangdong, China

Hyperparameter Optimization Markov Chain Monte Carlo Bayesian Inference Multi-Fidelity Optimization

Sep 1, 2026

When Metropolis and Hastings Meet Bradley and Terry: Exact MCMC From Preference Voting

Sampling from distributions conditioned on desired semantic properties is an emerging challenge in modern generative modeling. Metropolis-Hastings (MH) provides a principled route to conditional sampling, but requires access to exact pointwise target-density evaluations, which are not available in generative settings. Meanwhile, pairwise comparisons by humans or model "judge" are highly accessible and have proved valuable across diverse applications. We introduce Pref-MH, a general exact MH sampler for judge-induced conditional distributions using only stochastic binary pairwise comparisons. Our key observation is that the MH unnormalized density ratio matches the preference odds of the Bradley-Terry (BT) choice model. The central challenge is that while MH requires precise ratio computation, BT judges provide only sampled binary feedback. To this end, we develop a valid accept/reject rule whose resulting Markov chain provably converges to the target distribution. We further show that, for a fixed proposal kernel and budget, Pref-MH is optimal in the Peskun-Tierney sense among this class of exact reversible acceptance rules. Experiments on text generation and molecular design with LLM judges, as well as image generation with VLM judges, demonstrate that Pref-MH provides a practical and flexible approach to conditional sampling when comparative feedback is relatively easy to obtain.

Ariel Smogorghevski, Nir Rosenfeld, Yaniv Romano

Department of Data and Decision Sciences, Technion – Israel Institute of Technology · Department of Computer Science, Technion – Israel Institute of Technology · Department of Electrical and Computer Engineering, Technion – Israel Institute of Technology

Markov Chain Monte Carlo Markov Models Conditional Generative Modeling

Aug 31, 2026

Exact Global MCMC with Denoising Diffusion

This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.

Mitch Hill

Markov Chain Monte Carlo Markov Models Diffusion Models

Aug 28, 2026

On two proofs of $d^2$ mixing of weighted Dikin walks

We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance,

\barν

-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an

\widetilde O(d^2)

mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an

\widetilde O(d^4)

mixing bound for sampling from truncated PSD cones. Our second result establishes stronger

χ^2

-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an

\widetilde O(d^2)

mixing bound in

χ^2

-divergence, improving on the previous

\widetilde O(d^{9/4})

bound.

Yuansi Chen, Yunbum Kook

ETH ZÜrich · University of Michigan

Markov Chain Monte Carlo Log-Concave Sampling

Aug 27, 2026

Bayesian methods and Markov chain Monte Carlo algorithms for curve reconstruction and point cloud data analysis

Point-cloud data routinely captured by modern imaging and sensor technologies provide detailed geometric descriptions of objects and environments, but their analysis is hindered by large data volumes, localization noise, and missing information. In addition, existing point-cloud reconstruction pipelines typically return a single best-fit structure without uncertainty quantification. We introduce a fully Bayesian framework for representing point-cloud data and reconstructing closed curves, in which observed points are modeled as noisy perturbations of latent locations constrained to lie on the underlying curve that is regularized by a non-parametric prior. Posterior inference in our framework is carried out using a series of Markov chain Monte Carlo samplers tailored to point-cloud characteristics. Numerical experiments, including synthetic examples and real-world LiDAR datasets, show accurate reconstructions and quantified uncertainty over the recovered curves.

Asir Intesar Tushar, Ioannis Sgouralis

Department of Mathematics, University of Tennessee, Knoxville, TN, USA

Uncertainty Quantification Bayesian Inference Markov Chain Monte Carlo

Aug 25, 2026

Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincaré inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every

α\geq0

and

γ>0

and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive

α

and its benefit.

Wujun Lv, Xiaoyu Wang, Yingli Wang +1

School of Mathematics and Statistics, Donghua University, Shanghai, People’s Republic of China · Hong Kong University of Science and Technology (Guangzhou), Guangzhou, Guangdong Province, People’s Republic of China · School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China +1

Markov Chain Monte Carlo Langevin Dynamics

Aug 25, 2026

Provable Quantum-Classical Separation for Continuous Gibbs Sampling

We prove the first quantum-classical separation for a sampling problem over a continuous domain. For a class of Gibbs states

p\propto e^{-βE}

on the torus

\mathbb{T}^d

with smooth (

s

-Gevrey) potential and barrier amplitude

α=e^{βΔ}

, where

Δ= \max E-\min E

, every classical algorithm querying the value, gradient, or any higher-order derivatives of the log-density requires

Ω(α)

queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with

\tilde{O}\left(\sqrtα\right)

queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension,

e^{Ω(d)}

, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.

