Supervised machine learning describes the practice of fitting a parameterized model to labeled input-output data. Supervised machine learning methods have demonstrated promise in learning efficient surrogate models that can (partially) replace expensive high-fidelity models, making many-query analyses, such as optimization, uncertainty quantification, and inference, tractable. However, when training data must be obtained through the evaluation of an expensive model or experiment, the amount of training data that can be obtained is often limited, which can make learned surrogate models unreliable. However, in many engineering and scientific settings, cheaper \emph{low-fidelity} models may be available, for example arising from simplified physics modeling or coarse grids. These models may be used to generate additional low-fidelity training data. The goal of \emph{multifidelity} machine learning is to use both high- and low-fidelity training data to learn a surrogate model which is cheaper to evaluate than the high-fidelity model, but more accurate than any available low-fidelity model. This work proposes a new multifidelity training approach for Gaussian process regression which uses low-fidelity data to define additional features that augment the input space of the learned model. The approach unites desirable properties from two separate classes of existing multifidelity GPR approaches, cokriging and autoregressive estimators. Numerical experiments on several test problems demonstrate both increased predictive accuracy and reduced computational cost relative to the state of the art.

One of the most common machine learning setups is logistic regression. In many classification models, including neural networks, the final prediction is obtained by applying a logistic link function to a linear score. In binary logistic regression, the feedback can be either soft labels, corresponding to the true conditional probability of the data (as in distillation), or sampled hard labels (taking values $\pm 1$). We point out a fundamental problem that arises even in a particularly favorable setting, where the goal is to learn a noise-free soft target of the form $σ(\mathbf{x}^{\top}\mathbf{w}^{\star})$. In the over-constrained case (i.e. the number of samples $n$ exceeds the input dimension $d$) with examples $(\mathbf{x}_i,σ(\mathbf{x}_i^{\top}\mathbf{w}^{\star}))$, it is sufficient to recover $\mathbf{w}^{\star}$ and hence achieve the Bayes risk. However, we prove that when the examples are labeled by hard labels $y_i$ sampled from the same conditional distribution $σ(\mathbf{x}_i^{\top}\mathbf{w}^{\star})$ and $\mathbf{w}^{\star}$ is $s$-sparse, then rotation-invariant algorithms are provably suboptimal: they incur an excess risk $Ω\!\left(\frac{d-1}{n}\right)$, while there are simple non-rotation invariant algorithms with excess risk $O(\frac{s\log d}{n})$. The simplest rotation invariant algorithm is gradient descent on the logistic loss (with early stopping). A simple non-rotation-invariant algorithm for sparse targets that achieves the above upper bounds uses gradient descent on the weights $u_i,v_i$, where now the linear weight $w_i$ is reparameterized as $u_iv_i$.

The widespread use of AI and ML models in sensitive areas raises significant concerns about fairness. While the research community has introduced various methods for bias mitigation in binary classification tasks, the issue remains under-explored in multi-class classification settings. To address this limitation, in this paper, we first formulate the problem of fair learning in multi-class classification as a multi-objective problem between effectiveness (i.e., prediction correctness) and multiple linear fairness constraints. Next, we propose a Generalised Exponentiated Gradient (GEG) algorithm to solve this task. GEG is an in-processing algorithm that enhances fairness in binary and multi-class classification settings under multiple fairness definitions. We conduct an extensive empirical evaluation of GEG against six baselines across seven multi-class and three binary datasets, using four widely adopted effectiveness metrics and three fairness definitions. GEG overcomes existing baselines, with fairness improvements up to 92% and a decrease in accuracy up to 14%.

In many classification problems, the costs of misclassifying observations from different classes can be highly unequal. The Neyman-Pearson multiclass classification (NPMC) framework addresses this issue by minimizing a weighted misclassification risk while imposing upper bounds on class-specific error probabilities. Existing NPMC methods typically assume that training labels are correctly observed. In practice, however, labels are often corrupted due to measurement error or annotation, and the effect of such label noise on NPMC procedures remains largely unexplored. We study the NPMC problem when only noisy labels are available in the training data. We propose an empirical likelihood (EL)-based method that relates the distributions of noisy and true labels through an exponential tilting density ratio model. The resulting maximum EL estimators recover the class proportions and posterior probabilities of the clean labels required for error control. We establish consistency, asymptotic normality, and optimal convergence rates for these estimators. Under mild conditions, the resulting classifier satisfies NP oracle inequalities with respect to the true labels asymptotically. An expectation-maximization algorithm computes the maximum EL estimators. Simulations show that the proposed method performs comparably to the oracle classifier under clean labels and substantially improves over procedures that ignore label noise.

We introduce Generalized Discrete Diffusion from Snapshots (GDDS), a unified framework for discrete diffusion modeling that supports arbitrary noising processes over large discrete state spaces. Our formulation encompasses all existing discrete diffusion approaches, while allowing significantly greater flexibility in the choice of corruption dynamics. The forward noising process relies on uniformization and enables fast arbitrary corruption. For the reverse process, we derive a simple evidence lower bound (ELBO) based on snapshot latents, instead of the entire noising path, that allows efficient training of standard generative modeling architectures with clear probabilistic interpretation. Our experiments on large-vocabulary discrete generation tasks suggest that the proposed framework outperforms existing discrete diffusion methods in terms of training efficiency and generation quality, and beats autoregressive models for the first time at this scale. We provide the code along with a blog post on the project page : \href{https://oussamazekri.fr/gdds}{https://oussamazekri.fr/gdds}.

