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Theory · Probabilistic Methods

Yang Yang, Duo Zheng, Sandeep Jain, Kai Zhang, Ping-Shou Zhong

This paper introduces a new family of adaptive, distribution-free independence tests for multivariate random vectors based on binary expansion coefficients, supported by a tractable asymptotic theory. Our first key contribution establishes a general equivalence between independence testing and testing cross-covariances among exponentially many binary expansion interaction coefficients, applicable to broad sample spaces and not limited to kernel-induced representations. While this exponential interaction structure makes naive construction and computation infeasible, we overcome this challenge by reformulating the proposed tests as a class of U-statistics and deriving an explicit kernel representation that enables scalable and efficient computation. Exploiting the multiscale nature of binary expansions, the proposed framework automatically adapts to unknown dependence structures by selectively truncating higher-order interactions, yielding both strong power and clear interpretability. To further enhance power and computational efficiency, we introduce an adaptive weighted aggregation procedure, termed wa-dCoBET, which combines a baseline Covariance Binary Expansion Test (CoBET) with a distance-measure–based CoBET. Extensive simulations and a real-data application demonstrate that wa-dCoBET consistently matches or outperforms HSIC and distance covariance, particularly in higher-dimensional and non-monotone settings, while maintaining accurate type I error control.

Deep Learning · Large Language Models

XUCHEN, Dawei Yang, Zhixuan Chen

Diffusion-based large language models (DLLMs) have shown promise for non-autoregressive text generation, but their deployment is constrained by large model sizes and heavy computational costs. Post-training quantization (PTQ), a widely used method for compressing and accelerating Large Language Models (LLMs), suffers from severe accuracy degradation and reduced generalization performance when directly applied to DLLMs (e.g., AWQ suffers a 16% accuracy drop on LLADA under W4A4). This paper explores how the unique mechanisms of Dynamic Language Models (DLLMs) conflict with quantization, identifying three core issues: 1) During the iterative generation process of DLLMs, dynamic masking ratios are inherently involved, leading to notable differences in token distributions across decoding steps. Unfortunately, these distinct distributions are not sufficiently captured by current PTQ calibration approaches; 2) Quantization errors propogate and accumalte progressively during iterations in DLLMs, leading to a gradual decline in the performance of quantized models as decoding steps advance; 3) The stability of unmasked tokens, combined with the probabilistic nature of masked tokens, gives rise to an overall feature distribution that is uncoordinated and unsuitable for PTQ. To address these issues, we propose DLLMQuant, a PTQ framework tailored for DLLMs, which incorporates three novel techniques: 1) Temporal-Mask Adaptive Sampling (TMAS), a calibration method that accounts for both time and mask factors, with the capacity to capture distributions across timesteps. 2) Interaction-Aware Activation Quantization (IA-AQ), which utilizes bidirectional attention scores to identify important tokens, and prioritizes these tokens when minimizing quantization error. 3) Certainty-Guided Quantization (CGQ) incorporates mask status and token scores as core weighting criteria for error compensation, enabling PTQ to better align with the unique weight distribution of DLLMs. Experiments show that DLLMQuant achieves significant performance gains (e.g., over 10-point accuracy improvement on GSM8K for LLADA under 4-bit quantization) while enhancing efficiency.

Probabilistic Methods · Bayesian Models and Methods

Gyeonghun Kang, Changwoo Lee, Xiang Cheng

Prior-data fitted networks (PFNs) have recently emerged as a powerful approach for Bayesian prediction tasks, approximating the posterior predictive distribution (PPD) through in-context learning. Despite their strong empirical performance and ability to go beyond point predictions, theoretical understandings of the algorithmic capability of transformers to learn distributions in context are still lacking. Focusing on Gaussian process regression problems, we show by construction that transformers can implement a gradient descent algorithm targeting the posterior predictive mean and variance, followed by nonlinear mappings that yield binned probabilities of PPD. We study the error bounds of the approximated PPD in terms of attention depth and bin resolution. Based on these results, we further demonstrate the key role of normalization and the choice of attention depth in enabling the extrapolation abilities of transformers beyond the pretraining sample size range. We conduct simulations that corroborate our findings, providing insight into the expressivity of PFNs targeting PPDs and how architectural choices may influence generalization capabilities.

