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General Machine Learning · Probabilistic Methods

Dominik Meier, Shixing Yu, Sagnik Nandy, PROMIT GHOSAL, Kyra Gan

Single-cell RNA sequencing (scRNA-seq) enables the study of cellular heterogeneity. Yet, clustering accuracy, and with it downstream analyses based on cell labels, remain challenging due to measurement noise and biological variability. In standard latent spaces (e.g., obtained through PCA), data from different cell types can be projected close together, making accurate clustering difficult. We introduce a latent plug-and-play diffusion framework that separates the observation and denoising space. This separation is operationalized through a novel Gibbs sampling procedure: the learned diffusion prior is applied in a low-dimensional latent space to perform denoising, while to steer this process, noise is reintroduced into the original high-dimensional observation space. This unique ``input-space steering'' ensures the denoising trajectory remains faithful to the original data structure. Our approach offers three key advantages: (1) adaptive noise handling via a tunable balance between prior and observed data; (2) uncertainty quantification through principled uncertainty estimates for downstream analysis; and (3) generalizable denoising by leveraging clean reference data to denoise noisier datasets, and via averaging, improve quality beyond the training set. We evaluate robustness on both synthetic and real single-cell genomics data. Our method improves clustering accuracy on synthetic data across varied noise levels and dataset shifts. On real-world single-cell data, our method demonstrates improved biological coherence in the resulting cell clusters, with cluster boundaries that better align with known cell type markers and developmental trajectories.

Theory · Probabilistic Methods

Deepro Choudhury, Sinead Williamson, Adam Golinski, Ning Miao, Freddie Bickford Smith, Michael Kirchhof, Yizhe Zhang, Tom Rainforth

We propose a general-purpose approach for improving the ability of large language models (LLMs) to intelligently and adaptively gather information from a user or other external source using the framework of sequential Bayesian experimental design (BED). This enables LLMs to act as effective multi-turn conversational agents and interactively interface with external environments. Our approach, which we call BED-LLM (Bayesian experimental design with large language models), is based on iteratively choosing questions or queries that maximize the expected information gain (EIG) with respect to a variable of interest given the responses gathered previously. We show how this EIG can be formulated (and then estimated) in a principled way using a probabilistic model derived from the LLM's predictive distributions and provide detailed insights into key decisions in its construction and updating procedure. We find that BED-LLM achieves substantial gains in performance across a wide range of tests based on the 20 Questions game and using the LLM to actively infer user preferences, compared to purely prompting-based design generation and other adaptive design strategies.

General Machine Learning · Probabilistic Methods

Sébastien Melo, Gael Varoquaux, Marine Le Morvan

Distinguishing epistemic uncertainty (model ignorance) from aleatoric uncertainty (task randomness) is critical for reliable AI systems, yet standard confidence evaluation metrics capture different and incomplete aspects of uncertainty. While AUC and accuracy measure predictive signal, proper scoring rules assess overall uncertainty, and calibration metrics isolate part of the epistemic uncertainty but ignore within-bin heterogeneity of errors, known as grouping loss. We bridge this evaluation gap by introducing asymptotically consistent and sample-efficient estimators of the grouping loss and excess decision risk, providing a fine-grained assessment of epistemic uncertainty that complements existing calibration metrics. Applied to LLM question-answering with inherent aleatoric noise, our estimators reveal substantial grouping loss which decreases with model scale and instruction tuning. Their local nature enables automatic identification of subgroups with systematic over- or under-confidence, supporting interpretable confidence audits. Finally, we leverage these estimates to design LLM cascades that defer high excess decision risk predictions to stronger models, achieving higher accuracy at lower cost than competing approaches.

Deep Learning · Robustness

Andi Zhang, Xuan Ding, Steven McDonagh, Samuel Kaski

We propose a concept-based adversarial attack framework that extends beyond single-image perturbations by adopting a probabilistic perspective. Rather than modifying a single image, our method operates on an entire concept - represented by a distribution - to generate diverse adversarial examples. Preserving the concept is essential, as it ensures that the resulting adversarial images remain identifiable as instances of the original underlying category or identity. By sampling from this concept-based adversarial distribution, we generate images that maintain the original concept but vary in pose, viewpoint, or background, thereby misleading the classifier. Mathematically, this framework remains consistent with traditional adversarial attacks in a principled manner. Our theoretical and empirical results demonstrate that concept-based adversarial attacks yield more diverse adversarial examples and effectively preserve the underlying concept, while achieving higher attack efficiency.

