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General Machine Learning · Methodology

Daisuke Yamada, Qijun Zhang, Travis Pence, Barbara Bendlin, Federico Rey, Vikas Singh

Compositional data---vectors encoding relative proportions---arise across scientific domains, including ecology, geochemistry, and genomics. The features in these data often come with known hierarchical structure (e.g., taxonomies, phylogenies, ontologies), yet existing methods either ignore this structure, discard the intrinsic Aitchison geometry, require assumptions such as binary trees, or yield incomplete coordinate systems. We describe *PolyILR*, a canonical orthonormal decomposition of the Aitchison tangent space aligned with any tree topology. Our construction defines a weighted local geometry at each internal node capturing full branching structure, then lifts these to a global orthonormal basis where every coordinate corresponds to a specific tree location. On microbiome and single-cell benchmarks, PolyILR yields stable, interpretable features and enables inference at multiscale tree resolution. We also establish a novel theoretical connection to softmax classifiers, suggesting possible applications to probabilistic modeling.

Applications · Computer Vision

Jia-yu Li, Jiaxin Qi, Sheng Zhou, Jianqiang Huang, Xian-Sheng Hua

Contrastive vision-language models like CLIP exhibit remarkable zero-shot generalization. However, prompt tuning remains highly sensitive to label noise, as mislabeled samples generate disproportionately large gradients that can overwhelm pre-trained priors. We argue that because CLIP already provides a near-optimal initialization, adaptation should be inherently conservative, particularly against the extreme gradient updates common in noisy settings. To this end, we propose Double-Softmax Prompt Tuning (DSPT), a hyperparameter-free method for intrinsic gradient suppression. By applying a sequential probabilistic normalization, DSPT induces a self-adaptive saturation zone that suppresses gradients from high-error noisy samples while maintaining informative updates. We also provide both theoretical analysis and empirical evidence about how this mechanism achieves adaptive suppression. This design transforms ``gradient vanishing'', traditionally a training bottleneck, into a principled noise-filtering shield for label-noise prompt tuning. Extensive experiments confirm that this simple, drop-in design achieves state-of-the-art robustness across various noisy benchmarks, outperforming methods with complex architectures and handcrafted hyperparameters.

Theory · Probabilistic Methods

Zhengyan Wan, Yidong Ouyang, Qiang Yao, Liyan Xie, Fang Fang, Hongyuan Zha, Guang Cheng

Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error analysis remain largely unexplored. In this work, we develop a unified framework grounded in stochastic calculus theory to systematically investigate the theoretical properties of discrete flow models. Specifically, by leveraging a Girsanov-type theorem for the path measures of two continuous-time Markov chains (CTMCs), we present a comprehensive error analysis that accounts for both transition rate estimation error and early stopping error. In fact, the estimation error of transition rates has received little attention in existing works. Unlike discrete diffusion models, discrete flow incurs no initialization error caused by truncating the time horizon in the noising process. Building on generator matching and uniformization, we establish non-asymptotic error bounds for distribution estimation without the boundedness condition on oracle transition rates. Furthermore, we derive a faster rate of total variation convergence for the estimated distribution with the boundedness condition, yielding a nearly optimal rate in terms of sample size. Our results provide the first error analysis for discrete flow models. We also study model performance under different settings based on simulation results.

Theory · Probabilistic Methods

Zhengyan Wan, Yidong Ouyang, Liyan Xie, Hongyuan Zha, Fang Fang, Guang Cheng

Discrete flow models (DFMs) have been proposed to learn the data distribution on finite state space, offering a flexible framework as an alternative to discrete diffusion models. A line of recent work has studied samplers for discrete diffusion models, such as tau-leaping and Euler solver. However, these samplers require a large number of iterations to control discretization error, since the transition rates are frozen in time and evaluated at the initial state within each time interval. Moreover, theoretical results for these samplers often require boundedness conditions of the transition rate or they focus on a specific type of source distributions. To address those limitations, we establish non-asymptotic discretization error bounds for those samplers without any restriction on transition rates and source distributions, under the framework of discrete flow models. Furthermore, by analyzing a one-step lower bound of the Euler sampler, we propose two corrected samplers: \textit{time-corrected sampler} and \textit{location-corrected sampler}, which can reduce the discretization error of tau-leaping and Euler solver with almost no additional computational cost. We rigorously show that the location-corrected sampler has a lower complexity than existing parallel samplers. We validate the effectiveness of the proposed method by achieving better generation quality with reduced inference time on simulations and text-to-image generation tasks.