Deep Learning · Large Language Models

Hadi Reisizadeh, Jiajun Ruan, Yiwei Chen, Soumyadeep Pal, Sijia Liu, Mingyi Hong

Unlearning in large language models (LLMs) is critical for regulatory compliance and for building ethical generative AI systems that avoid producing private, toxic, illegal, or copyrighted content. Despite rapid progress, in this work we show that \textit{almost all} existing unlearning methods fail to achieve true forgetting in practice. Specifically, while evaluations of these `unlearned' models under deterministic (greedy) decoding often suggest successful knowledge removal using standard benchmarks (as has been done in the literature), we show that sensitive information reliably resurfaces when models are sampled with standard probabilistic decoding. To rigorously capture this vulnerability, we introduce \texttt{leak@$k$}, a new meta-evaluation metric that quantifies the likelihood of forgotten knowledge reappearing when generating $k$ samples from the model under realistic decoding strategies. Using three widely adopted benchmarks, TOFU, MUSE, and WMDP, we conduct the first large-scale, systematic study of unlearning reliability using our newly defined \texttt{leak@$k$} metric. Our findings demonstrate that knowledge leakage persists across methods and tasks, underscoring that current state-of-the-art unlearning techniques provide only limited forgetting and highlighting the urgent need for more robust approaches to LLM unlearning. We propose an algorithm, termed Robust Unlearning under LEak@$k$ metric (\texttt{RULE}), which serves as an initial step toward addressing this concern. We demonstrate that \texttt{RULE} provides an unlearned model for TOFU benchmark with no information leakage for a large number of generation samples.

Deep Learning · Generative Models and Autoencoders

Yangtian Zhang, Zhe Wang, Arthur Gretton, ZHITAO YING, David van Dijk, Michalis Titsias, Jiaxin Shi

Non-monotonic sequence generation methods, such as masked diffusion models, provide a flexible alternative to left-to-right autoregressive modeling by allowing tokens to be generated in non-fixed and prescribed orders. Despite their practical advantages, most existing non-monotonic models are order-agnostic and rely on a fixed-length masked token grid, limiting their ability to support variable-length generation and adaptive insertion order. In this work, we introduce a probabilistic framework for learning insertion order in variable-length insertion models. We formalize a bijective correspondence between insertion trajectories and permutations, which enables an exact reparameterization of the data likelihood as a sum over permutations. Building on this result, we propose the **Insertion Process (IP)**, a stochastic generative model that jointly learns *where* to insert, *what* to insert, and *when* to terminate, trained via permutation-based variational inference. Unlike prior masked or fixed-canvas approaches, IP natively supports variable-length generation and learns data-driven preferences over insertion orders. Experiments on planning benchmarks and molecular SMILES generation demonstrate that learning insertion order improves both modeling quality and generalization in domains without a canonical left-to-right structure.

Theory · Probabilistic Methods

Laura Lützow, Michael Eichelbeck, Mykel Kochenderfer, Matthias Althoff

Conformal prediction is a popular uncertainty quantification method that augments a base predictor to return sets of predictions with statistically valid coverage guarantees. However, current methods are often computationally expensive and data-intensive, as they require constructing an uncertainty model before calibration. Moreover, existing approaches typically represent the prediction sets with intervals, which limits their ability to capture dependencies in multi-dimensional outputs. We address these limitations by introducing zono-conformal prediction, a novel approach inspired by interval predictor models and reachset-conformant identification that constructs prediction zonotopes with assured coverage. By placing zonotopic uncertainty sets directly into the model of the base predictor, zono-conformal predictors can be identified via a single, data-efficient linear program. While we can apply zono-conformal prediction to arbitrary nonlinear base predictors, we focus on feed-forward neural networks in this work. Aside from regression tasks, we also construct optimal zono-conformal predictors in classification settings where the output of an uncertain predictor is a set of possible classes. We provide probabilistic coverage guarantees and present methods for detecting outliers in the identification data. In extensive numerical experiments, we show that zono-conformal predictors are less conservative than interval predictor models and standard conformal prediction methods, while achieving a similar coverage over the test data.