This paper proposes a new formulation of functional Gaussian Process regression in manifolds, based on an Empirical Bayes approach, in the spatiotemporal random field context. We apply the machinery of tight Gaussian measures in separable Hilbert spaces, exploiting the invariance property of covariance kernels under the group of isometries of the manifold. The identification of these measures with infinite-product Gaussian measures is then obtained via the eigenfunctions of the Laplace-Beltrami operator on the manifold. The involved time-varying angular spectra constitute the key tool for dimension reduction in the implementation of this regression approach, adopting a suitable truncation scheme depending on the functional sample size. The simulation study and synthetic data application undertaken illustrate the finite sample and asymptotic properties of the proposed functional regression predictor.

Existing machine learning frameworks for compliance monitoring -- Markov Logic Networks, Probabilistic Soft Logic, supervised models -- share a fundamental paradigm: they treat observed data as ground truth and attempt to approximate rules from it. This assumption breaks down in rule-governed domains such as taxation or regulatory compliance, where authoritative rules are known a priori and the true challenge is to infer the latent state of rule activation, compliance, and parametric drift from partial and noisy observations. We propose Rule-State Inference (RSI), a Bayesian framework that inverts this paradigm by encoding regulatory rules as structured priors and casting compliance monitoring as posterior inference over a latent rule-state space S = {(a_i, c_i, delta_i)}, where a_i captures rule activation, c_i models the compliance rate, and delta_i quantifies parametric drift. We prove three theoretical guarantees: (T1) RSI absorbs regulatory changes in O(1) time via a prior ratio correction, independently of dataset size; (T2) the posterior is Bernstein-von Mises consistent, converging to the true rule state as observations accumulate; (T3) mean-field variational inference monotonically maximizes the Evidence Lower BOund (ELBO). We instantiate RSI on the Togolese fiscal system and introduce RSI-Togo-Fiscal-Synthetic v1.0, a benchmark of 2,000 synthetic enterprises grounded in real OTR regulatory rules (2022-2025). Without any labeled training data, RSI achieves F1=0.519 and AUC=0.599, while absorbing regulatory changes in under 1ms versus 683-1082ms for full model retraining -- at least a 600x speedup.

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.

Physical AI agents, such as robots and other embodied systems operating under tight and fluctuating resource constraints, remain far less capable than biological agents in open-ended real-world environments. This paper argues that Active Inference (AIF), grounded in the Free Energy Principle, offers a principled foundation for closing that gap. We develop this argument from first principles, following a chain from probability theory through Bayesian machine learning and variational inference to active inference and reactive message passing. From the FEP perspective, systems that maintain their structural and functional integrity over time can, under suitable assumptions, be described as minimizing variational free energy (VFE), and AIF operationalizes this by unifying perception, learning, planning, and control within a single computational objective. We show that VFE minimization is naturally realized by reactive message passing on factor graphs, where inference emerges from local, parallel computations. This realization is well matched to the constraints of physical operation, including hard deadlines, asynchronous data, fluctuating power budgets, and changing environments. Because reactive message passing is event-driven, interruptible, and locally adaptable, performance degrades gracefully under reduced resources while model structure can adjust online. We further show that, under suitable coupling and coarse-graining conditions, coupled AIF agents can be described as higher-level AIF agents, yielding a homogeneous architecture based on the same message-passing primitive across scales. Our contribution is not empirical benchmarking, but a clear theoretical and architectural case for the engineering community.

Exploring the dynamic co-evolution of multiplex graphs and nodal attributes is a compelling question in criminal and terrorism networks. This article is motivated by the study of dynamically evolving interactions among prominent terrorist organizations, considering various organizational attributes like size, ideology, leadership, and operational capacity. Statistically principled integration of multiplex graphs with nodal attributes is significantly challenging due to the need to leverage shared information within and across layers, account for uncertainty in predicting unobserved links, and capture temporal evolution of node attributes. These difficulties increase when layers are partially observed, as in terrorism networks where connections are deliberately hidden to obscure key relationships. To address these challenges, we present a principled methodological framework to integrate the multiplex graph layers and nodal attributes. The approach employs time-varying stochastic latent factor models, leveraging shared latent factors to capture graph structure and its co-evolution with node attributes. Latent factors are modeled using Gaussian processes with an infinitely wide deep neural network-based covariance function, termed neural network Gaussian processes (NN-GP). The NN-GP framework on latent factors exploits the predictive power of Bayesian deep neural network architecture while propagating uncertainty for reliability. Simulation studies highlight superior performance of the proposed approach in achieving inferential objectives. The approach, termed as dynamic joint learner, enables predictive inference (with uncertainty) of diverse unobserved dynamic relationships among prominent terrorist organizations and their organization-specific attributes, as well as clustering behavior in terms of friend-and-foe relationships, which could be informative in counter-terrorism research.