Theory · Probabilistic Methods

Jan Boelts

Model misspecification is a critical challenge in simulation-based inference (SBI), particularly in neural SBI methods that use simulated data to train flexible neural density estimators. These methods typically assume that simulators faithfully represent the true data-generating process, an assumption that is often violated in practice. Resulting discrepancies can make observed data effectively out-of-distribution relative to the simulations, leading to biased posterior distributions and misleading uncertainty quantification. This post reviews recent work on model misspecification in neural SBI, covering formal definitions, methods for detection and mitigation, and their underlying assumptions. It also discusses practical implications for SBI workflows and outlines open challenges for developing robust SBI methods that remain reliable in realistic, imperfectly specified applications.

General Machine Learning · Probabilistic Methods

Jiajun He, Paul Jeha, Peter Potaptchik, Leo Zhang, José Miguel Hernández Lobato, Yuanqi Du, Saifuddin Syed, Francisco Vargas

Inference-time control of diffusion models aims to steer model outputs to satisfy new constraints without retraining. Previous approaches have mostly relied on heuristic guidance or have been coupled with Sequential Monte Carlo (SMC) for bias correction. In this paper, we propose a flexible alternative based on replica exchange, an algorithm designed initially for sampling problems. We refer to this method as CREPE (Controlling with REPlica Exchange). Unlike SMC, CREPE: (i) generates particles sequentially, (ii) maintains high diversity in the generated samples after a burn-in period, and (iii) enables online refinement or early termination. We demonstrate its versatility across various tasks, including temperature annealing, reward tilting, model composition and classifier-free guidance debiasing, with competitive performance compared to prior SMC methods.

General Machine Learning · Probabilistic Methods

Jiajun He, José Miguel Hernández Lobato, Yuanqi Du, Francisco Vargas

Diffusion models generate data by removing noise gradually, which corresponds to the time-reversal of a noising process. However, access to only the denoising kernels is often insufficient. In many applications, we need the knowledge of the marginal densities along the generation trajectory, which enables tasks such as inference-time control. To address this gap, in this paper, we introduce the Radon-Nikodym Estimator (RNE). Based on the concept of the density ratio between path distributions, it reveals a fundamental connection between marginal densities and transition kernels, providing a flexible plug-and-play framework that unifies (1) diffusion density estimation, (2) inference-time control, and (3) energy-based diffusion training under a single perspective. Experiments demonstrated that RNE delivers strong results in inference-time control applications, such as annealing and model composition, with promising inference-time scaling performance, and achieves simple yet efficient regularisation for training energy-based diffusion models. Additionally, our proposed RNE is modality-agnostic and applicable not only to continuous diffusion models but also to their discrete diffusion counterparts.

General Machine Learning · Probabilistic Methods

Kirill Tamogashev, Nikolay Malkin

The Schrödinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms enable the inference of such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schrödinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schrödinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method.

General Machine Learning · Probabilistic Methods

Minsu Kim, Jean-Pierre Falet, Oliver Richardson, Xiaoyin Chen, Moksh Jain, Sungjin Ahn, Sungsoo Ahn, Yoshua Bengio

Chain-of-Thought (CoT) reasoning has advanced the capabilities and transparency of language models (LMs); however, reasoning chains can contain inaccurate statements that reduce performance and trustworthiness. To address this, we propose to augment each reasoning step in a CoT with a latent veracity (or correctness) variable. To efficiently explore this expanded space, we introduce Veracity Search (VS), a discrete search algorithm over veracity assignments. It performs otherwise intractable inference in the posterior distribution over latent veracity values by leveraging the LM's joint likelihood over veracity and the final answer as a proxy reward. This efficient inference-time verification method facilitates supervised fine-tuning of an Amortized Veracity Inference (AVI) machine by providing pseudo-labels for veracity. AVI generalizes VS, enabling accurate zero-shot veracity inference in novel contexts. Empirical results demonstrate that VS reliably identifies errors in logical (ProntoQA), mathematical (GSM8K), and commonsense (CommonsenseQA) reasoning benchmarks, with AVI achieving comparable zero-shot accuracy. Finally, we demonstrate the utility of latent veracity inference for providing feedback during self-correction and self-improvement.