Probabilistic Methods · Monte Carlo and Sampling Methods

Mehmet Berk Sahin, Behzad Sharif, Abolfazl Hashemi

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities constitutes a fundamental challenge in machine learning, particularly when gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a robust mechanism for gradient-based sampling, its extension to the derivative-free setting is frequently compromised by high variance and a lack of rigorous convergence guarantees in non-convex landscapes. In this work, we propose a principled variance-reduced zeroth-order Langevin dynamics framework that addresses these limitations for both general non-log-concave black-box sampling and inverse problems utilizing pre-trained score-based generative priors. We introduce a novel gradient estimator that significantly mitigates the variance inherent in traditional zeroth-order methods, enabling stable navigation of complex, multimodal posterior distributions. Theoretically, we establish the first non-asymptotic complexity bounds for this class of algorithms, proving convergence to the target distribution in terms of $\varepsilon$-relative Fisher information, and, under a Poincare inequality, squared total variation distance, specifically for non-log-concave densities. We empirically validate our framework, demonstrating superior mixing and sampling accuracy on standard black-box benchmarks and establishing state-of-the-art performance for derivative-free linear and nonlinear inverse problems.

Probabilistic Methods · Bayesian Models and Methods

Luwei Wang, Dagmara Panas, Ke Wang, Sohan Seth, Bruce Guthrie

Constrained clustering incorporates prior knowledge in the form of pairwise constraints to guide data partitioning. While effective, existing Bayesian approaches are often limited in scalability to large datasets and provide weak interpretability due to the lack of explicit feature relevance modeling. We propose BASIL, a scalable Bayesian semi-supervised clustering framework that leverages stochastic variational inference to jointly infer cluster assignments and feature importance weights. This joint formulation enables the identification of discriminative features consistent with the imposed constraints. To robustly handle noisy or inconsistent supervision, BASIL introduces an adaptive constraint-weighting mechanism that down-weights unreliable constraints. Experiments on synthetic and real-world benchmarks demonstrate that our approach achieves competitive clustering performance while improving scalability and interpretability over existing baselines. We further demonstrate applicability to large-scale health data, including medical imaging and electronic health records.

Probabilistic Methods · Monte Carlo and Sampling Methods

Benjamin Patrick Evans, Sumitra Ganesh, Leo Ardon

Large language models (LLMs) achieve remarkable generative performance, yet their output quality is dependent on the decoding strategy. While sampling-based methods (e.g., top-$k$, nucleus) and search-and-select based methods (e.g., beam search, best-of-$n$, majority voting) can improve upon greedy decoding, both approaches suffer from limitations: sampling commits to a single path, while search often expends excessive computation regardless of task complexity. We introduce **Entropy-informed DEcodiNg** (EDEN), a plug-and-play, model-agnostic decoding framework that adaptively allocates computation based on the model’s own uncertainty, approximating higher width beam search with *fewer generations required*. At each generation step, EDEN estimates the entropy of the output token distribution and adjusts the branching factor monotonically with the entropy, expanding more candidates in high-entropy regions and following a greedier path in low-entropy regions, improving sample efficiency. Experiments across complex tasks, including mathematical reasoning, code generation, and scientific questions, demonstrate that EDEN consistently improves output quality over existing decoding strategies, achieving better trade-offs between accuracy and token generations than fixed beam search approaches. By treating next token selection as a noisy maximisation problem, we prove that branching factors monotone in entropy are guaranteed to find better (i.e. more probable) continuations than any fixed branching factor within the same total computation budget, motivating the dynamic branching.

