We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural ``Ville event'' $E_α$, this regret till time $T$ is bounded by $\ln^2(1/α)/V_T + \ln (1/α) + \ln \ln V_T$ up to universal constants, where $V_T$ is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to $\ln(1/α) + \ln \ln V_T$ if $V_T \geq \ln(1/α)$.) If the data were stochastic, then one can show that $E_α$ has probability at least $1-α$ under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event $E_0$ of probability one, the regret on every path in $E_0$ is eventually bounded by $\ln \ln V_T$ (up to constants). We explain how this work helps bridge the world of adversarial online learning (which usually deals with regret bounds for bounded data), with game-theoretic statistics (which can handle unbounded data, albeit using stochastic assumptions). In short, conditional regret bounds serve as a bridge between stochastic and adversarial betting.

We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.

Model selection is a pivotal process in the quantitative sciences, where researchers must navigate between numerous candidate models of varying complexity. Traditional information criteria, such as the corrected Akaike Information Criterion (AICc), Bayesian Information Criterion (BIC), and Mallows's $\mathit{C_p}$, are valuable tools for identifying optimal models. However, the exponential increase in candidate models with each additional model parameter renders the evaluation of these criteria for all models -- a strategy known as exhaustive, or brute-force, searches -- computationally prohibitive. Consequently, heuristic approaches like stepwise regression are commonly employed, albeit without guarantees of finding the globally-optimal model. In this study, we challenge the prevailing notion that non-monotonicity in information criteria precludes bounds on the search space. We introduce a simple but novel bound that enables the development of branch-and-bound algorithms tailored for these non-monotonic functions. We demonstrate that our approach guarantees identification of the optimal model(s) across diverse model classes, sizes, and applications, often with orders of magnitude computational speedups. For instance, in one previously-published model selection task involving $2^{32}$ (approximately 4 billion) candidate models, our method achieves a computational speedup exceeding 6,000. These findings have broad implications for the scalability and effectiveness of model selection in complex scientific domains.

Understanding and distinguishing temporal patterns in time series data is essential for scientific discovery and decision-making. For example, in biomedical research, uncovering meaningful patterns in physiological signals can improve diagnosis, risk assessment, and patient outcomes. However, existing methods for time series pattern discovery face major challenges, including high computational complexity, limited interpretability, and difficulty in capturing meaningful temporal structures. To address these gaps, we introduce a novel learning framework that jointly trains two Transformer models using complementary time series representations: shapelet-based representations to capture localized temporal structures and traditional feature engineering to encode statistical properties. The learned shapelets serve as interpretable signatures that differentiate time series across classification labels. Additionally, we develop a visual analytics system -- SigTIme -- with coordinated views to facilitate exploration of time series signatures from multiple perspectives, aiding in useful insights generation. We quantitatively evaluate our learning framework on eight publicly available datasets and one proprietary clinical dataset. Additionally, we demonstrate the effectiveness of our system through two usage scenarios along with the domain experts: one involving public ECG data and the other focused on preterm labor analysis.

Goal-Conditioned Reinforcement Learning (GCRL) mitigates the difficulty of reward design by framing tasks as goal reaching rather than maximizing hand-crafted reward signals. In this setting, the optimal goal-conditioned value function naturally forms a quasimetric, motivating Quasimetric RL (QRL), which constrains value learning to quasimetric mappings and enforces local consistency through discrete, trajectory-based constraints. We propose Eikonal-Constrained Quasimetric RL (Eik-QRL), a continuous-time reformulation of QRL based on the Eikonal Partial Differential Equation (PDE). This PDE-based structure makes Eik-QRL trajectory-free, requiring only sampled states and goals, while improving out-of-distribution generalization. We provide theoretical guarantees for Eik-QRL and identify limitations that arise under complex dynamics. To address these challenges, we introduce Eik-Hierarchical QRL (Eik-HiQRL), which integrates Eik-QRL into a hierarchical decomposition. Empirically, Eik-HiQRL achieves state-of-the-art performance in offline goal-conditioned navigation and yields consistent gains over QRL in manipulation tasks, matching temporal-difference methods.