General Machine Learning · Probabilistic Methods

Ilia Azizi, Juraj Bodik, Jakob M. Heiss, Bin Yu

Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters, $\gamma_1$ and $\gamma_2$, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability–Computability–Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3\% and 17.5\% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

General Machine Learning · Representation Learning

Kartik Nair, Indradyumna Roy, Soumen Chakrabarti, Anirban Dasgupta, Abir De

In this work, we discover a probabilistic symmetry, called as exchangeability in graph neural networks (GNNs). Specifically, we show that the trained node embedding computed using a large family of graph neural networks, learned under standard optimization tools, are exchangeable random variables. This implies that the probability density of the node embeddings remains invariant with respect to a permutation applied on their dimension axis. This results in identical distribution across the elements of the graph representations. Such a property enables approximation of transportation-based graph similarities by Euclidean similarities between order statistics. Leveraging this reduction, we propose a unified locality-sensitive hashing (LSH) framework that supports diverse relevance measures, including subgraph matching and graph edit distance. Experiments show that our method helps to do LSH more effectively than baselines.

General Machine Learning · Representation Learning

Kartik Nair, Indradyumna Roy, Soumen Chakrabarti, Anirban Dasgupta, Abir De

In this work, we discover a probabilistic symmetry, called as exchangeability in graph neural networks (GNNs). Specifically, we show that the trained node embedding computed using a large family of graph neural networks, learned under standard optimization tools, are exchangeable random variables. This implies that the probability density of the node embeddings remains invariant with respect to a permutation applied on their dimension axis. This results in identical distribution across the elements of the graph representations. Such a property enables approximation of transportation-based graph similarities by Euclidean similarities between order statistics. Leveraging this reduction, we propose a unified locality-sensitive hashing (LSH) framework that supports diverse relevance measures, including subgraph matching and graph edit distance. Experiments show that our method helps to do LSH more effectively than baselines.

Theory · Probabilistic Methods

Futoshi Futami, Atsushi Nitanda

Calibration is a critical requirement for reliable probabilistic prediction, especially in high-risk applications. However, the theoretical understanding of which learning algorithms can simultaneously achieve high accuracy and good calibration remains limited, and many existing studies provide empirical validation or a theoretical guarantee in restrictive settings. To address this issue, in this work, we focus on the smooth calibration error (CE) and provide a uniform convergence bound, showing that the smooth CE is bounded by the sum of the smooth CE over the training dataset and a generalization gap. We further prove that the functional gradient of the loss function can effectively control the training smooth CE. Based on this framework, we analyze three representative algorithms: gradient boosting trees, kernel boosting, and two-layer neural networks. For each, we derive conditions under which both classification and calibration performances are simultaneously guaranteed. Our results offer new theoretical insights and practical guidance for designing reliable probabilistic models with provable calibration guarantees.

Theory · Probabilistic Methods

Xinyu Zhang, Conor Hassan, Julien Martinelli, Daolang Huang, Samuel Kaski

Balancing competing objectives is omnipresent across disciplines, from drug design to autonomous systems. Multi-objective Bayesian optimization is a promising solution for such expensive, black-box problems: it fits probabilistic surrogates and selects new designs via an acquisition function that balances exploration and exploitation. In practice, it requires tailored choices of surrogate and acquisition that rarely transfer to the next problem, is myopic when multi-step planning is often required, and adds refitting overhead, particularly in parallel or time-sensitive loops. We present TAMO, a fully amortized, universal policy for multi-objective black-box optimization. TAMO uses a transformer architecture that operates across varying input and objective dimensions, enabling pretraining on diverse corpora and transfer to new problems without retraining: at test time, the pretrained model proposes the next design with a single forward pass. We pretrain the policy with reinforcement learning to maximize cumulative hypervolume improvement over full trajectories, conditioning on the entire query history to approximate the Pareto frontier. Across synthetic benchmarks and real tasks, TAMO produces fast proposals, reducing proposal time by 50–1000× versus alternatives while matching or improving Pareto quality under tight evaluation budgets. These results show that transformers can perform multi-objective optimization entirely in-context, eliminating per-task surrogate fitting and acquisition engineering, and open a path to foundation-style, plug-and-play optimizers for scientific discovery workflows.