Probabilistic Methods · Bayesian Models and Methods

Weifeng Zhang, Wenyuan Zhao, Amir Hossein Rahmati, Yucheng Wang, Zhiyuan Wang, Chao Tian, Xiaoning Qian

Principled uncertainty quantification (UQ) is increasingly recognized as essential for trustworthy artificial general intelligence (AGI). Bayesian Low-Rank Adaptation (LoRA) provides a principled mechanism for uncertainty-aware fine-tuning of large language models (LLMs). However, existing techniques either face scalability constraints, e.g. Laplace-LoRA, or rely on approximate inference schemes that lead to poorly calibrated posterior uncertainty, often manifesting as overconfident predictions under distribution shift. To address this challenge, we propose GPan-LoRA, the first scalable Gaussian Process (GP)-based framework for Bayesian LoRA, which integrates neural network-based sparse GP approximations with amortized variational inference. By preserving the Bayesian function prior and posterior semantics intrinsic to GPs, GPan-LoRA achieves a faithful balance between computational scalability and principled UQ. Empirically, GPan-LoRA produces well-calibrated uncertainty that remains reliable under distribution shift, mitigating overconfident failures while preserving competitive task performance.

Deep Learning · Robustness

Eitan Wagner, Omri Abend

Language modeling has shifted in recent years from a distribution over strings to prediction models with textual inputs and outputs for general-purpose tasks. This position paper highlights the often overlooked implications of this shift for the use of large language models (LLMs) as probability estimators, especially for world probabilities. In light of the theoretical distinction between distribution estimation and response prediction, we examine LLM training phases and common use cases for LLM output probabilities. We show that the different settings lead to distinct, potentially conflicting, desired output distributions. This lack of clarity leads to pitfalls when using output probabilities as event probabilities. Our position is that second-order prediction—incorporating probabilities as part of the output—is the only theoretically sound method. We conclude with suggestions for potential directions to improve the probabilistic soundness of this method.

Probabilistic Methods · Everything Else

Mengqi Chen, Thomas Berrett, Theodoros Damoulas, Michele Caprio

Distributionally robust optimisation (DRO) minimises the worst-case expected loss over an ambiguity set that can capture distributional shifts in out-of-sample environments. While Huber (linear-vacuous) contamination is a classical minimal-assumption model for an $\varepsilon$-fraction of arbitrary perturbations, including it in an ambiguity set can make the worst-case risk infinite and the DRO objective vacuous unless one imposes strong boundedness or support assumptions. We address these challenges by introducing bulk-calibrated credal ambiguity sets: we learn a high-mass bulk set from data while considering contamination inside the bulk and bounding the remaining tail contribution separately. This leads to a closed-form, finite $\mathrm{mean}+\sup$ robust objective and tractable linear or second-order cone programs for common losses and bulk geometries. Through this framework, we highlight and exploit the equivalence between the imprecise probability (IP) notion of upper expectation and the worst-case risk, demonstrating how IP credal sets translate into DRO objectives with interpretable tolerance levels. Experiments on heavy-tailed inventory control, geographically shifted house-price regression, and demographically shifted text classification show competitive robustness-accuracy trade-offs and efficient optimisation times, using Bayesian, frequentist, or empirical reference distributions.

Probabilistic Methods · Everything Else

Yingyan Zeng, Zipan Huang, Xiaoyu Chen

Existing online change-point detection (CPD) methods rely on fixed-dimensional Euclidean summaries, implicitly assuming that distributional changes are well captured by moment-based or feature-based representations. They can obscure important changes in distributional shape or geometry. We propose a geometry-aware CPD framework that treats streaming batch data as a stochastic process on the 2-Wasserstein space. Our method detects changes in the law of this process by mapping each empirical distribution to a tangent space relative to a pre-change Fréchet barycenter, yielding a reference-centered local linearization of 2-Wasserstein space. This representation enables sequential detectors by adapting classical multivariate monitoring statistics to tangent fields. We provide theoretical guarantees and demonstrate, via synthetic and real-world experiments, that our approach detects complex distributional shifts with reduced detection delay at matched $\mathrm{ARL}_0$ compared with moments-based and model-free baselines.

