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Probabilistic Methods · Monte Carlo and Sampling Methods

Jakob Robnik, Reuben Cohn-Gordon, Uros Seljak

Sampling from high dimensional distributions is a computational bottleneck in many scientific applications. Hamiltonian Monte Carlo (HMC), and in particular the No-U-Turn Sampler (NUTS), are widely used, yet they struggle on problems with a very large number of parameters or a complicated geometry. Microcanonical Langevin Monte Carlo (MCLMC) has been recently proposed as an alternative which shows striking gains in efficiency over NUTS, especially for high-dimensional problems. However, it produces biased samples, with a bias that is hard to control in general. We introduce the Metropolis-Adjusted Microcanonical sampler (MAMS), which relies on the same dynamics as MCLMC, but introduces a Metropolis-Hastings step and thus produces asymptotically unbiased samples. We develop an automated tuning scheme for the hyperparameters of the algorithm, making it applicable out of the box. We demonstrate that MAMS outperforms NUTS across the board on benchmark problems of varying complexity and dimensionality, achieving up to a factor of seven speedup.

Probabilistic Methods · Monte Carlo and Sampling Methods

Minkyu Kim, Kiyoung Seong, Dongyeop Woo, Sungsoo Ahn, Minsu Kim

We address the challenge of training diffusion models to sample from unnormalized energy distributions in the absence of data, the so-called diffusion samplers. Although these approaches have shown promise, they struggle to scale in more demanding scenarios where energy evaluations are expensive and the sampling space is high-dimensional. To address this limitation, we propose a scalable and sample-efficient framework that properly harmonizes the powerful classical sampling method and the diffusion sampler. Specifically, we utilize Monte Carlo Markov chain (MCMC) samplers with a novelty-based auxiliary energy as a Searcher to collect off-policy samples, using an auxiliary energy function to compensate for exploring modes the diffusion sampler rarely visits. These off-policy samples are then combined with on-policy data to train the diffusion sampler, thereby expanding its coverage of the energy landscape. Furthermore, we identify primacy bias, i.e., the preference of samplers for early experience during training, as the main cause of mode collapse during training, and introduce a periodic re-initialization trick to resolve this issue. Our method significantly improves sample efficiency on standard benchmarks for diffusion samplers and also excels at higher-dimensional problems and real-world molecular conformer generation.

Probabilistic Methods · Monte Carlo and Sampling Methods

Zijing Ou, Ruixiang Zhang, Yingzhen Li

Sampling from unnormalised discrete distributions is a fundamental problem across various domains. While Markov chain Monte Carlo offers a principled approach, it often suffers from slow mixing and poor convergence. In this paper, we propose Discrete Neural Flow Samplers (DNFS), a trainable and efficient framework for discrete sampling. DNFS learns the rate matrix of a continuous-time Markov chain such that the resulting dynamics satisfy the Kolmogorov equation. As this objective involves the intractable partition function, we then employ control variates to reduce the variance of its Monte Carlo estimation, leading to a coordinate descent learning algorithm. To further facilitate computational efficiency, we propose locally equivaraint Transformer, a novel parameterisation of the rate matrix that significantly improves training efficiency while preserving powerful network expressiveness. Empirically, we demonstrate the efficacy of DNFS in a wide range of applications, including sampling from unnormalised distributions, training discrete energy-based models, and solving combinatorial optimisation problems.

Probabilistic Methods · Monte Carlo and Sampling Methods

Imon Banerjee, Sayak Chakrabarty

The m-out-of-n bootstrap—proposed by \cite{bickel1992resampling}—approximates the distribution of a statistic by repeatedly drawing $m$ subsamples ($m \ll n$) without replacement from an original sample of size n; it is now routinely used for robust inference with heavy-tailed data, bandwidth selection, and other large-sample applications. Despite this broad applicability across econometrics, biostatistics, and machine-learning workflows, rigorous parameter-free guarantees for the soundness of the m-out-of-n bootstrap when estimating sample quantiles have remained elusive. This paper establishes such guarantees by analysing the estimator of sample quantiles obtained from m-out-of-n resampling of a dataset of length n. We first prove a central limit theorem for a fully data-driven version of the estimator that holds under a mild moment condition and involves no unknown nuisance parameters. We then show that the moment assumption is essentially tight by constructing a counter-example in which the CLT fails. Strengthening the assumptions slightly, we derive an Edgeworth expansion that delivers exact convergence rates and, as a corollary, a Berry–Esséen bound on the bootstrap approximation error. Finally, we illustrate the scope of our results by obtaining parameter-free asymptotic distributions for practical statistics, including the quantiles for random walk MH, and rewards of ergodic MDP's, thereby demonstrating the usefulness of our theory in modern estimation and learning tasks.

