In clustering, strong dominance in the size of a particular cluster is often undesirable, motivating a measure of cluster size uniformity that can be used to filter such partitions. A basic requirement of such a measure is stability: partitions that differ only slightly in their point assignments should receive similar uniformity scores. A difficulty arises because cluster labels are not fixed objects; algorithms may produce different numbers of labels even when the underlying point distribution changes very little. Measures defined directly over labels can therefore become unstable under label-count perturbations. I introduce the Mass Agreement Score (MAS), a point-centric metric bounded in [0, 1] that evaluates the consistency of expected cluster size as measured from the perspective of points in each cluster. Its construction yields fragment robustness by design, assigning similar scores to partitions with similar bulk structure while remaining sensitive to genuine redistribution of cluster mass.

Loss functions play a central role in supervised classification. Cross-entropy (CE) is widely used, whereas the mean absolute error (MAE) loss can offer robustness but is difficult to optimize. Interpolating between the CE and MAE losses, generalized cross-entropy (GCE) has recently been introduced to provide a trade-off between optimization difficulty and robustness. Existing formulations of GCE result in a non-convex optimization over classification margins that is prone to underfitting, leading to poor performances with complex datasets. In this paper, we propose a minimax formulation of generalized cross-entropy (MGCE) that results in a convex optimization over classification margins. Moreover, we show that MGCEs can provide an upper bound on the classification error. The proposed bilevel convex optimization can be efficiently implemented using stochastic gradient computed via implicit differentiation. Using benchmark datasets, we show that MGCE achieves strong accuracy, faster convergence, and better calibration, especially in the presence of label noise.

Data unfolding -- the removal of noise or artifacts from measurements -- is a fundamental task across the experimental sciences. Of particular interest in the present work are applications of data unfolding in physics, in which context the dominant approach is RichardsonLucy (RL) deconvolution. The classical RL approach aims to find denoised data that, once passed through the noise model, is as close as possible to the measured data, in terms of Kullback-Leibler (KL) divergence. Fundamental to this approach is the hypothesis that the support of the measured data overlaps with the output of the noise model, so that the KL divergence correctly captures their similarity. In practice, this hypothesis is typically enforced by binning the measured data and noise model, introducing numerical error into the unfolding process. As a counterpoint to classical binned methods for unfolding, the present work studies an alternative formulation of the unfolding problem, using a Wasserstein loss instead of the KL divergence to quantify the similarity between the measured data and the output of the noise model. We establish sharp conditions for existence and uniqueness of optimizers; as a consequence we answer open questions of Li, et al. [23], regarding necessary conditions for existence and uniqueness in the case of transport map noise models. Following these theoretical results, we then develop a provably convergent generalized Sinkhorn algorithm to compute approximate optimizers. Our algorithm requires only empirical observations of the noise model and measured data and scales with the size of the data, rather than the ambient dimension.

Population-based learning paradigms, including evolutionary strategies, Population-Based Training (PBT), and recent model-merging methods, combine fast within-model optimisation with slower population-level adaptation. Despite their empirical success, a general mathematical description of the resulting collective training dynamics remains incomplete. We introduce a theoretical framework for neural network training based on two-time-scale population dynamics. We model a population of neural networks as an interacting agent system in which network parameters evolve through fast noisy gradient updates of SGD/Langevin type, while hyperparameters evolve through slower selection--mutation dynamics. We prove the large-population limit for the joint distribution of parameters and hyperparameters and, under strong time-scale separation, derive a selection--mutation equation for the hyperparameter density. For each fixed hyperparameter, the fast parameter dynamics relaxes to a Boltzmann--Gibbs measure, inducing an effective fitness for the slow evolution. The averaged dynamics connects population-based learning with bilevel optimisation and classical replicator--mutator models, yields conditions under which the population mean moves toward the fittest hyperparameter, and clarifies the role of noise and diversity in balancing optimisation and exploration. Numerical experiments illustrate both the large-population regime and the reduced two-time-scale dynamics, and indicate that access to the effective fitness, either in closed form or through population-level estimation, can improve population-level updates.

Neural Collapse is a phenomenon that helps identify sparse and low rank structures in deep classifiers. Recent work has extended the definition of neural collapse to regression problems, albeit only measuring the phenomenon at the last layer. In this paper, we establish that Neural Regression Collapse (NRC) also occurs below the last layer across different types of models. We show that in the collapsed layers of neural regression models, features lie in a subspace that corresponds to the target dimension, the feature covariance aligns with the target covariance, the input subspace of the layer weights aligns with the feature subspace, and the linear prediction error of the features is close to the overall prediction error of the model. In addition to establishing Deep NRC, we also show that models that exhibit Deep NRC learn the intrinsic dimension of low rank targets and explore the necessity of weight decay in inducing Deep NRC. This paper provides a more complete picture of the simple structure learned by deep networks in the context of regression.