Enrico Olivucci, Mariia Sobchuk, Sehmimul Hoque +4

1Irr´eversible Inc., Sherbrooke, Qu´ebec, Canada · Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada · Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada +3

Markov Chain Monte Carlo

Aug 19, 2026

Heteroscedastic Neural Surrogate Modeling for Robust and Rapid Bayesian Inference in Fusion Plasma Diagnostics

Bayesian inference via Markov Chain Monte Carlo (MCMC) provides effective parameter estimation, but its real-time application in complex physical systems is hindered by heavy computational bottlenecks and extreme sensitivity to statistical noise. We address this by proposing a neural-network-based probabilistic surrogate framework for rapid and robust MCMC inference. Using fusion plasma Thomson scattering diagnostics as a challenging, noise-dominated testbed, our approach employs a dual-head architecture to simultaneously estimate the expected physical emission spectrum and the channel-wise intrinsic measurement noise variance. By optimizing a Gaussian Negative Log-Likelihood (GNLL) objective, the learned aleatoric uncertainty dynamically buffers the sampler against pathological shot noise. Evaluations demonstrate that this surrogate framework achieves > 1500x acceleration over exact physical forward models, while simultaneously reducing inference error (RMSE) by >20% compared to standard homoscedastic neural baselines, offering a highly promising paradigm for real-time physical analysis.

Liyun Zhang, Naoya Mamada, Kentaro Sakai +2

GSFS, The University of Tokyo Japan

Neural Surrogate Modeling Bayesian Inference Markov Chain Monte Carlo Heteroscedastic Regression

Aug 18, 2026

Nonlocal Transition Kernel for Efficient Learning of Restricted Boltzmann Machines

Learning restricted Boltzmann machines (RBMs) is computationally challenging because it requires expectations whose exact evaluation is generally intractable. The expectations are typically evaluated using a sampling approximation based on blocked Gibbs sampling (BGS), which is a local Markov chain Monte Carlo transition kernel. However, the locality of BGS can lead to poor sampling quality when the RBM has high energy barriers, thereby degrading learning performance. Deep tempering (DT), which performs parallel tempering over a sequence of learnable RBMs including the training RBM, alleviates this locality issue. However, DT algorithmically requires multiple steps to move through the RBM sequence to achieve a nonlocal transition. In this paper, we propose a transition kernel defined over the RBM sequence used in DT. The proposed kernel has a round-trip structure over the sequence, enabling nonlocal moves within a single transition while leaving the RBM sequence invariant. Numerical experiments show that the proposed kernel performs nonlocal transitions more frequently and achieves higher sampling quality with fewer transitions than BGS and DT. We further verify that learning based on the proposed kernel is more stable and mitigates the training failures observed with BGS- and DT-based learning.

Kaiji Sekimoto, Muneki Yasuda

Graduate School of Science and Engineering, Yamagata University, Japan

Restricted Boltzmann Machines Markov Chain Monte Carlo Parallel Tempering

Aug 17, 2026

Spectral Gaps of Hit-and-Run and Coordinate Hit-and-Run

For any convex body

\mathcal{K}\subset\mathbb{R}^{n}

containing a unit ball, the spectral gap of Hit-and-Run is

Ω(1/(n^2 C_{\mathsf{PI}}))

, where

C_{\mathsf{PI}}

is the Poincaré constant of the uniform distribution

π

over

\mathcal{K}

. This implies that Hit-and-Run converges to a distribution within

χ^2

-divergence

\varepsilon

of the uniform distribution

π

in

O(n^2 C_{\mathsf{PI}}\log(M/\varepsilon))

steps from any starting distribution

π_0

with

M=χ^2(π_{0}\,\|\,π)

, thus refining the known bound of

O(n^2 R^2 \log(M/\varepsilon))

by Lovász and Vempala (2004) in terms of the outer radius

R

; for nearly isotropic bodies, together with progress on the KLS conjecture, the complexity is

O(n^2\log n\log(M/\varepsilon))

, improving the dimension dependence from cubic to nearly quadratic while maintaining logarithmic dependence on the initial distance. It was an open problem to connect the convergence of Hit-and-Run to Poincaré/KLS constants as was done for the Ball walk by Kannan, Lovász and Simonovits (1997). Unlike Hit-and-Run, the Ball walk has an unavoidable linear dependence on (a stronger notion) of the initial warmness. We directly bound the spectral gap of the Hit-and-Run Markov chain by connecting it to functional isoperimetric constants, inspired by the recent analysis of In-and-Out. Rewriting the spectral gap in terms of dual certificates leads to the Babuška--Aziz constant studied in the analysis of PDEs; it is asymptotically bounded by the improved Poincaré constant, which we show can be bounded in terms of the usual Poincaré constant. The proof is based on duality and calculus, unlike known proofs of convergence for Hit-and-Run which are based on bounding the conductance. The same technique can be applied to Coordinate Hit-and-Run, resulting in a much improved mixing time of

O(n^3C_{\mathsf{PI}}\log(M/\varepsilon))

.