Probabilistic Methods · Monte Carlo and Sampling Methods

Wenda Chu, Zihui Wu, Yifan Chen, Yang Song, Yisong Yue

We study the problem of posterior sampling in discrete-state spaces using discrete diffusion models. While posterior sampling methods for continuous diffusion models have achieved remarkable progress, analogous methods for discrete diffusion models remain challenging. In this work, we introduce a principled plug-and-play discrete diffusion posterior sampling algorithm based on split Gibbs sampling, which we call SGDD. Our algorithm enables reward-guided generation and solving inverse problems in discrete-state spaces. We demonstrate the convergence of SGDD to the target posterior distribution and verify this through controlled experiments on synthetic benchmarks. Our method enjoys state-of-the-art posterior sampling performance on a range of benchmarks for discrete data, including DNA sequence design, discrete image inverse problems, and music infilling, achieving more than 30% improved performance compared to existing baselines.

Probabilistic Methods · Monte Carlo and Sampling Methods

Guan-Horng Liu, Jaemoo Choi, Yongxin Chen, Benjamin K Miller, Ricky T. Q. Chen

Computational methods for learning to sample from the Boltzmann distribution—where the target distribution is known only up to an unnormalized energy function—have advanced significantly recently. Due to the lack of explicit target samples, however, prior diffusion-based methods, known as _diffusion samplers_, often require importance-weighted estimation or complicated learning processes. Both trade off scalability with extensive evaluations of the energy and model, thereby limiting their practical usage. In this work, we propose **Adjoint Schrödinger Bridge Sampler (ASBS)**, a new diffusion sampler that employs simple and scalable matching-based objectives yet without the need to estimate target samples during training. ASBS is grounded on a mathematical model—the Schrödinger Bridge—which enhances sampling efficiency via kinetic-optimal transportation. Through a new lens of stochastic optimal control theory, we demonstrate how SB-based diffusion samplers can be learned at scale via Adjoint Matching and prove convergence to the global solution. Notably, ASBS generalizes the recent Adjoint Sampling (Havens et al., 2025) to arbitrary source distributions by relaxing the so-called memoryless condition that largely restricts the design space. Through extensive experiments, we demonstrate the effectiveness of ASBS on sampling from classical energy functions, amortized conformer generation, and molecular Boltzmann distributions. Codes are available at https://github.com/facebookresearch/adjoint_samplers

Probabilistic Methods · Monte Carlo and Sampling Methods

Yvann Le Fay, Nicolas Chopin, Simon Barthelmé

Variational inference seeks the best approximation of a target distribution within a chosen family, where "best" means minimizing Kullback-Leibler divergence. When the approximation family is exponential, the optimal approximation satisfies a fixed-point equation. We introduce LSVI (Least Squares Variational Inference), a gradient-free, Monte Carlo-based scheme for the fixed-point recursion, where each iteration boils down to performing ordinary least squares regression on tempered log-target evaluations under the variational approximation. We show that LSVI is equivalent to biased stochastic natural gradient descent and use this to derive convergence rates with respect to the numbers of samples and iterations. When the approximation family is Gaussian, LSVI involves inverting the Fisher information matrix, whose size grows quadratically with dimension $d$. We exploit the regression formulation to eliminate the need for this inversion, yielding $O(d^3)$ complexity in the full-covariance case and $O(d)$ in the mean-field case. Finally, we numerically demonstrate LSVI’s performance on various tasks, including logistic regression, discrete variable selection, and Bayesian synthetic likelihood, showing competitive results with state-of-the-art methods, even when gradients are unavailable.

Probabilistic Methods · Bayesian Models and Methods

Lucas Berry, David Meger

This work introduces a novel approach, Pairwise Epistemic Estimators (PairEpEsts), for epistemic uncertainty estimation in ensemble models for regression tasks using pairwise-distance estimators (PaiDEs). By utilizing the pairwise distances between model components, PaiDEs establish bounds on entropy. We leverage this capability to enhance the performance of Bayesian Active Learning by Disagreement (BALD). Notably, unlike sample-based Monte Carlo estimators, PairEpEsts can estimate epistemic uncertainty up to 100 times faster and demonstrate superior performance in higher dimensions. To validate our approach, we conducted a varied series of regression experiments on commonly used benchmarks: 1D sinusoidal data, *Pendulum*, *Hopper*, *Ant*, and *Humanoid*, demonstrating PairEpEsts’ advantage over baselines in high-dimensional regression active learning.

