This paper introduces a projected functional gradient descent algorithm (P-FGD) for training nonparametric additive quantile regression models in online settings. This algorithm extends the functional stochastic gradient descent framework to the pinball loss. An advantage of P-FGD is that it does not need to store historical data while maintaining $O(J_t\ln J_t)$ computational complexity per step where $J_t$ denotes the number of basis functions. Besides, we only need $O(J_t)$ computational time for quantile function prediction at time $t$. These properties show that P-FGD is much better than the commonly used RKHS in online learning. By leveraging a novel Hilbert space projection identity, we also prove that the proposed online quantile function estimator (P-FGD) achieves the minimax optimal consistency rate $O(t^{-\frac{2s}{2s+1}})$ where $t$ is the current time and $s$ denotes the smoothness degree of the quantile function. Extensions to mini-batch learning are also established.

Computational models support high-stakes decisions across engineering and science, and practitioners increasingly seek probabilistic predictions to quantify uncertainty in such models. Existing approaches generate predictions either by sampling input parameter distributions or by augmenting deterministic outputs with uncertainty representations, including distribution-free and distributional methods. However, sampling-based methods are often computationally prohibitive for real-time applications, and many existing uncertainty representations either ignore input dependence or rely on restrictive Gaussian assumptions that fail to capture asymmetry and heavy-tailed behavior. Therefore, we extend the ACCurate and Reliable Uncertainty Estimate (ACCRUE) framework to learn input-dependent, non-Gaussian uncertainty distributions, specifically two-piece Gaussian and asymmetric Laplace forms, using a neural network trained with a loss function that balances predictive accuracy and reliability. Through synthetic and real-world experiments, we show that the proposed approach captures an input-dependent uncertainty structure and improves probabilistic forecasts relative to existing methods, while maintaining flexibility to model skewed and non-Gaussian errors.

We develop a theoretical framework for generalization in the interpolating regime of statistical learning. The central question is why highly overparameterized estimators can attain zero empirical risk while still achieving nontrivial predictive accuracy, and how to characterize the boundary between benign and destructive overfitting. We introduce a spectral-transport stability framework in which excess risk is controlled jointly by the spectral geometry of the data distribution, the sensitivity of the learning rule under single-sample replacement, and the alignment structure of label noise. This leads to a scale-dependent Fredriksson index that combines effective dimension, transport stability, and noise alignment into a single complexity parameter for interpolating estimators. We prove finite-sample risk bounds, establish a sharp benign-overfitting criterion through the vanishing of the index along admissible spectral scales, and derive explicit phase-transition rates under polynomial spectral decay. For a model-specific specialization, we obtain an explicit theorem for polynomial-spectrum linear interpolation, together with a proof of the resulting rate. The framework also clarifies implicit regularization by showing how optimization dynamics can select interpolating solutions of minimal spectral-transport energy. These results connect algorithmic stability, double descent, benign overfitting, operator-theoretic learning theory, and implicit bias within a unified structural account of modern interpolation.

This article proposes a biconvex modification to convex biclustering in order to improve its performance in high-dimensional settings. In contrast to heuristics that discard a subset of noisy features a priori, our method jointly learns and accordingly weighs informative features while discovering biclusters. Moreover, the method is adaptive to the data, and is accompanied by an efficient algorithm based on proximal alternating minimization, complete with detailed guidance on hyperparameter tuning and efficient solutions to optimization subproblems. These contributions are theoretically grounded; we establish finite-sample bounds on the objective function under sub-Gaussian errors, and generalize these guarantees to cases where input affinities need not be uniform. Extensive simulation results reveal our method consistently recovers underlying biclusters while weighing and selecting features appropriately, outperforming peer methods. An application to a gene microarray dataset of lymphoma samples recovers biclusters matching an underlying classification, while giving additional interpretation to the mRNA samples via the column groupings and fitted weights.

The Hierarchical Kernel Transformer (HKT) is a multi-scale attention mechanism that processes sequences at L resolution levels via trainable causal downsampling, combining level-specific score matrices through learned convex weights. The total computational cost is bounded by 4/3 times that of standard attention, reaching 1.3125x for L = 3. Four theoretical results are established. (i) The hierarchical score matrix defines a positive semidefinite kernel under a sufficient condition on the symmetrised bilinear form (Proposition 3.1). (ii) The asymmetric score matrix decomposes uniquely into a symmetric part controlling reciprocal attention and an antisymmetric part controlling directional attention; HKT provides L independent such pairs across scales, one per resolution level (Propositions 3.5-3.6). (iii) The approximation error decomposes into three interpretable components with an explicit non-Gaussian correction and a geometric decay bound in L (Theorem 4.3, Proposition 4.4). (iv) HKT strictly subsumes single-head standard attention and causal convolution (Proposition 3.4). Experiments over 3 random seeds show consistent gains over retrained standard attention baselines: +4.77pp on synthetic ListOps (55.10+-0.29% vs 50.33+-0.12%, T = 512), +1.44pp on sequential CIFAR-10 (35.45+-0.09% vs 34.01+-0.19%, T = 1,024), and +7.47pp on IMDB character-level sentiment (70.19+-0.57% vs 62.72+-0.40%, T = 1,024), all at 1.31x overhead.

