Deep neural networks are renowned for their ability to generalise well across diverse tasks, even when heavily overparameterized. Existing works offer only partial explanations (for example, the NTK-based task-model alignment explanation neglects feature learning). Here, we provide an end-to-end, analytically tractable case study that links a network's inductive prior, its training dynamics including feature learning, and its eventual generalisation. Specifically, we exploit the one-to-one correspondence between depth-2 discrete fully connected networks and disjunctive normal form (DNF) formulas by training on Boolean functions. Under a Monte Carlo learning algorithm, our model exhibits predictable training dynamics and the emergence of interpretable features. This framework allows us to trace, in detail, how inductive bias and feature formation drive generalisation.

Kernel method-based intensity estimators, formulated within reproducing kernel Hilbert spaces (RKHSs), and classical kernel intensity estimators (KIEs) have been among the most easy-to-implement and feasible methods for estimating the intensity functions of inhomogeneous Poisson processes. While both approaches share the term "kernel", they are founded on distinct theoretical principles, each with its own strengths and limitations. In this paper, we propose a novel regularized kernel method for Poisson processes based on the least squares loss and show that the resulting intensity estimator involves a specialized variant of the representer theorem: it has the dual coefficient of unity and coincides with classical KIEs. This result provides new theoretical insights into the connection between classical KIEs and kernel method-based intensity estimators, while enabling us to develop an efficient KIE by leveraging advanced techniques from RKHS theory. We refer to the proposed model as the kernel method-based kernel intensity estimator (K$^2$IE). Through experiments on synthetic datasets, we show that K$^2$IE achieves comparable predictive performance while significantly surpassing the state-of-the-art kernel method-based estimator in computational efficiency.

Multimodal contrastive learning is a methodology for linking different data modalities; the canonical example is linking image and text data. The methodology is typically framed as the identification of a set of encoders, one for each modality, that align representations within a common latent space. In this work, we focus on the bimodal setting and interpret contrastive learning as the optimization of (parameterized) encoders that define conditional probability distributions, for each modality conditioned on the other, consistent with the available data. This provides a framework for multimodal algorithms such as crossmodal retrieval, which identifies the mode of one of these conditional distributions, and crossmodal classification, which is similar to retrieval but includes a fine-tuning step to make it task specific. The framework we adopt also gives rise to crossmodal generative models. This probabilistic perspective suggests two natural generalizations of contrastive learning: the introduction of novel probabilistic loss functions, and the use of alternative metrics for measuring alignment in the common latent space. We study these generalizations of the classical approach in the multivariate Gaussian setting. In this context we view the latent space identification as a low-rank matrix approximation problem. This allows us to characterize the capabilities of loss functions and alignment metrics to approximate natural statistics, such as conditional means and covariances; doing so yields novel variants on contrastive learning algorithms for specific mode-seeking and for generative tasks. The framework we introduce is also studied through numerical experiments on multivariate Gaussians, the labeled MNIST dataset, and on a data assimilation application arising in oceanography.

Variational inference often struggles with the posterior geometry exhibited by complex hierarchical Bayesian models. Recent advances in flow-based variational families and Variationally Inferred Parameters (VIP) each address aspects of this challenge, but their formal relationship is unexplored. Here, we prove that the combination of VIP and a full-rank Gaussian can be represented exactly as a forward autoregressive flow augmented with a translation term and input from the model's prior. Guided by this theoretical insight, we introduce the Model-Informed Flow (MIF) architecture, which adds the necessary translation mechanism, prior information, and hierarchical ordering. Empirically, MIF delivers tighter posterior approximations and matches or exceeds state-of-the-art performance across a suite of hierarchical and non-hierarchical benchmarks.

In this paper, we investigate a class of approximate Gaussian processes (GP) obtained by taking a linear combination of compactly supported basis functions with the basis coefficients endowed with a dependent Gaussian prior distribution. This general class includes a popular approach that uses a finite element approximation of the stochastic partial differential equation (SPDE) associated with Matérn GP. We explored another scalable alternative popularly used in the computer emulation literature where the basis coefficients at a lattice are drawn from a Gaussian process with an inverse-Gamma bandwidth. For both approaches, we study concentration rates of the posterior distribution. We demonstrated that the SPDE associated approach with a fixed smoothness parameter leads to a suboptimal rate despite how the number of basis functions and bandwidth are chosen when the underlying true function is sufficiently smooth. On the flip side, we showed that the later approach is rate-optimal adaptively over all smoothness levels of the underlying true function if an appropriate prior is placed on the number of basis functions. Efficient computational strategies are developed and numerics are provided to illustrate the theoretical results.