Probabilistic Methods · Monte Carlo and Sampling Methods

Sebastian Sanokowski, Lukas Gruber, Christoph Bartmann, Sepp Hochreiter, Sebastian Lehner

Diffusion bridges are a promising class of deep-learning methods for sampling from unnormalized distributions. Recent works show that the Log Variance (LV) loss consistently outperforms the reverse Kullback-Leibler (rKL) loss when using the reparametrization trick to compute rKL-gradients. While the on-policy LV loss yields identical gradients to the rKL loss when combined with the log-derivative trick for diffusion samplers with non-learnable forward processes, this equivalence does not hold for diffusion bridges or when diffusion coefficients are learned. Based on this insight we argue that for diffusion bridges the LV loss does not represent an optimization objective that can be motivated like the rKL loss via the data processing inequality. Our analysis shows that employing the rKL loss with the log-derivative trick (rKL-LD) does not only avoid these conceptual problems but also consistently outperforms the LV loss. Experimental results with different types of diffusion bridges on challenging benchmarks show that samplers trained with the rKL-LD loss achieve better performance. From a practical perspective we find that rKL-LD requires significantly less hyperparameter optimization and yields more stable training behavior.

Probabilistic Methods · Monte Carlo and Sampling Methods

James Cuin, Davide Carbone, O. Deniz Akyildiz

We utilise a sampler originating from nonequilibrium statistical mechanics, termed here Jarzynski-adjusted Langevin algorithm (JALA), to build statistical estimation methods in latent variable models. We achieve this by leveraging Jarzynski’s equality and developing algorithms based on a weighted version of the unadjusted Langevin algorithm (ULA) with recursively updated weights. Adapting this for latent variable models, we develop a sequential Monte Carlo (SMC) method that provides the maximum marginal likelihood estimate of the parameters, termed JALA-EM. Under suitable regularity assumptions on the marginal likelihood, we provide a nonasymptotic analysis of the JALA-EM scheme implemented with stochastic gradient descent and show that it provably converges to the maximum marginal likelihood estimate. We demonstrate the performance of JALA-EM on a variety of latent variable models and show that it performs comparably to existing methods in terms of accuracy and computational efficiency. Importantly, the ability to recursively estimate marginal likelihoods—an uncommon feature among scalable methods—makes our approach particularly suited for model selection, which we validate through dedicated experiments.

Probabilistic Methods · Monte Carlo and Sampling Methods

Luhuan Wu, Yi Han, Christian Andersson Naesseth, John Cunningham

We propose a novel sequential Monte Carlo (SMC) method for sampling from unnormalized target distributions based on a reverse denoising diffusion process. While recent diffusion-based samplers simulate the reverse diffusion using approximate score functions, they can suffer from accumulating errors due to time discretization and imperfect score estimation. In this work, we introduce a principled SMC framework that formalizes diffusion-based samplers as proposals while systematically correcting for their biases. The core idea is to construct informative intermediate target distributions that progressively steer the sampling trajectory toward the final target distribution. Although ideal intermediate targets are intractable, we develop \emph{exact approximations} using quantities from the score estimation-based proposal, without requiring additional model training or inference overhead. The resulting sampler, termed \textit{\ourmethodfull}, enables consistent sampling and unbiased estimation of the target's normalization constant under mild conditions. We demonstrate the effectiveness of our method on a range of synthetic targets and real-world Bayesian inference problems.

Applications · Chemistry and Drug Discovery

Charlie Tan, Majdi Hassan, Leon Klein, Saifuddin Syed, Dominique Beaini, Michael Bronstein, Alexander Tong, Kirill Neklyudov