Yunbum Kook, Santosh S. Vempala

University of Michigan · Georgia Tech

Markov Chain Monte Carlo Markov Models

Aug 13, 2026

FLARE MCMC: Fidelity-based Layer-Adaptive REcursive proposals for MCMC

Markov chain Monte Carlo (MCMC) requires only the ability to evaluate the likelihood, making it a common technique for inference in complex models. However, it can have a slow mixing rate, requiring the generation of many samples to obtain good estimates and an overall high computational cost. FLARE MCMC is a multi-fidelity layered MCMC method that exploits lower-fidelity approximations of the true likelihood calculation to improve mixing and leads to overall faster performance. Such lower-fidelity likelihoods are commonly available in scientific and engineering applications where the model involves a simulation whose resolution or accuracy can be tuned. Our technique uses recursive, layered chains with simple layer tuning; it does not require the likelihood to take any form or have any particular internal mathematical structure. We demonstrate experimentally that FLARE MCMC achieves larger effective sample sizes for the same computational time across different scientific domains including hydrology and cosmology.

Harini Venkatesan, Christian Shelton, Ming-Feng Ho +2

Department of Computer Science and Engineering, University of California, Riverside · Leinweber Center for Theoretical Physics, University of Michigan · Department of Physics and Astronomy, University of California, Riverside

Markov Chain Monte Carlo Posterior Sampling Multifidelity Modeling

Aug 9, 2026

LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.

Maria-Nicoleta Crăciun, C. -H. Luke Ong, Tom Schrijvers +1

University of Oxford, UK · Nanyang Technological University, Singapore, Singapore · KU Leuven, Leuven, Belgium

Markov Chain Monte Carlo Automatic Differentiation Bayesian Inference Probabilistic Programming

Aug 4, 2026

A Direct Route to Markov Chain Convergence via Asymptotic Equivalence with the Target

For a Markov kernel

T

with an invariant probability measure

π

, we give a self-contained proof of the Markov chain convergence theorem via a criterion called asymptotic equivalence with the target. It assumes two parts about the Lebesgue decompositions of

T^{n}_{x}

and

π

for every starting point

x

: 1.) asymptotic absolute continuity: the singular mass sing

(T^{n}_{x}\midπ)

tends to

0

; 2.) asymptotic domination of the target: the singular mass sing

(π\mid T^{n}_{x})

tends to

0

, as

n \to \infty

. This criterion, on countably generated measurable spaces, is both sufficient and necessary for the Markov chain convergence. A density version of this criterion is verified on general measurable spaces in three cases: (i)

T

has a positive transition density wrt

π

; (ii)

T

consists of an absolutely continuous part with positive transition density together with an atom at the starting point, which covers the Metropolis--Hastings algorithm; (iii) the transition density is positive only after a finite number of steps that may depend on the starting point

x

. To demonstrate our general criterion, we investigate the Gibbs sampler with random scan and the parallel tempering algorithm. Furthermore, we show that in all mentioned settings Birkhoff's ergodic theorem applies, so as to obtain the strong law of large numbers. Throughout this paper, neither irreducibility, nor aperiodicity, nor recurrence, nor couplings, nor splitting constructions, nor small sets are used. In most results, the state space is a general measurable space, which carries no structure beyond a

σ

-algebra. Countable generation is only assumed where the density-free form of the criterion is stated. None of the theorems proved here is new; what is offered is a short route to a single, widely applicable Markov chain convergence criterion, which is both sufficient and necessary.

Patrick Forré

AI4Science Lab Korteweg-de Vries Institute for Mathematics University of Amsterdam

Markov Chain Monte Carlo Convergence Analysis Parallel Tempering

Aug 3, 2026

Wasserstein mixing time of the unadjusted Langevin algorithm

We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order

κ\sqrt{d}/\varepsilon

, where

κ

is the condition number,

d

is the dimension, and

\varepsilon

is the target precision: this improves by a factor of

\sqrt{d}/\varepsilon

over the previous state-of-the-art results.