Deep Learning · Foundation Models

Mehmet Ozgur Turkoglu, Dominik J. Mühlematter, Alexander Becker, Konrad Schindler, Helge Aasen

Foundation models have become a dominant paradigm in machine learning, achieving remarkable performance across diverse tasks through large-scale pretraining. However, these models often yield overconfident, uncalibrated predictions. The standard approach to quantifying epistemic uncertainty, training an ensemble of independent models, incurs prohibitive computational costs that scale linearly with ensemble size, making it impractical for large foundation models. We propose Singular Value Ensemble (SVE), a parameter-efficient implicit ensemble method that builds on a simple, but powerful core assumption: namely, that the singular vectors of the weight matrices constitute meaningful subspaces of the model's knowledge. Pretrained foundation models encode rich, transferable information in their weight matrices. If the singular vectors are indeed meaningful (orthogonal) "knowledge directions". To obtain a model ensemble, we modulate only how strongly each direction contributes to the output. Rather than learning entirely new parameters, we freeze the singular vectors and only train per-member singular values that rescale the contribution of each direction in that shared knowledge basis. Ensemble diversity emerges naturally as stochastic initialization and random sampling of mini-batches during joint training cause different members to converge to different combinations of the same underlying knowledge. SVE achieves uncertainty quantification comparable to explicit deep ensembles while increasing the parameter count of the base model by less than 1%, making principled uncertainty estimation accessible in resource-constrained settings. We validate SVE on NLP and vision tasks with various different backbones and show that it improves calibration while maintaining predictive accuracy.

Probabilistic Methods · Bayesian Models and Methods

Zijian Chen, Archana Venkataraman

Human ratings are central to learning and inference across several application domains, but they are also subject to inter-rater biases and judgment errors. Quantifying the uncertainty of these human ratings would require repeated measurements, which are expensive and rarely available at scale. We propose a Bayesian graphical model to estimate the instance-level and item-level uncertainty of (subjective) human ratings by leveraging auxiliary (objective) data. Our model learns a shared latent content representation that explains factors common to both the human rating and auxiliary data and a latent uncertainty variable that captures fluctuations in the human assessments via a data-conditioned prior. We develop a scalable amortized variational inference procedure that uses modality-appropriate neural encoders and decoders to represent the posterior factors. Experiments on synthetic data demonstrate that our framework can accurately recover the latent uncertainty under targeted ablations and stress tests. We further demonstrate our approach on a real-world dataset of paired functional MRI scans and behavioral testing for autism, thus highlighting the need for uncertainty quantification.

Probabilistic Methods · Gaussian Processes

Hui Li, Huafeng Liu, Chenguang Li, Tianxiao Zhang, Yajun Yang, Liping Jing

Parametric partial differential equations (PDEs) serve as fundamental models across science and engineering, yet constructing fast and accurate surrogate models from sparse, irregularly sampled observations with reliable uncertainty quantification remains challenging. Existing approaches struggle to simultaneously handle variable observation patterns, preserve physics consistency, and provide well-calibrated predictive uncertainty. We introduce Bias-Spectrum Neural Processes (BSNP), a unified meta-learning framework that systematically integrates weak structural priors (translation equivariance, locality) with strong physical priors (governing equations and boundary conditions). BSNP addresses two critical obstacles: discretization overfitting through stochastic collocation that resamples residual evaluation points, and uncertainty collapse through mean-field enforcement that applies PDE constraints only to predictive means while preserving learned uncertainty. Comprehensive experiments on nonlinear Poisson equations, Burgers dynamics, and Navier-Stokes flows demonstrate that BSNP achieves superior accuracy and well-calibrated uncertainty quantification in sparse-data regimes.

Social Aspects · Accountability, Transparency, and Interpretability

Peng Cui, Jiahao Zhang, Lijie Hu

While Contrastive Learning (CL) has revolutionized self-supervised representation learning, its latent representations remain highly entangled and opaque, limiting their interpretability in safety-critical applications. We identify that a fundamental cause of this entanglement is the reliance on deterministic similarity measures, which treat all feature dimensions equally. In compositional scenes, this creates an Optimization Conflict: common background features, such as, "blue sky", are encouraged to align in positive pairs but simultaneously repelled in negative pairs, causing gradient oscillations that hinder precise semantic disentanglement. To address this, we propose **BayesNCL** (Bayesian Gated Non-Negative Contrastive Learning). Unlike standard approaches, BayesNCL introduces a probabilistic gating mechanism that dynamically filters out task-irrelevant, high-frequency common features while selectively retaining discriminative semantics. By formalizing feature selection as a variational inference problem with a sparse Bernoulli prior, our method effectively resolves the optimization conflict. Empirical experimental results on benchmark datasets demonstrate that BayesNCL achieves a remarkable 142.1\% improvement in semantic consistency compared to state-of-the-art baselines, yielding highly interpretable representations without compromising downstream task performance.

