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Probabilistic Methods · Gaussian Processes

Tim Weiland, Philipp Hennig

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. Our method uses recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we match the accuracy of classical solvers while providing structurally meaningful uncertainty. On inverse problems, we recover posteriors over nonparametric source fields in seconds --- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Probabilistic Methods · Monte Carlo and Sampling Methods

Chenyang Wang, Weizhong Wang, Yinuo Ren, Jose Blanchet, Yiping Lu

Diffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

Probabilistic Methods · Graphical Models

Mustafa Seddiqi, Tiberiu Popa, Marta Kersten-Oertel

Learning effective value heuristics for combinatorial search is difficult, as prior methods rely on surrogate supervision or costly downstream search to assess progress. We introduce a trajectory-aware probabilistic framework that models uncertainty in cost-to-go labels instead of treating them as fixed targets. Heuristic learning is cast as inference over state trajectories using an HMM-style model (Rabiner, 1989), where estimated depth-change dynamics define transitions and forward–backward inference yields soft supervision. To evaluate heuristic quality without search, we propose a large-scale local ranking metric that measures a model’s ability to order neighboring states. On the Rubik’s Cube, our approach consistently improves local ranking accuracy and downstream search performance under matched computational budgets.

Deep Learning · Large Language Models

Wenwen Qiang, Ziyin Gu, Jiahuan Zhou, Jie Hu, Jingyao Wang, Changwen Zheng, Hui Xiong

Training stability remains a critical bottleneck for Group Relative Policy Optimization (GRPO), often manifesting as a trade-off between reasoning plasticity and general capability retention. We identify a root cause as the geometric conflict between plasticity and stability gradients, which leads to destructive interference. Crucially, we argue that deterministic projection methods are suboptimal for GRPO as they overlook the intrinsic stochasticity of group-based gradient estimates. To address this, we propose Probabilistic Conflict Resolution (PCR), a Bayesian framework that models gradients as random variables. PCR dynamically arbitrates conflicts via an uncertainty-aware ``soft projection'' mechanism, optimizing the signal-to-noise ratio. Extensive experiments demonstrate that PCR significantly smooths the training trajectory and achieves superior performance in various reasoning tasks.

Probabilistic Methods · Monte Carlo and Sampling Methods

Wei Yuan, Guanyang Wang

In sampling tasks, it is common for target distributions to be known up to a normalizing constant. However, in many situations, even evaluating the unnormalized distribution can be costly or infeasible. This issue arises in scenarios such as sampling from the Bayesian posterior for tall datasets and the 'doubly-intractable' distributions. In this paper, we begin by observing that seemingly different Markov chain Monte Carlo (MCMC) algorithms, such as the exchange algorithm, PoissonMH, and TunaMH, can be unified under a simple common procedure. We then extend this procedure into a novel framework that allows the use of auxiliary variables in both the proposal and the acceptance--rejection step. Several new MCMC algorithms emerge from this framework that uses estimated gradients to guide the proposal moves. They have demonstrated significantly better performance than existing methods on both synthetic and real datasets. We also develop theory for the new framework and use it to simplify and extend results for existing algorithms.

Probabilistic Methods · Everything Else

Philip Mortimer, Cristiana Diaconu, Tommy Rochussen, Bruno Mlodozeniec, Richard E Turner

Neural Processes (NPs), and specifically Transformer Neural Processes (TNPs), have demonstrated remarkable performance across tasks ranging from spatiotemporal forecasting to tabular data modelling. However, many of these applications are inherently sequential, involving continuous data streams such as real-time sensor readings or database updates. In such settings, models should support cheap, incremental updates rather than recomputing internal representations from scratch for every new observation—a capability existing TNP variants lack. Drawing inspiration from Large Language Models, we introduce the Incremental TNP ($\texttt{incTNP}$). By leveraging causal masking, Key-Value (KV) caching, and a data-efficient autoregressive training strategy, $\texttt{incTNP}$ matches the predictive performance of standard TNPs while reducing the computational cost of updates from quadratic to linear time complexity. We empirically evaluate our model on a range of synthetic and real-world tasks, including tabular regression and temperature prediction. Our results show that, surprisingly, $\texttt{incTNP}$ delivers performance comparable to—or better than—non-causal TNPs while unlocking orders-of-magnitude speedups for sequential inference. Finally, we assess the consistency of the model's updates---by adapting a metric of "implicit Bayesianness", we show that $\texttt{incTNP}$ retains a prediction rule as implicitly Bayesian as standard non-causal TNPs, demonstrating that $\texttt{incTNP}$ achieves the computational benefits of causal masking without sacrificing the consistency required for streaming inference.