Formal model evaluation methods typically certify that a model satisfies a prescribed target key performance indicator (KPI) level. However, in many applications, the relevant target KPI level may not be known a priori, and the user may instead wish to compare candidate models by analyzing the full trade-offs between performance and reliability achievable at test time by the models. This task, requiring the reliable estimate of the test-time KPI distributions, is made more complicated by the fact that the same data must often be used both to pre-select a subset of candidate models and to estimate their KPI distributions, causing a potential post-selection bias. In this work, we introduce post-selection distributional model evaluation (PS-DME), a general framework for statistically valid distributional model assessment after arbitrary data-dependent model pre-selection. Building on e-values, PS-DME controls post-selection false coverage rate (FCR) for the distributional KPI estimates and is proved to be more sample efficient than a baseline method based on sample splitting. Experiments on synthetic data, text-to-SQL decoding with large language models, and telecom network performance evaluation demonstrate that PS-DME enables reliable comparison of candidate configurations across a range of reliability levels, supporting the statistically reliable exploration of performance--reliability trade-offs.

Most methods for learning with noisy labels require privileged knowledge such as noise transition matrices, clean subsets or pretrained feature extractors, resources typically unavailable when robustness is most needed. We propose Conformal Margin Risk Minimization (CMRM), a plug-and-play envelope framework that improves any classification loss under label noise by adding a single quantile-calibrated regularization term, with no privileged knowledge or training pipeline modification. CMRM measures the confidence margin between the observed label and competing labels, and thresholds it with a conformal quantile estimated per batch to focus training on high-margin samples while suppressing likely mislabeled ones. We derive a learning bound for CMRM under arbitrary label noise requiring only mild regularity of the margin distribution. Across five base methods and six benchmarks with synthetic and real-world noise, CMRM consistently improves accuracy (up to +3.39%), reduces conformal prediction set size (up to -20.44%) and does not hurt under 0% noise, showing that CMRM captures a method-agnostic uncertainty signal that existing mechanisms did not exploit.

We propose an adaptive MCMC method that learns a linear preconditioner which is dense in its off-diagonal elements but sparse in its parametrisation. Due to this sparsity, we achieve a per-iteration computational complexity of $O(m^2d)$ for a user-determined parameter $m$, compared with the $O(d^2)$ complexity of existing adaptive strategies that can capture correlation information from the target. Diagonal preconditioning has an $O(d)$ per-iteration complexity, but is known to fail in the case that the target distribution is highly correlated, see \citet[Section 3.5]{hird2025a}. Our preconditioner is constructed using eigeninformation from the target covariance which we infer using online principal components analysis on the MCMC chain. It is composed of a diagonal matrix and a product of carefully chosen reflection matrices. On various numerical tests we show that it outperforms diagonal preconditioning in terms of absolute performance, and that it outperforms traditional dense preconditioning and multiple diagonal plus low-rank alternatives in terms of time-normalised performance.

Time-series analysis is often affected by missing data, a common problem across several fields, including healthcare and environmental monitoring. Multiple Imputation by Chained Equations (MICE) has been prominent for imputing missing values through "fully conditional specification". We extend MICE using the Bayesian framework (tBayes-MICE), utilising Bayesian inference to impute missing values via Markov Chain Monte Carlo (MCMC) sampling to account for uncertainty in MICE model parameters and imputed values. We also include temporally informed initialisation and time-lagged features in the model to respect the sequential nature of time-series data. We evaluate the tBayes-MICE method using two real-world datasets (AirQuality and PhysioNet), and using both the Random Walk Metropolis (RWM) and the Metropolis-Adjusted Langevin Algorithm (MALA) samplers. Our results demonstrate that tBayes-MICE reduces imputation errors relative to the baseline methods over all variables and accounts for uncertainty in the imputation process, thereby providing a more accurate measure of imputation error. We also found that MALA mixed better than RWM across most variables, achieving comparable accuracy while providing more consistent posterior exploration. Overall, these findings suggest that the tBayes-MICE framework represents a practical and efficient approach to time-series imputation, balancing increased accuracy with meaningful quantification of uncertainty in various environmental and clinical settings.

While both classical and neural network classifiers can achieve high accuracy, they fall short on offering uncertainty bounds on their predictions, making them unfit for safety-critical applications. Existing kernel-based classifiers that provide such bounds scale with $\mathcal O (n^{\sim3})$ in time, making them computationally intractable for large datasets. To address this, we propose a novel, computationally efficient classification algorithm based on the Nadaraya-Watson estimator, for whose estimates we derive frequentist uncertainty intervals. We evaluate our classifier on synthetically generated data and on electrocardiographic heartbeat signals from the MIT-BIH Arrhythmia database. We show that the method achieves competitive accuracy $>$\SI{96}{\percent} at $\mathcal O(n)$ and $\mathcal O(\log n)$ operations, while providing actionable uncertainty bounds. These bounds can, e.g., aid in flagging low-confidence predictions, making them suitable for real-time settings with resource constraints, such as diagnostic monitoring or implantable devices.