Probabilistic Methods · Monte Carlo and Sampling Methods

Xiaoning Wang, Huo Yuyang, Liuhua Peng, Changliang Zou

$U$-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for $U$-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for $U$-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting $U$-statistic, which is proposed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to $U$-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.

Deep Learning · Generative Models and Autoencoders

Andreas Grivas, Lorenzo Loconte, Emile van Krieken, Piotr Nawrot, Yu Zhao, Euan Wielewski, Pasquale Minervini, Edoardo Ponti, Antonio Vergari

Multi-token prediction (MTP) is a prominent strategy to significantly speed up generation in large language models (LLMs), especially in byte-level LLMs, which are tokeniser-free but prohibitively slow. However, existing MTP methods often sacrifice expressiveness by assuming independence between future tokens. In this work, we investigate the trade-off between expressiveness and latency in MTP within the framework of probabilistic circuits (PCs). Our framework, named MTPC, allows one to explore different ways to encode the joint distributions over future tokens by selecting different circuit architectures, generalising classical models such as (hierarchical) mixture models, hidden Markov models, and tensor networks. We show the efficacy of MTPC by retrofitting existing byte-level LLMs, such as EvaByte, and byte-fied subword models, such as Llama3.2 3B. Our experiments show that, when combined with speculative decoding, MTPC substantially speeds up generation compared to MTP with independence assumptions, while guaranteeing to retain the performance of the original verifier LLM. We also rigorously study the optimal trade-off between expressiveness and latency when exploring the possible parameterisations of MTPC, such as PC architectures and partial layer sharing between the verifier and draft LLMs.

Applications · Language, Speech and Dialog

Clara Meister, Ahmetcan Yavuz, Pietro Lesci, Tiago Pimentel

Language Identification (LID) is an important component of many multilingual natural language processing pipelines, where it facilitates corpus curation, training data analysis, and cross-lingual evaluation of large language models. Despite near-perfect performance on high-resource languages, existing systems remain brittle in low-resource and closely related language settings. We introduce a simple and efficient LID method based on the probabilistic framing, parameter estimation technique and inference strategy of the UnigramLM tokenization algorithm. In short, we learn language-conditional unigram distributions over a shared tokenizer vocabulary but treat segmentation as a language-specific phenomenon. Our formulation is data- and compute-efficient, supports incremental addition of new languages without retraining existing models, and can naturally be integrated into existing language model tokenization pipelines. Empirical evaluations against widely used baselines, including fastText, GlotLID-M, and CLD3, show that UniLID achieves competitive performance on standard benchmarks, substantially improves sample efficiency in low-resource settings--surpassing 70% accuracy with as few as five labeled samples per language---and delivers large gains on fine-grained dialect identification.

Social Aspects · Fairness

Li Zhang, Yuyuan Li, XiaoHua Feng, Jiaming Zhang, Fengyuan Yu, Chaochao Chen

Ensuring fair and equitable treatment across diverse groups, particularly in multi-class classification tasks, poses a significant challenge due to the persistent biases inherent in machine learning models. Most existing bias mitigation techniques are tailored to binary settings, and the presence of multi-dimensional outputs and complex fairness mechanisms makes their extension to multi-class scenarios neither straightforward nor effective. In this paper, we investigate two fundamental, unresolved challenges in fair classification: (i) *characterizing the optimal accuracy-fairness frontier in multi-class settings*, and (ii) *designing practical algorithms that attain this optimum across different training phases*. To tackle these challenges, we first specify an analytically tractable probabilistic formulation of the optimal classifier under fairness constraints. Building upon this, we propose two attribute-blind algorithms to enforce fairness requirements in practice: an in-processing approach for fairness intervention during training via the reduction approach, and a post-processing approach for fine-tuning output probabilities with plug-in estimation. Theoretical analysis reveals that both methods converge to the optimal accuracy-fairness Pareto frontier. Experiments conducted on multiple datasets demonstrate the superior performance of our methods in balancing accuracy and fairness.

