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Evaluating large generative models across benchmarks is time-consuming and costly, motivating methods that can estimate full benchmark performance by evaluating models on only a subset of items (coreset). While current literature mostly requires the practitioner to input a coreset size, we introduce BayesAME, a sequential active Bayesian framework that uses historical data and an information-gain criterion to dynamically select items to include in the coreset thus allowing automatic determination of its size. Additionally, we show how a multi-target extension further reduces coreset size by leveraging model correlations. Extensive experiments demonstrate that BayesAME consistently outperforms existing methods. Crucially, our analysis resolves recent skepticism in the literature by showing that a) non-random coreset selection is advantageous over random selection and b) one can reliably improve upon the random sample mean baseline.

This paper is a follow up to an article by two of the authors dedicated to the study of Poincaré and logarithmic Sobolev inequalities for measures of the form dμ= e^-U dνwhere e^-U is seen as a perturbation of dν. Application to the same functional inequalities for convolution products are then discussed. In the present paper we investigate similar problems for weaker functional inequalities, namely weak Poincaré, weighted Poincaré, weak log-Sobolev and weighted log-Sobolev inequalities.

We develop diffusion-based samplers for target distributions known up to a normalising constant. To this end, we rely on the well-known diffusion path that smoothly interpolates between a simple base distribution and the target, popularised by diffusion models. We tackle the score estimation problem by developing an efficient sequential Monte Carlo sampler that evolves auxiliary variables from conditional distributions along the path, providing principled score and density estimates for time-varying distributions. To control the variance of score estimates, we further propose practical control variate schedules that incur minimal overhead. We adapt this general framework to paths induced by the Ornstein-Uhlenbeck (OU) time-reversal process, stochastic interpolants, and diffusion annealed Langevin dynamics, outlining their trade-offs. Finally, we provide theoretical guarantees and empirically demonstrate the effectiveness of our method on several synthetic and real-world datasets.

Modern post-training methods for large language models learn a single adapted parameter vector, forcing heterogeneous data, conflicting preferences, and epistemic uncertainty into one behavioural compromise. We propose an Renyi variational framework for learning distributions over post-training parameters, with a specific focus on uncertainty quantification for large language models, offering a principled alternative to deep ensemble approaches. The resulting objective interpolates between classical variational Bayes and predictively oriented posterior learning. The interpolation parameter balances globally plausible individual models against complementary specialists. We analyse this objective identifying local stability criteria which demonstrate how model misspecification can make non-degenerate posterior spread locally favourable, manifesting contradictory or conflicting data as predictive uncertainty. We apply our framework to LLM post-training by learning an ensemble of LoRA adapters attached to a shared, frozen base model. We develop a scalable, finite-particle training procedure for both supervised fine-tuning and preference optimization. This approach enables training examples to be softly routed across particles, explicitly promoting model specialization and improving uncertainty awareness across different experiments.

Recent advances such as self-consistency and test-time reinforcement learning (TTRL) improve the reliability of large language models (LLMs) without additional supervision, yet their underlying mechanisms and statistical guarantees remain poorly understood. We present a unified framework for certifiable inference in LLMs, showing that majority voting provides a statistical certificate of self-consistency: under mild assumptions, the aggregated answer coincides with the mode of the model’s terminal distribution with high probability. We derive finite-sample and anytime-valid concentration bounds that quantify this confidence, and introduce the Martingale Majority Certificate (MMC), a sequential stopping rule that adaptively determines when sufficient samples have been drawn. We further prove that label-free post-training methods such as TTRL implicitly sharpen the answer distribution by exponentially tilting it toward its mode, thereby reducing the number of samples required for certification. Building on this insight, we propose new post-training objectives that explicitly optimise this trade-off between sharpness and bias. Together, these results explain and connect two central test-time scaling strategies, self-consistency and TTRL, within a single statistical framework for label-free, certifiable reliability in reasoning LLMs.

In this work we study the diffusion annealed Langevin dynamics, a score-based diffusion process recently introduced in the theory of generative models and which is an alternative to the classical overdamped Langevin diffusion. Our goal is to provide a rigorous construction and to study the theoretical efficiency of these models for general base distribution as well as target distribution. As a matter of fact these diffusion processes are a particular case of Nelson processes i.e. diffusion processes with a given flow of time marginals. Providing existence and uniqueness of the solution to the annealed Langevin diffusion leads to proving a Poincaré inequality for the conditional distribution of X knowing X+Z=y uniformly in y, as recently observed by one of us and her coauthors. Part of this work is thus devoted to the study of such Poincaré inequalities. Additionally we show that strengthening the Poincaré inequality into a logarithmic Sobolev inequality improves the efficiency of the model.

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.

We introduce a class of algorithms, termed Proximal Interacting Particle Langevin Algorithms (PIPLA), for inference and learning in latent variable models whose joint probability density is non-differentiable. Leveraging proximal Markov chain Monte Carlo (MCMC) techniques and the recently introduced interacting particle Langevin algorithm (IPLA), we propose several variants within the novel proximal IPLA family, tailored to the problem of estimating parameters in a non-differentiable statistical model. We prove nonasymptotic bounds for the parameter estimates produced by multiple algorithms in the strongly log-concave setting and provide comprehensive numerical experiments on various models to demonstrate the effectiveness of the proposed methods. In particular, we demonstrate the utility of the proposed family of algorithms on a toy hierarchical example where our assumptions can be checked, as well as on the problems of sparse Bayesian logistic regression, sparse Bayesian neural network, and sparse matrix completion. Our theory and experiments together show that PIPLA family can be the de facto choice for parameter estimation problems in latent variable models for non-differentiable models.

Selecting cost-effective optimal sensor configurations for subsequent inference of parameters in black-box stochastic systems faces significant computational barriers. We propose a novel and robust approach, modelling the joint distribution over input parameters and solution with a joint energy-based model, trained on simulation data. Unlike existing simulation-based inference approaches, which must be tied to a specific set of point evaluations, we learn a functional representation of parameters and solution. This is used as a resolution-independent plug-and-play surrogate for the joint distribution, which can be conditioned over any set of points, permitting an efficient approach to sensor placement. We demonstrate the validity of our framework on a variety of stochastic problems, showing that our method provides highly informative sensor locations at a lower computational cost compared to conventional approaches.