Theory · Probabilistic Methods

Preetum Nakkiran, Arwen Bradley, Adam Golinski, Eugene Ndiaye, Michael Kirchhof, Sinead Williamson

Large Language Models (LLMs) often lack meaningful confidence estimates for the semantic content of their outputs. While base LLMs are known to exhibit next-token calibration, it remains unclear whether they can assess confidence in the actual meaning of their responses beyond the token level. We find that, when using a certain sampling-based notion of semantic calibration, base LLMs are remarkably well-calibrated: they can meaningfully assess confidence in various open-ended question-answering tasks, despite training only on next-token prediction. To formalize this phenomenon, we introduce "$B$-calibration," a notion of calibration parameterized by the choice of equivalence classes. Our main theoretical contribution establishes a mechanism for why semantic calibration emerges in base LLMs, leveraging a recent connection between calibration and local loss optimality. This theoretical mechanism leads to a testable prediction: base LLMs will be semantically calibrated when they can easily predict their own distribution over semantic answer classes before generating a response. We state three implications of this prediction, which we validate through experiments: (1) Base LLMs are semantically calibrated across question-answering tasks, (2) instruction-tuning procedures systematically break this calibration, and (3) chain-of-thought reasoning breaks calibration (intuitively because models cannot predict their final answers before completing their generation). To our knowledge, our work provides the first principled explanation of when and why semantic calibration emerges in LLMs.

General Machine Learning · Probabilistic Methods

Michael Kirchhof, Luca Füger, Adam Golinski, Eeshan Gunesh Dhekane, Arno Blaas, Seong Joon Oh, Sinead Williamson

The common approach to communicate a large language model's (LLM) uncertainty is to add a percentage number or a hedging word to its response. But is this all we can do? Instead of generating a single answer and then hedging it, an LLM that is fully transparent to the user needs to be able to reflect on its internal belief distribution and output a summary of all options it deems possible, and how likely they are. To test whether LLMs possess this capability, we develop the SelfReflect metric, an information-theoretic distance between a given summary and a distribution over answers. In interventional and human studies, we find that SelfReflect indicates even slight deviations, yielding a fine measure of faithfulness between a summary string and an LLM's actual internal distribution over answers. With SelfReflect, we make a resounding negative observation: modern LLMs are, across the board, incapable of revealing what they are uncertain about, neither through reasoning, nor chains-of-thoughts, nor explicit finetuning. However, we do find that LLMs are able to generate faithful summaries of their uncertainties if we help them by sampling multiple outputs and feeding them back into the context. This simple approach shines a light at the universal way of communicating LLM uncertainties whose future development the SelfReflect score enables. To support the development of this universal form of LLM uncertainties, we publish the code that implements our metric for arbitrary LLMs under https://github.com/apple/ml-selfreflect .

General Machine Learning · Probabilistic Methods

Mykyta Ielanskyi, Kajetan Schweighofer, Lukas Aichberger, Sepp Hochreiter

Hallucinations are a common issue that undermine the reliability of large language models (LLMs). Recent studies have identified a specific subset of hallucinations, known as confabulations, which arise due to predictive uncertainty of LLMs. To detect confabulations, various methods for estimating predictive uncertainty in natural language generation (NLG) have been developed. These methods are typically evaluated by correlating uncertainty estimates with the correctness of generated text, with question-answering (QA) datasets serving as the standard benchmark. However, commonly used approximate correctness functions have substantial disagreement between each other and, consequently, in the ranking of the uncertainty estimation methods. This allows one to inflate the apparent performance of uncertainty estimation methods. We propose using several alternative risk indicators for risk correlation experiments that improve robustness of empirical assessment of UE algorithms for NLG. For QA tasks, we show that marginalizing over multiple LLM-as-a-judge variants leads to reducing the evaluation biases. Furthermore, we explore structured tasks as well as out of distribution and perturbation detection tasks which provide robust and controllable risk indicators. Finally, we propose to use an Elo rating of uncertainty estimation methods to give an objective summarization over extensive evaluation settings.