General Machine Learning · Representation Learning

Luyao Tang, Jiewei Zheng, Kunze Huang, Chaoqi Chen, Yue Huang, Cheng Chen

Generalized Category Discovery (GCD) assigns unlabeled instances, mixed with labeled data, to known or novel categories, requiring human-like compositional reasoning: reusing primitives learned from known classes and deciding when new combinations imply new categories. Existing GCD methods operate on unstructured token features and struggle to extrapolate to novel compositions. We propose CoGe-GCD, which rethinks GCD through compositional generalization with two coupled stages. (i) Compositional Perception structures patch tokens by mapping them to a small vocabulary of primitives and refining token embeddings via competitive token-primitive assignment and information passing, yielding coherent groups for discovery. (ii) Generalizing Induction exploits the induced geometric structure and applies a geometric-structure-preserving calibration over spatial relations, maintaining probabilistic semantics while improving extrapolation to unseen primitive combinations. CoGe-GCD is implemented as an inductive-bias module between backbone and projection head, without modifying heads or losses, and can be plugged into diverse GCD frameworks. On standard benchmarks, it consistently improves all-class accuracy, unknown-class number estimation, and geometric quality, with marginal computational overhead.

Social Aspects · Safety

Rico Angell, Raghav Singhal, Zachary Horvitz, Zhou Yu, Rajesh Ranganath, Kathleen McKeown, He He

Language models are increasingly capable and are being rapidly deployed on a population-level scale. As a result, the safety of these models is increasingly high-stakes. Fortunately, advances in alignment have significantly reduced the likelihood of harmful model outputs. However, when models are queried billions of times in a day, even rare worst-case behaviors will occur. Current safety evaluations focus on capturing the distribution of inputs that yield harmful outputs. These evaluations disregard the probabilistic nature of models and their tail output behavior. To measure this tail risk, we propose a method to efficiently estimate the probability of harmful outputs for any input query. Instead of naive brute-force sampling from the target model, where harmful outputs could be rare, we operationalize importance sampling by creating unsafe versions of the target model. These unsafe versions enable sample-efficient estimation by making harmful outputs more probable. On benchmarks measuring misuse and misalignment, these estimates match brute-force Monte Carlo estimates using 10–20× fewer samples. For example, we can estimate probability of harmful outputs on the order of $10^{−4}$ with just 500 samples. Additionally, we find that these harmfulness estimates can reveal the sensitivity of models to perturbations in model input and predict deployment risks. Our work demonstrates that rare-event estimation is both critical and feasible for safety evaluations.

Theory · Probabilistic Methods

Dario Rancati, Jan Maas, Francesco Locatello

Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (${W}_2$) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the increasing popularity of diffusion models on discrete spaces using continuous-time Markov chains, a parallel theoretical framework based on gradient flows has remained elusive due to intrinsic challenges in translating the ${W}_2$ distance directly into these settings. In this work, we propose the first successful approach addressing these challenges, leveraging an appropriate metric $W_K$ on the simplex of probability distributions, which enables us to interpret widely used discrete diffusion paths, such as the discrete heat equation, as gradient flows of specific free-energy functionals. Through this theoretical insight, we introduce a novel methodology for learning diffusion dynamics over discrete spaces, which recovers the underlying functional directly by leveraging first-order optimality conditions for the JKO scheme. The resulting method optimizes a simple quadratic loss, trains extremely fast, does not require individual sample trajectories, and only needs a numerical preprocessing computing $W_K$-geodesics. We validate our method through extensive numerical experiments on synthetic data, showing that we can recover the underlying functional for a variety of graph classes.

Theory · Probabilistic Methods

Tal Wagner

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{\mathrm{k}}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The best known bounds for the Gaussian kernel are $O(d/\varepsilon^2)$, $\widetilde O(d+1/\varepsilon^4)$, and $\widetilde O(d+\Delta^2/\varepsilon^2)$, where $\Delta$ is the diameter of a region containing the points. We prove the new bound $\tilde O(d+\varepsilon\Delta^2+1/\varepsilon^3)$, which improves over the previous ones in regimes with small error $\varepsilon$ and intermediate diameter $\Delta$. At the center of our proof is a new fast spherical embedding theorem in the sense introduced by Bartal, Recht and Schulman (2011), which limits the embedded data diameter while preserving local Euclidean distances and avoiding ``distance collapse'' at larger scales. This fast embedding theorem may be of independent interest.