Bayesian models based on Gaussian processes (GPs) offer a flexible framework to predict spatially distributed variables with uncertainty. But the use of non-stationary priors, often necessary for capturing complex spatial patterns, makes sampling from the predictive posterior distribution (PPD) computationally intractable. In this paper, we propose a two-step approach based on diffusion generative models (DGMs) to mimic PPDs associated with non-stationary GP priors: we replace the GP prior by a DGM surrogate, and leverage recent advances on training-free guidance algorithms for DGMs to sample from the desired posterior distribution. We apply our approach to a rich non-stationary GP prior from which exact posterior sampling is untractable and validate that the issuing distributions are close to their GP counterpart using several statistical metrics. We also demonstrate how one can fine-tune the trained DGMs to target specific parts of the GP prior. Finally we apply the proposed approach to solve inverse problems arising in environmental sciences, thus yielding state-of-the-art predictions.

Current deep reinforcement learning (DRL) approaches achieve state-of-the-art performance in various domains, but struggle with data efficiency compared to human learning, which leverages core priors about objects and their interactions. Active inference offers a principled framework for integrating sensory information with prior knowledge to learn a world model and quantify the uncertainty of its own beliefs and predictions. However, active inference models are usually crafted for a single task with bespoke knowledge, so they lack the domain flexibility typical of DRL approaches. To bridge this gap, we propose a novel architecture that integrates a minimal yet expressive set of core priors about object-centric dynamics and interactions to accelerate learning in low-data regimes. The resulting approach, which we call AXIOM, combines the usual data efficiency and interpretability of Bayesian approaches with the across-task generalization usually associated with DRL. AXIOM represents scenes as compositions of objects, whose dynamics are modeled as piecewise linear trajectories that capture sparse object-object interactions. The structure of the generative model is expanded online by growing and learning mixture models from single events and periodically refined through Bayesian model reduction to induce generalization. AXIOM masters various games within only 10,000 interaction steps, with both a small number of parameters compared to DRL, and without the computational expense of gradient-based optimization.

Sample efficiency remains a major obstacle for real world adoption of reinforcement learning (RL): success has been limited to settings where simulators provide access to essentially unlimited environment interactions, which in reality are typically costly or dangerous to obtain. Offline RL in principle offers a solution by exploiting offline data to learn a near-optimal policy before deployment. In practice, however, current offline RL methods rely on extensive online interactions for hyperparameter tuning, and have no reliable bound on their initial online performance. To address these two issues, we introduce two algorithms. Firstly, SOReL: an algorithm for safe offline reinforcement learning. Using only offline data, our Bayesian approach infers a posterior over environment dynamics to obtain a reliable estimate of the online performance via the posterior predictive uncertainty. Crucially, all hyperparameters are also tuned fully offline. Secondly, we introduce TOReL: a tuning for offline reinforcement learning algorithm that extends our information rate based offline hyperparameter tuning methods to general offline RL approaches. Our empirical evaluation confirms SOReL's ability to accurately estimate regret in the Bayesian setting whilst TOReL's offline hyperparameter tuning achieves competitive performance with the best online hyperparameter tuning methods using only offline data. Thus, SOReL and TOReL make a significant step towards safe and reliable offline RL, unlocking the potential for RL in the real world. Our implementations are publicly available: https://github.com/CWibault/sorel\_torel.

We study the robustness of Transformer language models under semantic out-of-distribution (OOD) shifts, where training and test data lie in disjoint latent spaces. Using Wasserstein-1 distance and Gevrey-class smoothness, we derive sub-exponential upper bounds on prediction error. Our theoretical framework explains how smoothness governs generalization under distributional drift. We validate these findings through controlled experiments on arithmetic and Chain-of-Thought tasks with latent permutations and scalings. Results show empirical degradation aligns with our bounds, highlighting the geometric and functional principles underlying OOD generalization in Transformers.

Large vision-language models have become widely adopted to advance in various domains. However, developing a trustworthy system with minimal interpretable characteristics of large-scale models presents a significant challenge. One of the most prevalent terms associated with the fallacy functions caused by these systems is hallucination, where the language model generates a response that does not correspond to the visual content. To mitigate this problem, several approaches have been developed, and one prominent direction is to ameliorate the decoding process. In this paper, we propose a new Bijective Maximum Likelihood Learning (BIMA) approach to hallucination mitigation using normalizing flow theories. The proposed BIMA method can efficiently mitigate the hallucination problem in prevailing vision-language models, resulting in significant improvements. Notably, BIMA achieves the average F1 score of 85.06% on POPE benchmark and remarkably reduce CHAIRS and CHAIRI by 7.6% and 2.6%, respectively. To the best of our knowledge, this is one of the first studies that contemplates the bijection means to reduce hallucination induced by large vision-language models.