Theory · Probabilistic Methods

Lukas Aichberger, Kajetan Schweighofer, Sepp Hochreiter

Large Language Models (LLMs) are increasingly employed in real-world applications, driving the need to evaluate the trustworthiness of their generated text. To this end, reliable uncertainty estimation is essential. Leading uncertainty estimation methods generate and analyze multiple output sequences, which is computationally expensive and impractical at scale. In this work, we inspect the theoretical foundations of these methods and explore new directions to enhance computational efficiency. Building on the framework of proper scoring rules, we find that the negative log-likelihood of the most likely output sequence constitutes a theoretically principled uncertainty measure. To approximate this alternative measure, we propose G-NLL, obtained using a single output sequence from greedy decoding. This approach streamlines uncertainty estimation while preserving theoretical rigor. Empirical results demonstrate that G-NLL achieves state-of-the-art performance across various scenarios. Our work lays the theoretical foundation for efficient and reliable uncertainty estimation in natural language generation, challenging the necessity of the prevalent methods that are more complex and resource-intensive.

Theory · Probabilistic Methods

Nikos Tsikouras, Yorgos Pantis, Ioannis Mitliagkas, Christos Tzamos

Understanding the dynamics of feature learning in neural networks (NNs) remains a significant challenge. The work of (Mousavi-Hosseini et al., 2023) analyzes a multiple index teacher-student setting and shows that a two-layer student attains a low-rank structure in its first-layer weights when trained with stochastic gradient descent (SGD) and a strong regularizer. This structural property is known to reduce sample complexity of generalization. Indeed, in a second step, the same authors establish algorithm-specific learning guarantees under additional assumptions. In this paper, we focus exclusively on the structure discovery aspect and study it under weaker assumptions, more specifically: we allow (a) NNs of arbitrary size and depth, (b) with all parameters trainable, (c) under any smooth loss function, (d) tiny regularization, and (e) trained by any method that attains a second-order stationary point (SOSP), e.g. perturbed gradient descent (PGD). At the core of our approach is a key $\textit{derandomization}$ lemma, which states that optimizing the function $E_{x} \left[g_{\theta}(Wx + b)\right]$ converges to a point where $W = 0$, under mild conditions. The fundamental nature of this lemma directly explains structure discovery and has immediate applications in other domains including an end-to-end approximation for MAXCUT, and computing Johnson-Lindenstrauss embeddings.

General Machine Learning · Probabilistic Methods

Paul Hofman, Timo Löhr, Maximilian Muschalik, Yusuf Sale, Eyke Hüllermeier

A reliable representation of uncertainty is essential for the application of modern machine learning methods in safety-critical settings. In this regard, the use of credal sets (i.e., convex sets of probability distributions) has recently been proposed as a suitable approach to representing epistemic uncertainty. However, as with other approaches to epistemic uncertainty, training credal predictors is computationally complex and usually involves (re-)training an ensemble of models. The resulting computational complexity prevents their adoption for complex models such as foundation models and multi-modal systems. To address this problem, we propose an efficient method for credal prediction that is grounded in the notion of relative likelihood and inspired by techniques for the calibration of probabilistic classifiers. For each class label, our method predicts a range of plausible probabilities in the form of an interval. To produce the lower and upper bounds of these intervals, we propose a technique that we refer to as decalibration. Extensive experiments show that our method yields credal sets with strong performance across diverse tasks, including coverage–efficiency evaluation, out-of-distribution detection, and in-context learning. Notably, we demonstrate credal prediction on models such as TabPFN and CLIP—architectures for which the construction of credal sets was previously infeasible.