We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $λ$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.

In statistical learning, models are classified as regular or singular depending on whether the mapping from parameters to probability distributions is injective. Most models with hierarchical structures or latent variables are singular, for which conventional criteria such as the Akaike Information Criterion and the Bayesian Information Criterion are inapplicable due to the breakdown of normal approximations for the likelihood and posterior. To address this, the Widely Applicable Information Criterion (WAIC) and the Widely Applicable Bayesian Information Criterion (WBIC) have been proposed. Since WAIC and WBIC are computed using posterior distributions at different temperature settings, separate posterior sampling is generally required. In this paper, we theoretically derive an asymptotic equation that links WAIC and WBIC, despite their dependence on different posteriors. This equation yields an asymptotically unbiased expression of WAIC in terms of the posterior distribution used for WBIC. The result clarifies the structural relationship between these criteria within the framework of singular learning theory, and deepens understanding of their asymptotic behavior. This theoretical contribution provides a foundation for future developments in the computational efficiency of model selection in singular models.

Large Language Models (LLMs) are increasingly leveraged for translation tasks but often fall short when translating inclusive language -- such as texts containing the singular 'they' pronoun or otherwise reflecting fair linguistic protocols. Because these challenges span both computational and societal domains, it is imperative to critically evaluate how well LLMs handle inclusive translation with a well-founded framework. This paper presents FairTranslate, a novel, fully human-annotated dataset designed to evaluate non-binary gender biases in machine translation systems from English to French. FairTranslate includes 2418 English-French sentence pairs related to occupations, annotated with rich metadata such as the stereotypical alignment of the occupation, grammatical gender indicator ambiguity, and the ground-truth gender label (male, female, or inclusive). We evaluate four leading LLMs (Gemma2-2B, Mistral-7B, Llama3.1-8B, Llama3.3-70B) on this dataset under different prompting procedures. Our results reveal substantial biases in gender representation across LLMs, highlighting persistent challenges in achieving equitable outcomes in machine translation. These findings underscore the need for focused strategies and interventions aimed at ensuring fair and inclusive language usage in LLM-based translation systems. We make the FairTranslate dataset publicly available on Hugging Face, and disclose the code for all experiments on GitHub.

We show that the behavior of stochastic gradient descent is related to Bayesian statistics by showing that SGD is effectively diffusion on a fractal landscape, where the fractal dimension can be accounted for in a purely Bayesian way. By doing this we show that SGD can be regarded as a modified Bayesian sampler which accounts for accessibility constraints induced by the fractal structure of the loss landscape. We verify our results experimentally by examining the diffusion of weights during training. These results offer insight into the factors which determine the learning process, and seemingly answer the question of how SGD and purely Bayesian sampling are related.

This study explores the capacity of large language models (LLMs) for explicit learning, a process involving the assimilation of metalinguistic explanations to carry out language tasks. Using constructed languages generated by cryptographic means as controlled test environments, we designed experiments to assess an LLM's ability to explicitly learn and apply grammar rules. Our results demonstrate that while LLMs possess a measurable capacity for explicit learning, this ability diminishes as the complexity of the linguistic phenomena at hand increases. Supervised fine-tuning on chains of thought significantly enhances LLM performance but struggles to generalize to typologically novel or more complex linguistic features. These findings point to the need for more diverse training sets and alternative fine-tuning strategies to further improve explicit learning by LLMs.

Gender-inclusive language is often used with the aim of ensuring that all individuals, regardless of gender, can be associated with certain concepts. While psycholinguistic studies have examined its effects in relation to human cognition, it remains unclear how Large Language Models (LLMs) process gender-inclusive language. Given that commercial LLMs are gaining an increasingly strong foothold in everyday applications, it is crucial to examine whether LLMs in fact interpret gender-inclusive language neutrally, because the language they generate has the potential to influence the language of their users. This study examines whether LLM-generated coreferent terms align with a given gender expression or reflect model biases. Adapting psycholinguistic methods from French to English and German, we find that in English, LLMs generally maintain the antecedent's gender but exhibit underlying masculine bias. In German, this bias is much stronger, overriding all tested gender-neutralization strategies.

This paper presents a complete explainable system that interprets a set of data, abstracts the underlying features and describes them in a natural language of choice. The system relies on two crucial stages: (i) identifying emerging properties from data and transforming them into abstract concepts, and (ii) converting these concepts into natural language. Despite the impressive natural language generation capabilities demonstrated by Large Language Models, their statistical nature and the intricacy of their internal mechanism still force us to employ these techniques as black boxes, forgoing trustworthiness. Developing an explainable pipeline for data interpretation would allow facilitating its use in safety-critical environments like processing medical information and allowing non-experts and visually impaired people to access narrated information. To this end, we believe that the fields of knowledge representation and automated reasoning research could present a valid alternative. Expanding on prior research that tackled the first stage (i), we focus on the second stage, named Concept2Text. Being explainable, data translation is easily modeled through logic-based rules, once again emphasizing the role of declarative programming in achieving AI explainability. This paper explores a Prolog/CLP-based rewriting system to interpret concepts-articulated in terms of classes and relations, plus common knowledge-derived from a generic ontology, generating natural language text. Its main features include hierarchical tree rewritings, modular multilingual generation, support for equivalent variants across semantic, grammar, and lexical levels, and a transparent rule-based system. We outline the architecture and demonstrate its flexibility through some examples capable of generating numerous diverse and equivalent rewritings based on the input concept.

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.