Efficient equilibrium sampling of molecular conformations remains a core challenge in computational chemistry and statistical inference. Classical approaches such as molecular dynamics or Markov chain Monte Carlo inherently lack amortization; the computational cost of sampling must be paid in-full for each system of interest. The widespread success of generative models has inspired interest into overcoming this limitation through learning sampling algorithms. Despite performing on par with conventional methods when trained on a single system, learned samplers have so far demonstrated limited ability to transfer across systems. We prove that deep learning enables the design of scalable and transferable samplers by introducing Ensemble, a 200 million parameter all-atom transferable normalizing flow trained on a corpus of peptide molecular dynamics trajectories up to 8 residues in length. Ensemble draws zero-shot uncorrelated proposal samples for arbitrary peptide systems, achieving the previously intractable transferability across sequence length, whilst retaining the efficient likelihood evaluation of normalizing flows. Through extensive empirical evaluation we demonstrate the efficacy of Ensemble as a proposal for a variety of sampling algorithms, finding a simple importance sampling-based finetuning procedure to achieve superior performance to established methods such as sequential Monte Carlo. We open-source the Ensemble codebase, model weights, and training dataset, to further stimulate research into amortized sampling methods and finetuning objectives.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jose Blanchet, Yassine Hamoudi, Mario Szegedy, Guanyang Wang

The mean of a random variable can be understood as a *linear* functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this paper, we investigate whether a similar quadratic speedup is achievable for estimating *non-linear* functionals of probability distributions. We propose a \textit{quantum-inside-quantum} algorithm that achieves this speedup for the broad class of nonlinear estimation problems known as nested expectations. Our algorithm improves upon the direct application of the quantum-accelerated multilevel Monte Carlo algorithm introduced by An et. al.. The existing lower bound indicates that our algorithm is optimal up to polylogarithmic factors. A key innovation of our approach is a new sequence of multilevel Monte Carlo approximations specifically designed for quantum computing, which is central to the algorithm's improved performance.

Probabilistic Methods · Monte Carlo and Sampling Methods

Yixiu Zhao, Jiaxin Shi, Feng Chen, Shaul Druckmann, Lester Mackey, Scott Linderman

Discrete diffusion has emerged as a powerful framework for generative modeling in discrete domains, yet efficiently sampling from these models remains challenging. Existing sampling strategies often struggle to balance computation and sample quality when the number of sampling steps is reduced, even when the model has learned the data distribution well. To address these limitations, we propose a predictor-corrector sampling scheme where the corrector is informed by the diffusion model to more reliably counter the accumulating approximation errors. To further enhance the effectiveness of our informed corrector, we introduce complementary architectural modifications based on hollow transformers and a simple tailored training objective that leverages more training signal. We use a synthetic example to illustrate the failure modes of existing samplers and show how informed correctors alleviate these problems. On the Text8 dataset, the informed corrector improves sample quality by generating text with significantly fewer errors than the baselines. On tokenized ImageNet 256x256, this approach consistently produces superior samples with fewer steps, achieving improved FID scores for discrete diffusion models. These results underscore the potential of informed correctors for fast and high-fidelity generation using discrete diffusion.

Theory · Reinforcement Learning and Planning

Yang Xu, Washim Mondal, Vaneet Aggarwal

We present the first finite-sample analysis of policy evaluation in robust average-reward Markov Decision Processes (MDPs). Prior work in this setting have established only asymptotic convergence guarantees, leaving open the question of sample complexity. In this work, we address this gap by showing that the robust Bellman operator is a contraction under a carefully constructed semi-norm, and developing a stochastic approximation framework with controlled bias. Our approach builds upon Multi-Level Monte Carlo (MLMC) techniques to estimate the robust Bellman operator efficiently. To overcome the infinite expected sample complexity inherent in standard MLMC, we introduce a truncation mechanism based on a geometric distribution, ensuring a finite expected sample complexity while maintaining a small bias that decays exponentially with the truncation level. Our method achieves the order-optimal sample complexity of $\tilde{\mathcal{O}}(\epsilon^{-2})$ for robust policy evaluation and robust average reward estimation, marking a significant advancement in robust reinforcement learning theory.

Probabilistic Methods · Monte Carlo and Sampling Methods

Adrien Vacher, Omar Chehab, Anna Korba

Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions --- including all Gaussian mixtures --- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.

