hybrid mcmc sampler

This work addresses the inefficiency and lack of a unified theoretical understanding of gradient-based MCMC methods when sampling from anisotropic and hierarchical posteriors. Starting from continuous-time dynamics, the authors systematically derive algorithms such as HMC, MALA, NUTS, and MAKLA through numerical discretization and Metropolis correction, establishing a cohesive theoretical framework. They introduce two key innovations: a globally whitened mass matrix and a stochastic step-size strategy, which together mitigate sampling difficulties arising from state-dependent curvature. The proposed approach substantially improves sampling efficiency and demonstrates superior convergence and stability in large-scale Bayesian inference and hierarchical models.

This paper proposes the first quantum-accelerated algorithmic framework with rigorous theoretical guarantees for two fundamental computational problems: (1) Markov chain Monte Carlo (MCMC) sampling from exponential-family distributions (π ∝ e⁻ᶠ), and (2) empirical risk minimization (ERM) over nonsmooth approximately convex functions. Methodologically, it integrates quantum stochastic gradient estimation, quantum Gibbs sampling, and phase estimation to design novel quantum variants of Hamiltonian and Langevin Monte Carlo under a stochastic potential oracle and a two-point joint query model. Key contributions include: (1) the first polynomial quantum speedup in dimension *d*, accuracy *ε*, and Lipschitz constant *L*; (2) a two-point synchronous quantum stochastic gradient estimator that substantially reduces gradient/function query complexity; and (3) provable quantum acceleration for nonsmooth ERM, achieving an Ω(√*n*) improvement in convergence rate over classical optimal algorithms.

This work addresses efficient gradient-driven sampling from discrete distributions. We propose a novel Hamiltonian Monte Carlo (HMC) method tailored for discrete domains. Our key innovation is the first incorporation of continuous momentum variables and Hamiltonian dynamics into discrete sampling, achieved by constructing an augmented Hamiltonian distribution that satisfies generalized detailed balance; under a linear potential, this yields rejection-free, irreversible state transitions. The method integrates auxiliary-variable proposals, momentum reversal with gradient correction, and over-relaxation of state variables to yield a differentiable, fully discrete sampling procedure. Experiments demonstrate substantial improvements in sampling efficiency and convergence speed on binary and ordinal distribution tasks, alongside enhanced exploration of target distributions. Our approach establishes a new paradigm for discrete Bayesian inference.

Unadjusted Hamiltonian Monte Carlo (HMC) and underdamped Langevin algorithms suffer from asymptotic bias in high-dimensional sampling due to numerical integration errors and lack automatic step-size adaptation. Method: This paper establishes, for the first time, a quantitative relationship between Hamiltonian energy error and asymptotic bias, and proposes the first black-box, provably bounded step-size adaptation scheme that eliminates the need for Metropolis–Hastings (MH) correction. Contribution/Results: The method rigorously controls asymptotic bias within any user-specified tolerance. Theoretical analysis—validated on Gaussian and canonical Bayesian models—confirms strong bias controllability. Empirical evaluation demonstrates several-fold speedup over MH-adjusted samplers in high dimensions, alongside markedly improved stability and overall performance. This breakthrough overcomes a key practical barrier to deploying unadjusted samplers in real-world applications.

Markov chain Monte Carlo (MCMC) sampling suffers from slow convergence in high-dimensional, non-compact spaces. Method: This paper proposes a general acceleration mechanism based on radial coordinate transformation (‖x‖ = f(z)), which reshapes the potential energy via an exponential radial reparameterization adapted to the target potential V, transforming the problem into an exponential-growth potential field in z-space—enabling exponential convergence even with standard Gaussian proposals. Contribution/Results: It provides the first universal, analytically constructible radial update scheme for arbitrary V; extends exponential convergence guarantees from Hamiltonian Monte Carlo to generalized MCMC frameworks; and rigorously quantifies how suboptimal radial transformations degrade convergence rates. Experiments demonstrate speedups of several orders of magnitude for heavy-tailed distributions; in d dimensions, isotropic Gaussian updates with scale σ ≈ 1/√d achieve near-optimal performance, with theoretically guaranteed dimension-free convergence rates.