Francesco Pedrotti, Peter A. Whalley

ETH Zürich

Markov Chain Monte Carlo Langevin Dynamics Log-Concave Sampling Wasserstein Distance

Aug 3, 2026

Tunneling the Loss Landscape: Bypassing Memorization with Monte Carlo Parameter Swapping

Grokking is a striking phenomenon in neural network training, where a model can undergo a prolonged period of pure memorization before abrupt generalization. While previous works have attempted to interpret it through classical machine learning mechanisms like weight norm, recent research draws an analogy from statistical physics, framing grokking as a form of computational glass relaxation. This theory defines the initial memorization as a result of fast cooling' where the training loss is reduced so quickly that a glass state is formed, followed by a slow relaxation' towards final generalization. Although providing a unifying framework for representative grokking theories, this perspective has remained largely at the theoretical on macroscopic level without direct empirical validation on training dynamics. Here we introduce a three-component framework to directly characterize the training dynamics via parameter mobility (PM), and two representative measurements from glassy dynamics: replica correlation (RC) and fractal dimension (FD). We demonstrate that standard optimization presents clear signatures of glass dynamics and inherently traps the grokking network in a kinetic arrested memorization state with a collapsed mobility, strong history dependence, and channel-like motions. This quantitative agreement motivates us to introduce State-Aware Monte Carlo Parameter Swapping (SAM-Swap), an optimization plug-in that can accelerate generalization, inspired by swap Monte Carlo algorithm widely used in glass dynamics. Comparing SAM-Swap, weight decay, and Gaussian gradient noise, we find that accelerated generalization is consistently associated with random exploration in the parameter space, similar to diffusion in physics.

Lai Shun Chan, Xiaotian Zhang, Yue Shang +2

Department of Physics, City University of Hong Kong, Hong Kong, China · Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA · 3Innovation Campus Delaware, Air Liquide, Newark, DE, USA

Neural Network Generalization Markov Chain Monte Carlo Neural Network Memorization Neural Network Training Dynamics Grokking

Jul 30, 2026

Windowed thinning and query complexity for the bouncy particle and Zigzag samplers

Let

μ(d x)\propto e^{-U(x)} d x

on

\R^d

, where

U

is

m

-strongly convex and

L

-smooth, and denote by

κ=L/m

the condition number. We consider windowed thinning, an exact simulation method for the bouncy particle sampler and the coordinate Zigzag process. The method divides a trajectory into deterministic windows and uses a gradient evaluation at the beginning of each window to construct a tractable local envelope for the event rate. Combining this construction with quantitative mixing estimates and finite-time bounds on the expected numbers of bounces and flips yields query complexity guarantees from a Gaussian cold start. For total-variation error

\varepsilon

, the expected query counts are

O(κ^{1/2}d\,(d\logκ+\log\frac1\varepsilon))

gradient queries for the bouncy particle sampler and

O(κd^{1/4}(d\logκ+\log\frac1\varepsilon))

full-gradient equivalents for Zigzag, where

d

coordinate-partial queries count as one equivalent.

Jianfeng Lu, Yinchen Luo

Markov Chain Monte Carlo Query Complexity

Jul 30, 2026

Accelerated Random-Sweep Gibbs Sampling for Gaussian Graphical Models via Dual Normal Factor Graphs

We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous

k

-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.

Borna Khodabandeh, Mehdi Molkaraie

Department of Statistics, University of Oxford, Oxford OX1 3LB, UK · Department of Signal Theory and Communications, UPC, 08034 Barcelona, Spain · Department of Statistical Sciences, University of Toronto, ON M5G 1X6, Canada

Factor Graph Optimization Markov Chain Monte Carlo Probabilistic Graphical Models

Jul 29, 2026

Equilibrium Training of Energy-Based Models with Parallel Trajectory Tempering

Energy-Based Models (EBMs) provide an interpretable framework for generative modeling of scientific data, but poor Markov Chain Monte Carlo mixing often limits their reliability. We introduce a training algorithm based on Parallel Trajectory Tempering (PTT), which exploits the continuity of the optimization path to maintain equilibrium sampling throughout learning. This enables stable and fast training on highly multimodal and data-scarce scientific datasets. Combined with reservoir sampling and adaptive optimization, PTT has a computational cost comparable to Persistent Contrastive Divergence, making it a practical replacement for standard training methods. It also provides direct estimates of thermalization times, equilibrium samples from trained models, and accurate log-likelihoods at essentially no additional cost. Experiments on Restricted Boltzmann Machines show that PTT consistently outperforms existing EBM training approaches. On discrete tabular data, it also surpasses state-of-the-art deep generative models, yielding higher-quality samples and greater robustness to overfitting and limited data. Our results make equilibrium maximum-likelihood training of EBMs practical and computationally efficient.

Nicolas Béreux, Aurélien Decelle, Cyril Furtlehner +1

Université Paris-Saclay, CNRS, INRIA, LISN, 91190 Gif-sur-Yvette, France · Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid, 28006 Madrid, Spain · Departamento de Física Teórica & IPARCOS, Universidad Complutense de Madrid, 28040 Madrid, Spain.

Markov Chain Monte Carlo Energy-Based Models Parallel Tempering