Probabilistic Methods · Monte Carlo and Sampling Methods

Yunbum Kook, Shunshi Zhang

We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $π$ satisfies a Poincaré inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.

Probabilistic Methods · Everything Else

Ian Li, Zilei Shao, Benjie Wang, Rose Yu, Guy Van den Broeck, Anji Liu

Diffusion language models theoretically allow for efficient parallel generation but are practically hindered by the "factorization barrier": the assumption that simultaneously predicted tokens are independent. This limitation forces a trade-off: models must either sacrifice speed by resolving dependencies sequentially or suffer from incoherence due to factorization. We argue that this barrier arises not from limited backbone expressivity, but from a structural misspecification: models are restricted to fully factorized outputs because explicitly parameterizing a joint distribution would require the Transformer to output a prohibitively large number of parameters. We propose **Co**upled **D**iscrete **D**iffusion (**CoDD**), a hybrid framework that breaks this barrier by replacing the fully-factorized output distribution with a lightweight, tractable probabilistic inference layer. This formulation yields a distribution family that is significantly more expressive than standard factorized priors, enabling the modeling of complex joint dependencies, yet remains compact enough to avoid the prohibitive parameter explosion associated with full joint modeling. Empirically, **CoDD** seamlessly enhances diverse diffusion language model architectures with negligible overhead, matching the reasoning performance of computationally intensive Reinforcement Learning baselines at a fraction of the training cost. Furthermore, it prevents performance collapse in few-step generation, enabling high-quality outputs at significantly reduced latencies.

Probabilistic Methods · Everything Else

Nabil Alami, Jad Zakharia, Souhaib Ben Taieb

Standard conformal prediction (CP) procedures are typically formulated in terms of $p$-values, but reliance on $p$-values alone limits flexibility, for example, when combining dependent evidence across models or data splits. Recent work has explored $e$-value formulations for conformal inference, yet a direct connection between $p$- and $e$-value formulations in CP has been missing, especially regarding their statistical efficiency. We first identify limitations of classical p-to-e calibrators in the CP setting, showing that they are not set-preserving and can lead to overly conservative prediction sets. To address this, we propose a novel P2E calibrator that converts conformal $p$-values into $e$-values without altering the prediction set induced by the original conformal $p$-value. We establish both theoretically and empirically that this calibrator yields substantial efficiency gains over existing p-to-e methods. This $e$-value formulation enables principled use of recent advances in $e$-value merging and randomization to improve conformal inference. We demonstrate its impact in two applications: cross-conformal prediction (CCP), whose variants typically provide only approximate $1-2\alpha$ coverage, and conformal aggregation (CA). In both cases, our $e$-value-based methods achieve exact $1-\alpha$ coverage while improving efficiency over standard baselines. More broadly, our approach expands the flexibility of CP and opens new directions for efficient, distribution-free uncertainty quantification.

Probabilistic Methods · Monte Carlo and Sampling Methods

Paweł Parys, Sairam Vaidya, Taylor Berg-Kirkpatrick, Loris D'Antoni

Language Models (LMs) are increasingly used in applications where generated outputs must satisfy strict semantic or syntactic constraints. Existing approaches to constrained generation fall along a spectrum: greedy constrained decoding methods enforce validity during decoding but distort the LM’s distribution, while rejection sampling (RS) preserves fidelity but wastes computation by discarding invalid outputs. Both extremes are problematic in domains such as program fuzzing, where both validity and diversity of samples are essential. We present Constrained Adaptive Rejection Sampling (CARS), an approach that strictly improves the sample-efficiency of RS without distributional distortion. CARS begins with unconstrained LM sampling and adaptively rules out constraint-violating continuations by recording them in a trie and subtracting their probability mass from future draws. This adaptive pruning ensures that prefixes proven invalid are never revisited, acceptance rates improve monotonically, and the resulting samples exactly follow the constrained distribution. In experiments on a variety of domains--e.g., program fuzzing and molecular generation--CARS consistently achieves higher efficiency--measured in the number of LM forward passes per valid sample--while also producing stronger sample diversity than both Greedy Constrained Decoding (GCD) and methods that approximate the LM's distribution.