Deep Learning · Generative Models and Autoencoders

Weizhong Huang, Yuxin Zhang, Xiawu Zheng, Fei Chao, Rongrong Ji, Liujuan Cao

Mixture-of-Experts (MoE) architectures enable efficient scaling of large language models but face prohibitive memory demands due to massive parameterization. Existing pruning methods rely on heuristic metrics or impractical enumeration of expert subsets, leading to suboptimal performance or scalability. In this paper, we propose Shapley-MoE, an efficient pruning method for MoE models inspired by cooperative game theory. By quantifying each expert’s contribution via Shapley value, our method identifies important experts without exhaustive combination evaluations. To overcome the NP-hard complexity of exact Shapley computation, we introduce a Monte Carlo sampling strategy for efficient approximation that reduces complexity to quadratic time. However, vanilla Monte Carlo sampling still faces issues of insufficient estimation accuracy and low sampling efficiency. To address these issues, we further propose two novel methods to improve sampling accuracy and efficiency: (1) Early Truncation, which early terminates unstable sampling steps caused by overly small expert subsets, and (2) Router-Guided Importance Sampling, which prioritize sampling important expert subsets using gating activation probabilities. Both theoretical and experimental analyses show that both methods can accelerate Shapley value estimation and improve accuracy. Extensive empirical evaluations demonstrate that our pruned MoE models outperform existing expert pruning methods. Notably, when applied to the Qwen2-57B-A14B model, our method reduces the number of experts by 25% with only a 0.92 increase in perplexity and over 96.4% of the average zero-shot accuracy is maintained.

Probabilistic Methods · Monte Carlo and Sampling Methods

Paula Cordero-Encinar, Andrew Duncan, Sebastian Reich, O. Deniz Akyildiz

We introduce a novel framework for efficient sampling from complex, unnormalised target distributions by exploiting multiscale dynamics. Traditional score-based sampling methods either rely on learned approximations of the score function or involve computationally expensive nested Markov chain Monte Carlo (MCMC) loops. In contrast, the proposed approach leverages stochastic averaging within a slow-fast system of stochastic differential equations (SDEs) to estimate intermediate scores along a diffusion path without training or inner-loop MCMC. Two algorithms are developed under this framework: *MultALMC*, which uses multiscale annealed Langevin dynamics, and *MultCDiff*, based on multiscale controlled diffusions for the reverse-time Ornstein-Uhlenbeck process. Both overdamped and underdamped variants are considered, with theoretical guarantees of convergence to the desired diffusion path. The framework is extended to handle heavy-tailed target distributions using Student’s t-based noise models and tailored fast-process dynamics. Empirical results across synthetic and real-world benchmarks, including multimodal and high-dimensional distributions, demonstrate that the proposed methods are competitive with existing samplers in terms of accuracy and efficiency, without the need for learned models.

Probabilistic Methods · Monte Carlo and Sampling Methods

Emile Anand, Sarah Liaw

Thompson Sampling (TS) is widely used to address the exploration/exploitation tradeoff in contextual bandits, yet recent theory shows that it does not explore aggressively enough in high-dimensional problems. Feel-Good Thompson Sampling (FG-TS) addresses this by adding an optimism bonus that biases toward high-reward models, and it achieves the asymptotically minimax-optimal regret in the linear setting when posteriors are exact. However, its performance with \emph{approximate} posteriors, common in large-scale or neural problems, has not been benchmarked. We provide the first systematic study of FG-TS and its smoothed variant (SFG-TS) across fourteen real-world and synthetic benchmarks. To evaluate their robustness, we compare performance across settings with exact posteriors (linear and logistic bandits) to approximate regimes produced by fast but coarse stochastic-gradient samplers. Ablations over preconditioning, bonus scale, and prior strength reveal a trade-off: larger bonuses help when posterior samples are accurate, but hurt when sampling noise dominates. FG-TS generally outperforms vanilla TS in linear and logistic bandits, but tends to be weaker in neural bandits. Nevertheless, because FG-TS and its variants are competitive and easy-to-use, we recommend them as baselines in modern contextual-bandit benchmarks. Finally, we provide source code for all our experiments in https://github.com/SarahLiaw/ctx-bandits-mcmc-showdown.

Huimin Xu, Xin Mao, Feng-Lin Li, Xiaobao Wu, Wang Chen, Wei Zhang, Anh Tuan Luu

Process Reward Models (PRMs) have demonstrated promising results in mathematical reasoning, but existing process annotation approaches, whether through human annotations or Monte Carlo simulations, remain computationally expensive. In this paper, we introduce Step COmpression for Process Estimation (SCOPE), a novel compression-based approach that significantly reduces annotation costs. We first translate natural language reasoning steps into code and normalize them through Abstract Syntax Tree, then merge equivalent steps to construct a prefix tree. Unlike simulation-based methods that waste numerous samples on estimation, SCOPE leverages a compression-based prefix tree where each root-to-leaf path serves as a training sample, reducing the complexity from O(NMK) to O(N) We construct a large-scale dataset containing 509K samples with only 5% of the computational resources required by previous methods. Empirical results demonstrate that PRMs trained on our dataset consistently outperform existing automated annotation approaches on both Best-of-N strategy and ProcessBench.