density-guided random-walk sampling

This paper addresses the optimal design of proposal densities in Monte Carlo importance sampling, particularly under challenging settings involving dynamic updates and noise—such as in Approximate Bayesian Computation (ABC) and policy evaluation in reinforcement learning. It provides the first unified theoretical analysis of the applicability boundaries of multiple optimality criteria—including minimum variance and KL-divergence minimization—while establishing a cross-framework evaluation framework that jointly ensures theoretical guarantees and empirically comparable performance. Methodologically, the work integrates variational inference, sequential importance resampling, and annealed posterior modeling to propose a multi-proposal adaptive mechanism. Key contributions are: (1) necessary and sufficient conditions for proposal optimality across frameworks, with convergence guarantees; (2) systematic empirical validation of trade-offs among model selection accuracy, noise robustness, and computational efficiency in adaptive proposal design; and (3) an open-source empirical benchmark enabling reproducible, standardized comparison of future proposal mechanisms.

This work addresses the challenge of verifying ergodicity for adaptive MCMC algorithms in non-compact state and parameter spaces, where traditional compactness assumptions fail. By abandoning such restrictive assumptions, the authors instead introduce probabilistic bounds on the sample and parameter sequences to formulate a new set of easily verifiable sufficient conditions. Integrating tools from MCMC theory, adaptive algorithm analysis, and concentration inequalities, they establish a novel ergodicity framework that operates without compactness requirements. This approach significantly enhances the practical applicability and tractability of convergence analysis in complex real-world settings where non-compactness is inherent.

This paper addresses the poor robustness of Markov chain Monte Carlo (MCMC) methods under two pathological target distributions: *roughness* (characterized by highly oscillatory gradients causing numerical instability) and *flatness* (marked by vanishing gradients hindering exploration). We propose a unified robust MCMC framework that formally characterizes both pathologies and integrates non-local jumps, adaptive step sizes, gradient regularization, and stability-driven proposal mechanisms. Theoretical analysis exposes the convergence failure mechanisms of conventional local MCMC algorithms under such pathologies. Empirical evaluation demonstrates substantial improvements in sampling efficiency and convergence speed on canonical pathological benchmarks. Key contributions include: (i) establishing quantitative, interpretable criteria for diagnosing roughness and flatness; (ii) systematically unifying anti-pathological design principles into a coherent framework; and (iii) providing a theoretically grounded yet practical sampling methodology applicable to high-dimensional, nonsmooth, and low signal-to-noise ratio settings.

To address the susceptibility of Markov chain Monte Carlo (MCMC) algorithms to local optima within the approximate Bayesian computation (ABC) framework, this paper proposes a global-local adaptive hybrid sampling scheme. The method iteratively constructs a dynamic global proposal distribution via importance resampling, while enhancing local move efficiency through Langevin dynamics integrated with common random numbers. It introduces, for the first time, a regularization-based normalizing flow approach to learn the importance sampling distribution, enabling adaptive optimization of its shape. Furthermore, an expected squared jump distance (ESJD)-driven adaptive coordination strategy is designed to balance exploration and exploitation. Experiments demonstrate substantial improvements in sampling efficiency and convergence reliability for complex multimodal posterior distributions. The implementation is publicly available on GitHub.

This paper addresses the theoretical performance limits of lifted samplers in Markov chain Monte Carlo (MCMC), specifically analyzing the asymptotic variance relative to conventional reversible base algorithms such as Metropolis–Hastings (MH). Method: Building upon Tierney’s (1998) Markov chain convergence theory, the authors develop a unified framework integrating lifted dynamics modeling, abstraction of direction-inducing mechanisms, and spectral analysis of transition operators. Contribution/Results: The work establishes, for the first time, a universal upper bound: for any target distribution, any direction construction, and any reversible base algorithm (e.g., MH or reversible jump MCMC), the asymptotic variance of a lifted sampler is at most twice that of its base counterpart. This bound is independent of state-space ordering, the specific form of direction definition, or implementation details of the base algorithm—and incurs no additional computational overhead. The result provides a rigorous, general guarantee that lifting yields bounded improvement with controllable risk.

研究解决了轻尾目标分布下Metropolis接受率不稳定的问题，通过无梯度随机游走Metropolis采样方法，在合理选择提议方差时保持了稳定的接受概率。

This work addresses the fundamental limitation of traditional Markov chain Monte Carlo (MCMC) integration, whose convergence rate is typically restricted to $ t^{-1/2} $, hindering efficient high-precision estimation. For the first time, we rigorously incorporate the true self-avoiding walk (TSAW) mechanism into the MCMC framework by dynamically adjusting transition probabilities through penalties on empirical visitation frequencies in a finite state space, thereby discouraging redundant sampling. Leveraging adaptive dynamics based on irreducible Markov kernels and a careful analysis of occupation measures, we establish that, for any bounded integrand, the integration error converges almost surely at a rate of $ O(\sqrt{\log t}/t) $. This rate substantially improves upon classical MCMC methods, yielding significantly enhanced sample efficiency and estimation accuracy.

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 study addresses the failure of classical Monte Carlo algorithms in prediction-oriented posteriors due to the absence of an explicit density. We propose a pointwise density approximation method that integrates Markov chain Monte Carlo with numerical approximation techniques, enabling posterior sampling with rapid error decay. This approach overcomes the sampling bottleneck for distributions lacking closed-form densities and effectively quantifies uncertainty even under model misspecification. The proposed method is validated across diverse applications, including epidemiology, spatial statistics, and nuclear physics, demonstrating both effectiveness and high precision. Ultimately, this work establishes a robust new paradigm for complex Bayesian inference.

This work addresses the challenges of sampling from high-dimensional probability distributions, which are often hindered by the curse of dimensionality and metastable multimodal traps. It introduces a novel approach that repurposes generative models—such as normalizing flows and diffusion models—from their conventional data-driven paradigm into data-free auxiliary tools for efficient and accurate sampling from target distributions known only up to an unnormalized density. By integrating Monte Carlo methods with enhanced sampling techniques, the authors develop a tailored training strategy and systematically formulate a unified framework for generative-model-assisted sampling. This contribution offers a theoretically grounded and practically implementable tutorial, serving as both a methodological guide and a springboard for interdisciplinary research at the intersection of physics and machine learning.