Theory · Probabilistic Methods

Shiyuan Zhang, Qiwei Di, Xuheng Li, Quanquan Gu

Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $\pi\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al., 2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework \citep{altschuler2025shifted} to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration complexity for underdamped Langevin Monte Carlo relative to overdamped Langevin methods in regimes where $\mathrm{tr}(\mathbf{H})\ll d$.

Probabilistic Methods · Monte Carlo and Sampling Methods

Lorenzo Baldassari, Josselin Garnier, Knut Solna, Maarten de Hoop

Designing algorithms that can explore multimodal target distributions accurately across successive refinements of an underlying high-dimensional problem is a central challenge in sampling. Annealed Langevin dynamics (ALD) is a widely used alternative to classical Langevin since it often yields much faster mixing on multimodal targets, but there is still a gap between this empirical success and existing theory: when, and under which design choices, can ALD be guaranteed to remain stable as dimension increases? In this paper, we help bridge this gap by providing a uniform-in-dimension analysis of continuous-time ALD for multimodal targets that can be well-approximated by Gaussian mixture models. Along an explicit annealing path obtained by progressively removing Gaussian smoothing of the target, we identify sufficient spectral conditions—linking smoothing covariance and the covariances of the Gaussian components of the mixture—under which ALD achieves a prescribed accuracy within a single, dimension-uniform time horizon. We then establish dimension-robustness to imperfect initialization and score approximation: under a misspecified-mixture score model, we derive explicit conditions showing that preconditioning the ALD algorithm with a sufficiently decaying spectrum is necessary to prevent error terms from accumulating across coordinates and destroying dimension-uniform control. Finally, numerical experiments illustrate and validate the theory.

Probabilistic Methods · Bayesian Models and Methods

Elizabeth Baker, Alexander Denker, Jes Frellsen

Score-based diffusion models have recently been extended to infinite-dimensional function spaces, with uses such as inverse problems arising from partial differential equations. In the Bayesian formulation of inverse problems, the aim is to sample from a posterior distribution over functions obtained by conditioning a prior on noisy observations. While diffusion models provide expressive priors in function space, the theory of conditioning them to sample from the posterior remains open. We address this, assuming that either the prior lies in the Cameron-Martin space, or is absolutely continuous with respect to a Gaussian measure. We prove that the models can be conditioned using an infinite-dimensional extension of Doob's $h$-transform, and that the conditional score decomposes into an unconditional score and a guidance term. As the guidance term is intractable, we propose a simulation-free score matching objective (called *Supervised Guidance Training*) enabling efficient and stable posterior sampling. We illustrate the theory with numerical examples on Bayesian inverse problems in function spaces. In summary, our work offers the first function-space method for fine-tuning trained diffusion models to accurately sample from a posterior.

Shishang Wu, Bingjing Tang, Vinayak A Rao

Generative models such as diffusion models and transformers are powerful tools for learning complex data distributions and generating new samples. However, their black-box nature limits interpretability, and the learned distributions may violate side knowledge arising from domain expertise. We represent such side knowledge as probability distributions over noisy functions of the modeled objects and seek to minimally adjust the generative model to satisfy such constraints. Our approach is to optimize the dual of the corresponding constrained optimization problem, encoding the infinite-dimensional dual variable using a neural network. We introduce a simple and efficient score-based method for fitting the parameters of this neural network, and for simulating from the resulting adjusted distribution. We evaluate our approach on a number of synthetic tasks, as well on two real-world problems: a regularized nonparametric maximum likelihood estimation problem, and the incorporation of class-level fairness constraints into image diffusion models.

Probabilistic Methods · Everything Else

Wenbin Zhou, Shixiang Zhu

Robust optimization safeguards decisions against uncertainty by optimizing against worst-case scenarios, yet their effectiveness hinges on a prespecified robustness level that is often chosen ad hoc, leading to either insufficient protection or overly conservative and costly solutions. Recent approaches using conformal prediction construct data-driven uncertainty sets with finite-sample coverage guarantees, but they still fix coverage targets a priori and offer little guidance for selecting robustness levels. We propose a new framework that provides distribution-free, finite-sample guarantees on both miscoverage and regret for any family of robust predict-then-optimize policies. Our method constructs valid estimators that trace out the miscoverage--regret Pareto frontier, enabling decision-makers to reliably evaluate and calibrate robustness levels according to their cost--risk preferences. The framework is simple to implement, broadly applicable across classical optimization formulations, and achieves sharper finite-sample performance. This paper offers a principled data-driven methodology for guiding robustness selection and empowers practitioners to balance robustness and conservativeness in high-stakes decision-making.