Probabilistic Methods · Variational Inference

Safa Messaoud, Skander Charni, Elaa Bouazza, Ali Pourghasemi, Halima Bensmail

Computing the differential entropy of distributions known only up to a normalization constant is a long-standing challenge with broad theoretical and practical significance. While variational inference is the most scalable approach for density approximation _from samples_, its potential in settings where _only the unnormalized density_ is available remains largely under-explored. The central difficulty lies in constructing variational distributions that simultaneously ($i$) exploit the structure of the unnormalized density, ($ii$) are expressive enough to capture complex target distributions, ($iii$) remain computationally tractable, and ($iv$) support efficient sampling. Recently, \citet{messaoud2024s} introduced _P-SVGD_, a particle-based variational method that leverages Stein Variational Gradient Descent dynamics, satisfies all of these constraints and demonstrates promising results in low-dimensional setups. We show, however, that _P-SVGD_ does not scale to high dimensions due to _fundamental algorithmic flaws_: ($i$) misdiagnosed sensitivity to \textit{SVGD} hyperparameters, ($ii$) violation of the global invertibility assumption in the entropy derivation, ($iii$) omission of a critical trace-of-Hessian term, ($iv$) along with suboptimal heuristics, including a divergence-based sampling check that induces mode collapse and loose informal bounds with no practical value. These issues severely limit both the correctness and the scalability of the approach. We propose _MET-SVGD_, a principled extension of _P-SVGD_ that addresses these flaws by providing a general framework for \textit{SVGD} hyperparameters selection with global invertibility and convergence guarantees. This enabled more accurate and scalable entropy estimation in high-dimensional settings. Empirically, in entropy estimation benchmarks, _MET-SVGD_ achieves accuracy improvements of up to 12$\times$ and 16$\times$ over _P-SVGD_ and baselines from the _SVGD_ literature, respectively. On CIFAR-10 Energy-Based image generation, it improves FID by $80.4$% compared to _P-SVGD_ and achieves 64$\times$ higher training stability. In Maximum-Entropy reinforcement learning, _MET-SVGD_ yields up to $16$% better returns than _P-SVGD_. We will make our code publicly available at https://tinyurl.com/2esyfx8j.

Probabilistic Methods · Bayesian Models and Methods

Niels Bracher, Lars Kühmichel, Desi Ivanova, Xavier Intes, Paul Buerkner, Stefan Radev

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences without density evaluations. Inference networks are instantiated with diffusion models that can approximate high-dimensional and multimodal posteriors at every experimental step. JADAI achieves superior or competitive performance across adaptive design benchmarks.

Probabilistic Methods · Graphical Models

Juha Harviainen, Frank Sommer, Manuel Sorge

Polytrees are a subclass of Bayesian networks that seek to capture the conditional dependencies between a set of $n$ variables as a directed forest and are motivated by their more efficient inference and improved interpretability. Since the problem of learning the best polytree is NP-hard, we study which restrictions make it more tractable by considering for example in-degree bounds, properties of score functions measuring the quality of a polytree, and approximation algorithms. We devise an algorithm that finds the optimal polytree in time $\mathcal{O}\big((2+\epsilon)^n\big)$ for arbitrarily small $\epsilon > 0$ and any constant in-degree bound $k$, improving over the fastest previously known algorithm of time complexity $\mathcal{O}\big(3^n\big)$. We further give polynomial-time algorithms for finding a polytree whose score is within a factor of $k$ from the optimal one for arbitrary scores and a factor of $2$ for additive ones. Many of the results are complemented by (nearly) tight lower bounds for either the time complexity or the approximation factors.

Probabilistic Methods · Bayesian Models and Methods

Yaohong Yang, Sammie Katt, Samuel Kaski

Multi-objective Bayesian optimization (MOBO) provides a principled framework for optimizing expensive black-box functions with multiple objectives. However, existing MOBO methods often struggle with coverage, scalability with respect to the number of objectives, and integrating constraints and preferences. In this work we propose STAGE-BO, Sequential Targeting Adaptive Gap-Filling $\varepsilon$-Constraint Bayesian Optimization, that explicitly targets under-explored regions of the Pareto front. By analyzing the coverage of the approximate Pareto front, our method identifies the largest geometric gaps. These gaps are then used as constraints, which transforms the problem into a sequence of inequality-constrained subproblems, efficiently solved via constrained expected improvement acquisition. Our approach provides a uniform Pareto coverage without hypervolume computation and naturally applies to constrained, and preference-based settings. Experiments on synthetic and real-world benchmarks demonstrate superior coverage and competitive hypervolume performance against state-of-the-art baselines.