Understanding how biomarker distributions evolve over time is a central challenge in digital health and chronic disease monitoring. In diabetes, changes in the distribution of glucose measurements can reveal patterns of disease progression and treatment response that conventional summary measures miss. Motivated by a 26-week clinical trial comparing the closed-loop insulin delivery system t:slim X2 with standard therapy in children with type 1 diabetes, we propose a probabilistic framework to model the continuous-time evolution of time-indexed distributions using continuous glucose monitoring data (CGM) collected every five minutes. We represent the glucose distribution as a Gaussian mixture, with time-varying mixture weights governed by a neural ODE. We estimate the model parameter using a distribution-matching criterion based on the maximum mean discrepancy. The resulting framework is interpretable, computationally efficient, and sensitive to subtle temporal distributional changes. Applied to CGM trial data, the method detects treatment-related improvements in glucose dynamics that are difficult to capture with traditional analytical approaches.

Privacy and algorithmic fairness have become two central issues in modern machine learning. Although each has separately emerged as a rapidly growing research area, their joint effect remains comparatively under-explored. In this paper, we systematically study the joint impact of differential privacy and fairness on classification in a federated setting, where data are distributed across multiple servers. Targeting demographic disparity constrained classification under federated differential privacy, we propose a two-step algorithm, namely FDP-Fair. In the special case where there is only one server, we further propose a simple yet powerful algorithm, namely CDP-Fair, serving as a computationally-lightweight alternative. Under mild structural assumptions, theoretical guarantees on privacy, fairness and excess risk control are established. In particular, we disentangle the source of the private fairness-aware excess risk into a) intrinsic cost of classification, b) cost of private classification, c) non-private cost of fairness and d) private cost of fairness. Our theoretical findings are complemented by extensive numerical experiments on both synthetic and real datasets, highlighting the practicality of our designed algorithms.

Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has emerged as a rapidly growing alternative framework that is particularly well suited to modern applications involving high-dimensional data and complex machine learning models. Its appeal stems from being both distribution-free -- relying mainly on symmetry assumptions such as exchangeability -- and model-agnostic, treating the learning algorithm as a black box. Even under such limited assumptions, conformal prediction provides exact finite-sample guarantees, though these are typically of a marginal nature that requires careful interpretation. This paper explains the core ideas of conformal prediction and reviews selected methods. Rather than offering an exhaustive survey, it aims to provide a clear conceptual entry point and a pedagogical overview of the field.

We introduce Exponential Family Discriminant Analysis (EFDA), a unified generative framework that extends classical Linear Discriminant Analysis (LDA) beyond the Gaussian setting to any member of the exponential family. Under the assumption that each class-conditional density belongs to a common exponential family, EFDA derives closed-form maximum-likelihood estimators for all natural parameters and yields a decision rule that is linear in the sufficient statistic, recovering LDA as a special case and capturing nonlinear decision boundaries in the original feature space. We prove that EFDA is asymptotically calibrated and statistically efficient under correct specification, and we generalise it to $K \geq 2$ classes and multivariate data. Through extensive simulation across five exponential-family distributions (Weibull, Gamma, Exponential, Poisson, Negative Binomial), EFDA matches the classification accuracy of LDA, QDA, and logistic regression while reducing Expected Calibration Error (ECE) by $2$-$6\times$, a gap that is structural: it persists for all $n$ and across all class-imbalance levels, because misspecified models remain asymptotically miscalibrated. We further prove and empirically confirm that EFDA's log-odds estimator approaches the Cramér-Rao bound under correct specification, and is the only estimator in our comparison whose mean squared error converges to zero. Complete derivations are provided for nine distributions. Finally, we formally verify all four theoretical propositions in Lean 4, using Aristotle (Harmonic) and OpenGauss (Math, Inc.) as proof generators, with all outputs independently machine-checked by AXLE (Axiom).

Despite the strong predictive performance achieved by machine learning models across many application domains, assessing their trustworthiness through reliable estimates of predictive confidence remains a critical challenge. This issue arises in scenarios where the likelihood of error inferred from learned representations follows a bimodal distribution, resulting from the coexistence of confident and ambiguous predictions. Standard regression approaches often struggle to adequately express this predictive uncertainty, as they implicitly assume unimodal Gaussian noise, leading to mean-collapse behavior in such settings. Although Mixture Density Networks (MDNs) can represent different distributions, they suffer from severe optimization instability. We propose a family of distribution-aware loss functions integrating normalized RMSE with Wasserstein and Cramér distances. When applied to standard deep regression models, our approach recovers bimodal distributions without the volatility of mixture models. Validated across four experimental stages, our results show that the proposed Wasserstein loss establishes a new Pareto efficiency frontier: matching the stability of standard regression losses like MSE in unimodal tasks while reducing Jensen-Shannon Divergence by 45% on complex bimodal datasets. Our framework strictly dominates MDNs in both fidelity and robustness, offering a reliable tool for aleatoric uncertainty estimation in trustworthy AI systems.