Deep Learning · Large Language Models

Ziqin Wang, Hengyuan Zhao, Qixin Sun, Yilin Li, Kaiyou Song, Xiaolin Hu, Qingpei Guo, Linjiang Huang, Si Liu

Mixture of Experts architectures have recently advanced the scalability and adaptability of Large Language Models for continual multimodal learning. However, extending these models to accommodate sequential tasks remains challenging. As new tasks arrive, naive model expansion leads to rapid parameter growth, while modifying shared routing components often causes catastrophic forgetting, undermining previously learned knowledge. To address these issues, we propose CoPE, a continual learning framework for LLMs that requires no replay data of previous tasks and ensures both parameter efficiency and robust knowledge retention. Our approach introduces the Probe-Guided Knowledge Extension mechanism, which uses probe experts to dynamically determine when and where new experts should be added, enabling adaptive and minimal parameter expansion tailored to task complexity. To support inference without task labels, we further incorporate a Probabilistic Task Locator that dynamically matches inputs to the correct task-specific components. To handle the practical issue that task labels are unknown during inference, we leverage a VAE-based reconstruction strategy to identify the most suitable router by matching input distributions, allowing automatic and accurate expert allocation. This design mitigates routing conflicts and catastrophic forgetting, enabling robust continual learning without explicit task labels. Extensive experiments on the CoIN benchmark, covering eight diverse VQA tasks, demonstrate that CoPE delivers strong continual learning performance with a compact model size, significantly reducing forgetting and parameter overhead compared to prior methods. These results showcase the effectiveness and scalability of our approach for parameter-efficient continual learning in large language models. Our code will be open-sourced soon.

Probabilistic Methods · Spectral Methods

Genki Osada

While diffusion models enable new approaches for estimating Local Intrinsic Dimension (LID), existing methods fail in high-dimensional spaces where noise from vast normal directions overwhelms the tangent signal. We propose Local Hessian Spectral Dimension (LHSD), which resolves this by applying spectral filtering to the log-density Hessian, explicitly cutting off large eigenvalues associated with normal directions to count zero-curvature tangent directions. Implemented using Stochastic Lanczos Quadrature (SLQ), LHSD avoids full Hessian construction, achieving linear scalability with dimension $D$. Experiments on synthetic and real data confirm LHSD’s superior robustness and its utility in detecting memorization in large-scale diffusion models.

General Machine Learning · Kernel methods

Johannes Teutsch, Oleksii Molodchyk, Marion Leibold, Timm Faulwasser, Armin Lederer

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

Theory · Probabilistic Methods

Tom Yates, Yuzhou Cheng, Ignacio Alzugaray, Danyal Akarca, Pedro Mediano, Andrew Davison

Belief Propagation (BP) is a powerful algorithm for distributed inference in probabilistic graphical models, however it quickly becomes infeasible for practical compute and memory budgets. Many efficient, non-parametric forms of BP have been developed, but the most popular is Gaussian Belief Propagation (GBP), a variant that assumes all distributions are locally Gaussian. GBP is widely used due to its efficiency and empirically strong performance in applications like computer vision or sensor networks – even when modelling non-Gaussian problems. In this paper, we seek to provide a theoretical guarantee for when Gaussian approximations are valid in highly non-Gaussian, sparsely-connected factor graphs performing BP (common in Spatial AI). We leverage the Central Limit Theorem to prove mathematically that variables’ beliefs under BP converge to a Gaussian distribution in complex, loopy factor graphs obeying our 4 key assumptions. We then confirm experimentally that variable beliefs become increasingly Gaussian after just a few BP iterations in a stereo depth estimation task.

Probabilistic Methods · Monte Carlo and Sampling Methods

Hoang Phuc Hau Luu, Zhongjian Wang

We study a sampling problem whose target distribution is $\pi \propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line of DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $\pi$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Moreover, numerical experiments show that DC-LA produces accurate distributions in synthetic settings and reliably provides uncertainty quantification in a real-world Computed Tomography application.