Explainable machine learning techniques have gained increasing attention in engineering applications, especially in aerospace design and analysis, where understanding how input variables influence data-driven models is essential. Partial Dependence Plots (PDPs) are widely used for interpreting black-box models by showing the average effect of an input variable on the prediction. However, their global sensitivity metric can be misleading when strong interactions are present, as averaging tends to obscure interaction effects. To address this limitation, we propose a global sensitivity metric based on Individual Conditional Expectation (ICE) curves. The method computes the expected feature importance across ICE curves, along with their standard deviation, to more effectively capture the influence of interactions. We provide a mathematical proof demonstrating that the PDP-based sensitivity is a lower bound of the proposed ICE-based metric under truncated orthogonal polynomial expansion. In addition, we introduce an ICE-based correlation value to quantify how interactions modify the relationship between inputs and the output. Comparative evaluations were performed on three cases: a 5-variable analytical function, a 5-variable wind-turbine fatigue problem, and a 9-variable airfoil aerodynamics case, where ICE-based sensitivity was benchmarked against PDP, SHapley Additive exPlanations (SHAP), and Sobol' indices. The results show that ICE-based feature importance provides richer insights than the traditional PDP-based approach, while visual interpretations from PDP, ICE, and SHAP complement one another by offering multiple perspectives.

In Data Science, entities are typically represented by single valued measurements. Symbolic Data Analysis extends this framework to more complex structures, such as intervals and histograms, that express internal variability. We propose an extension of multiclass Fisher's Discriminant Analysis to interval-valued data, using Moore's interval arithmetic and the Mallows' distance. Fisher's objective function is generalised to consider simultaneously the contributions of the centres and the ranges of intervals and is numerically maximised. The resulting discriminant directions are then used to classify interval-valued observations.To support visual assessment, we adapt the class map, originally introduced for conventional data, to classifiers that assign labels through minimum distance rules. We also extend the silhouette plot to this setting and use stacked mosaic plots to complement the visual display of class assignments. Together, these graphical tools provide insight into classifier performance and the strength of class membership. Applications to real datasets illustrate the proposed methodology and demonstrate its value in interpreting classification results for interval-valued data.

We introduce Guided Harmonic Path-Integral Diffusion (GH-PID), a linearly-solvable framework for guided Stochastic Optimal Transport (SOT) with a hard terminal distribution and soft, application-driven path costs. A low-dimensional guidance protocol shapes the trajectory ensemble while preserving analytic structure: the forward and backward Kolmogorov equations remain linear, the optimal score admits an explicit Green-function ratio, and Gaussian-Mixture Model (GMM) terminal laws yield closed-form expressions. This enables stable sampling and differentiable protocol learning under exact terminal matching. We develop guidance-centric diagnostics -- path cost, centerline adherence, variance flow, and drift effort -- that make GH-PID an interpretable variational ansatz for empirical SOT. Three navigation scenarios illustrated in 2D: (i) Case A: hand-crafted protocols revealing how geometry and stiffness shape lag, curvature effects, and mode evolution; (ii) Case B: single-task protocol learning, where a PWC centerline is optimized to minimize integrated cost; (iii) Case C: multi-expert fusion, in which a commander reconciles competing expert/teacher trajectories and terminal beliefs through an exact product-of-experts law and learns a consensus protocol. Across all settings, GH-PID generates geometry-aware, trust-aware trajectories that satisfy the prescribed terminal distribution while systematically reducing integrated cost.

Diffusion-based samplers -- Score Based Diffusions, Bridge Diffusions and Path Integral Diffusions -- match a target at terminal time, but the real leverage comes from choosing the schedule that governs the intermediate-time dynamics. We develop a path-wise schedule -- selection gramework for Harmonic PID with a time-varying stiffness, exploiting Piece-Wise-Constant(PWC) parametrizations and a simple hierarchical refinement. We introduce schedule-sensitive Quality-of-Sampling (QoS) diagnostics. Assuming a Gaussian-Mixture (GM) target, we retain closed-form Green functions' ration and numerically stable, Neural-Network free oracles for predicted-state maps and score. Experiments in 2D show that QoS driven PWC schedules consistently improve early-exit fidelity, tail accuracy, conditioning of the dynamics, and speciation (label-selection) timing at fixed integration budgets.

Graph Neural Networks (GNNs) are characterized by their capacity of processing graph-structured data. However, due to the sparsity of labels under semi-supervised learning, they have been found to exhibit biased performance on specific nodes. This kind of bias has been validated to correlate with topological structure and is considered as a bottleneck of GNNs' performance. Existing work focuses on the study of homogeneous GNNs and little attention has been given to topological bias in Heterogeneous Graph Neural Networks (HGNNs). In this work, firstly, in order to distinguish distinct meta relations, we apply meta-weighting to the adjacency matrix of a heterogeneous graph. Based on the modified adjacency matrix, we leverage PageRank along with the node label information to construct a projection. The constructed projection effectively maps nodes to values that strongly correlated with model performance when using datasets both with and without intra-type connections, which demonstrates the universal existence of topological bias in HGNNs. To handle this bias, we propose a debiasing structure based on the difference in the mapped values of nodes and use it along with the original graph structure for contrastive learning. Experiments on three public datasets verify the effectiveness of the proposed method in improving HGNNs' performance and debiasing.