Applications · Computer Vision

Rongjin Guo, Guan Huankang, Rynson Lau

Salient Object Ranking (SOR) aims to study how humans visually explore complex scenes by predicting an ordered sequence of objects that attracts our attention. Existing SOR approaches typically model this ranking deterministically, assuming a single, fixed ranking sequence of attention. However, such deterministic SOR fails to capture the true nature of human attention. We observe that human attention shifts exhibit variability and stochasticity, showing that the next object of fixation is not a definitive choice but rather a probability distribution. Yet, existing SOR methods and evaluation metrics do not account for this inherent randomness. To address this fundamental problem, we propose ProbSOR, a novel Probabilistic Salient Object Ranking model that explicitly learns the uncertainty of attention shifts by incorporating Group Relative Policy Optimization (GRPO). We leverage a Vision-Language Model (VLM) as the foundation for ProbSOR to identify salient objects and infer their ranked order, utilizing a segmentation decoder for precise object extraction. We also propose a new metric tailored to ProbSOR, as existing SOR metrics only support deterministic rankings. Further, we construct a ProbSOR dataset comprising 15,000 probabilistic SOR samples, to support both model training and evaluation. Extensive experiments demonstrate that ProbSOR achieves strong performances in salient object ranking under both our proposed and traditional benchmarks.

Probabilistic Methods · Everything Else

Cristiana Diaconu, Miles Cranmer, Richard E Turner, Tanya Marwah, Payel Mukhopadhyay

Dominant approaches for modelling Partial Differential Equations (PDEs) rely on deterministic predictions, yet many physical systems of interest are inherently chaotic and uncertain. While training probabilistic models from scratch is possible, it is computationally expensive and fails to leverage the significant resources already invested in high-performing deterministic backbones. In this work, we adopt a training-efficient strategy to transform pre-trained deterministic models into probabilistic ones via retrofitting with a proper scoring rule: the Continuous Ranked Probability Score (CRPS). Crucially, this approach is architecture-agnostic: it applies the same adaptation mechanism across distinct model backbones with minimal code modifications. The method proves highly effective across different scales of pre-training: for models trained on single dynamical systems, we achieve $20–54\%$ reductions in rollout CRPS and up to $30\%$ improvements in variance-normalised RMSE (VRMSE) relative to compute-matched deterministic fine-tuning. We further validate our approach on a PDE foundation model, trained on multiple systems and retrofitted on the dataset of interest, to show that our probabilistic adaptation yields an improvement of up to $40\%$ in CRPS and up to $15\%$ in VRMSE compared to deterministic fine-tuning. Validated across diverse architectures and dynamics, our results show that probabilistic PDE modelling need not require retraining from scratch, but can be unlocked from existing deterministic backbones with modest additional training cost.

Probabilistic Methods · Monte Carlo and Sampling Methods

Emanuel Sommer, Kangning Diao, Jakob Robnik, Uros Seljak, David Rügamer

Scaling inference methods such as Markov chain Monte Carlo to high-dimensional models remains a central challenge in Bayesian deep learning. A promising recent proposal, microcanonical Langevin Monte Carlo, has shown state-of-the-art performance across a wide range of problems. However, its reliance on full-dataset gradients makes it prohibitively expensive for large-scale problems. This paper addresses a fundamental question: Can microcanonical dynamics effectively leverage mini-batch gradient noise? We provide the first systematic study of this problem, establishing a novel continuous-time theoretical analysis of stochastic-gradient microcanonical dynamics. We reveal two critical failure modes: a theoretically derived bias due to anisotropic gradient noise and numerical instabilities in complex high-dimensional posteriors. To tackle these issues, we propose a principled gradient noise preconditioning scheme shown to significantly reduce this bias and develop a novel, energy-variance-based adaptive tuner that automates step size selection and dynamically informs numerical guardrails. The resulting algorithm is a robust and scalable microcanonical Monte Carlo sampler that achieves state-of-the-art performance on challenging high-dimensional inference tasks like Bayesian neural networks. Combined with recent ensemble techniques, our work unlocks a new class of stochastic microcanonical Langevin ensemble (SMILE) samplers for large-scale Bayesian inference.