This work addresses the slow convergence and strong dimension dependence of traditional MCMC methods in high- and infinite-dimensional Bayesian posterior sampling by proposing and analyzing two novel multi-proposal preconditioned Crank–Nicolson algorithms, termed mpCN and MTpCN. Leveraging parallelized proposal mechanisms, these algorithms achieve enhanced sampling efficiency and are shown to converge under non-convex, high-dimensional settings. The study establishes, for the first time, rigorous dimension-independent and proposal-number-uniform exponential convergence rates for both methods. By innovatively constructing two coupling schemes, the authors derive Wasserstein contraction, an $L^2$ spectral gap, and non-asymptotic statistical guarantees. The theory demonstrates that dimension-independent mixing is attainable without convexity assumptions, provided the log-likelihood is bounded and Lipschitz. Numerical experiments confirm faster warm-up and more robust parameter tuning, significantly outperforming standard pCN and independent parallel-chain approaches.

This work addresses the challenge of posterior inference in chained Markov fusion models, where shared variables between adjacent submodels complicate computation. To overcome this, the authors propose a multi-stage sequential Monte Carlo (SMC) sampler based on a divide-and-conquer strategy. This approach introduces divide-and-conquer SMC into the chained Markov fusion framework for the first time, decomposing the model using a tree structure to enable flexible composition of an arbitrary number of heterogeneous submodels and efficient Bayesian inference. Experimental results demonstrate that the method accurately estimates key parameters—such as immigration and reproduction rates—in both a synthetic example comprising 11 submodels and an ecological integrated population model, significantly improving inference efficiency and scalability.

This study addresses the high variance inherent in Markov chain Monte Carlo (MCMC) sampling and the limitation of control variate methods that rely on analytical solutions to the Poisson equation, which restricts their applicability to general target distributions. To overcome these challenges, this work proposes a variance reduction framework based on normalizing flows. By leveraging bijective transformations to map the target distribution into a reference latent space, the authors derive the transformed Markov kernel and an explicit solution to the corresponding Poisson equation. This approach extends exact control variates to arbitrary target distributions and unifies the theoretical frameworks of importance sampling and control variates. Experimental results demonstrate that the proposed method significantly outperforms state-of-the-art samplers and existing control variate techniques on both synthetic and real-world posterior distributions.

This work addresses the challenge of efficient sampling from finite mixtures of Boltzmann–Gibbs distributions by proposing a switching Hamiltonian Monte Carlo method that accurately simulates state-switching dynamics through the coupling of a symmetric numerical integrator with a Poisson jump process. Theoretically, it establishes for the first time that the integrator incurs a second-order bias and develops an error estimation framework based on the discrete Poisson equation. Furthermore, geometric ergodicity of the associated Markov chain is rigorously proven. Experimental results confirm that the method’s convergence rate aligns with theoretical predictions, significantly enhancing both the efficiency and accuracy of sampling from mixture distributions.

Traditional MCMC methods suffer from low sampling efficiency in complex distributions, while multi-proposal MCMC (MP-MCMC) offers parallelization potential but lacks a clear theoretical understanding of its behavior and optimization mechanisms under a large number of proposals. This work establishes a general theoretical framework for multi-proposal involutive MCMC in abstract state spaces, systematically analyzing the properties of transition kernels under various proposal and acceptance schemes. It introduces three novel algorithms (Algs. 1.1, 3.3, 3.4), unifies existing MP-MCMC approaches by revealing their intrinsic connections, and eliminates ineffective strategies. Through asymptotic analysis in the large-proposal limit and a unified modeling perspective, the study clarifies convergence and efficiency properties under high parallelism, providing both theoretical foundations and practical algorithms for large-scale parallel MCMC.