Probabilistic Methods · Variational Inference

James Odgers, Ben Riegler, Siddharth Swaroop, Vincent Fortuin

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian linear regression, we show that this characterisation is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on i.i.d. data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

Probabilistic Methods · Everything Else

Daniel Csillag, Pedro Dall’Antonia, Claudio Struchiner, Guilherme Tegoni Goedert

Prediction-powered inference is a recent methodology for the safe use of black-box ML models to impute missing data, strengthening inference of statistical parameters. However, many applications require strong properties besides valid inference, such as privacy, robustness or validity under continuous distribution shifts; deriving prediction-powered methods with such guarantees is generally an arduous process, and has to be done case by case. In this paper, we resolve this issue by connecting prediction-powered inference with conformal prediction: by performing imputation through a calibrated set-predictor, we attain validity while achieving additional guarantees in a natural manner. We instantiate our procedure for the inference of means, Z- and M-estimation, as well as e-values and e-value-based procedures. Furthermore, in the case of e-values, ours is the first general prediction-powered procedure that operates off-line. We demonstrate these advantages by applying our method on private and time-series data. Both tasks are nontrivial within the standard prediction-powered framework but become natural under our method.

Probabilistic Methods · Everything Else

Jinzong Dong, Zhaohui Jiang, Bo Yang

Confidence calibration for classification models is vital in safety-critical decision-making scenarios and has received extensive attention. General confidence calibration methods assume training and test data are independent and identically distributed ($i.i.d.$), limiting their effectiveness under covariate shifts. Previous calibration methods under covariate shift struggle with class-wise or canonical calibrations and often rely on unstable importance weighting when density ratios are large or unbounded. Given the above limitations, this paper rethinks confidence calibration under covariate shifts. First, we derive a necessary and sufficient condition for confidence calibration under covariate shifts, named Expectation consistency condition, which reveals covariate shifts do not necessarily lead to uncalibrated confidence and provides a weaker condition for confidence calibration than global covariate distribution alignment. Then, utilizing Expectation consistency condition, this paper proposes an unsupervised domain adaptation loss to calibrate confidence of the target domain, named Expectation consistency loss (ECL), which is compatible with canonical calibration, class-wise calibration, and top-label calibration. Third, we prove that computing ECL loss has the same sample complexity as Expected Calibration Error (ECE) and provide a theoretically grounded mini-batch trainable scheme for ECL loss. Finally, we validate the effectiveness of our method on both simulated and real-world covariate shift datasets.

Probabilistic Methods · Structure Learning

Sobihan Surendran, Adeline Fermanian, Sylvain Le Corff

Combinatorial Optimization problems are widespread in domains such as logistics, manufacturing, and drug discovery, yet their NP-hard nature makes them computationally challenging. Recent Neural Combinatorial Optimization (NCO) methods leverage deep learning to learn policies for constructing solutions, trained via Supervised or Reinforcement Learning. While promising, these approaches often rely on task-specific augmentations, perform poorly on out-of-distribution instances, and lack robust inference mechanisms. Moreover, existing latent space models either require labeled data or use an instance-independent latent distribution. In this work, we propose LGS-Net, a novel latent space model that conditions on problem instances, and introduce an efficient inference method, Latent Guided Sampling (LGS), based on Markov Chain Monte Carlo and Stochastic Approximation. We show that the iterations of our method form a time-inhomogeneous Markov Chain and provide rigorous theoretical convergence guarantees. Empirical results on benchmark routing tasks show that our method achieves state-of-the-art performance among NCO baselines.