Papers

Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning that departs from traditional density-based approaches by explicitly modeling the intrinsic curvature of the data manifold. The method relies on a discrete approximation of the shape operator, estimated from local k-nearest neighbor patches, to compute pointwise mean curvature without requiring explicit manifold parametrization. The key insight of MCBP is to use mean curvature as a principled descriptor of boundary structure: high-curvature regions naturally correspond to transitions between clusters, geometric irregularities, and low-density interfaces. This yields a unified geometric interpretation of boundary, outlier, and transition points. We further introduce an adaptive percentile-based thresholding scheme that enables multiscale boundary extraction without relying on ad hoc density parameters. Beyond detection, we propose a curvature-driven data decomposition that separates samples into smooth (low-curvature) and boundary (high-curvature) subsets, effectively acting as a non-linear geometric filtering mechanism. This representation enhances cluster separability and improves the robustness of downstream unsupervised algorithms. Extensive experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance, particularly in complex and high-dimensional scenarios. These results position MCBP as a concrete contribution to Geometric Machine Learning, highlighting the potential of curvature-aware analysis as a unifying paradigm bridging differential geometry and data-driven modeling.

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $\beta_1$, and a preconditioner perturbation governed by $\beta_2$. We characterize the burn-in time required for the transient terms to decay to the asymptotic tracking bound under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise–drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can yield favorable upper guarantees, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can enlarge Adam's tracking guarantee, potentially allowing vanilla SGD to attain a smaller tracking error. Our explicit $(\beta_1,\beta_2,\epsilon)$-dependent bounds identify mechanisms through which adaptive step-sizing can help or hurt under nonstationarity and provide theoretical explanations consistent with Adam's empirical instability and stabilization under distribution shift.

Reinforcement learning from human feedback (RLHF) typically assumes a static or non-strategic reward model (RM). In iterative deployment, however, the policy generates the data on which the RM is retrained, creating a feedback loop. Building on the Stackelberg game formulation of this interaction, we derive an analytical decomposition of the policy's true optimization gradient into a standard policy gradient and a parameter-steering term that captures the policy's influence on the RM's future parameters. We show that standard iterative RLHF, which drops this steering term entirely, suffers from alignment collapse: the policy systematically exploits the RM's blind spots, producing low-quality, high-reward outputs whose feedback reinforces the very errors it exploits. To mitigate this, we propose foresighted policy optimization (FPO), a mechanism-design intervention that restores the missing steering term by regularizing the policy's parameter-steering effect on RM updates. We instantiate FPO via a scalable first-order approximation and demonstrate that it prevents alignment collapse on both controlled environments and an LLM alignment pipeline using Llama-3.2-1B.

Many machine learning problems involve data supported on curved spaces such as spheres, rotation groups, hyperbolic spaces, and general Riemannian manifolds, where Euclidean geometry can distort distances, averages, and the resulting optimal transport (OT) problem. Existing manifold OT methods have pursued amortized out-of-sample maps, while entropic regularization has made discrete OT more scalable, but these advantages have remained largely disjoint. We propose Entropic Riemannian Neural Optimal Transport (Entropic RNOT), a unified framework that combines intrinsic entropic OT with amortized out-of-sample evaluation on Riemannian manifolds. Our method learns a single target-side Schrödinger potential through a neural pullback parameterization, recovers the induced Gibbs coupling, and uses the resulting conditional laws to construct intrinsic transport surrogates. These include barycentric projections on Cartan-Hadamard manifolds and heat-smoothed conditional surrogates on stochastically complete manifolds, the latter turning possibly atomic target laws into absolutely continuous ones. For fixed regularization $\varepsilon>0$, we prove that the proposed hypothesis class recovers the entropic optimal coupling in strong probabilistic metrics. As consequences, barycentric surrogates converge in $L^2$, while heat-smoothed surrogates are stable at fixed heat time and asymptotically unbiased as the heat time vanishes. The guarantees hold for compactly supported data on possibly noncompact manifolds. Empirically, our method matches or improves over Euclidean, tangent-space, and log-Euclidean baselines on benchmarks over $\mathbb{S}^2$, $\mathrm{SO}(3)$, $\mathrm{SPD}(3)$, $\mathrm{SE}(3)$, and $\mathbb{H}^2$, scales favorably relative to discrete manifold Sinkhorn, and in a protein-ligand docking application, refines poses on $\mathrm{SE}(3)$ without retraining or per-instance optimization.

Zero-inflated outcomes, where responses are zero with positive probability and otherwise continuous, are common in biomedical, environmental, and social science studies. We propose a conformal prediction based framework that provides distribution-free uncertainty quantification tailored to such outcomes. Standard conformal methods often ignore strong predictors distinguishing zero from non-zero outcomes, leading to overly conservative and unnecessarily long prediction sets. Our method integrates a classification step to identify zero outcomes and applies conformal inference to the non-zero part, producing prediction sets that are either ${0}$ or an interval. Under exchangeability, we establish that the proposed procedure attains the target marginal coverage and achieves asymptotically minimal interval length within this framework, regardless of the choice of classification or regression models. Extensive simulations and real-data application demonstrate the superior performance of our approach.

Inferring higher-order interaction structure from observations of dynamics is a central challenge in complex systems, particularly when data are scarce, noisy, or concentrated in lower-dimensional regions of state space. We develop Bayes-THIS, a Bayesian extension of Taylor-based Hypergraph Inference using SINDy (THIS) that replaces threshold-based sparse regression with sparse Bayesian regression using automatic relevance determination. This yields adaptive, term-wise shrinkage and a posterior over inferred Taylor expansion coefficients that encode the presence or absence of interactions. Across numerical experiments, Bayes-THIS improves inference in low-data, high-noise, and ill-conditioned regimes. The Bayesian formulation also enables two capabilities absent from the original THIS framework: a posterior predictive diagnostic for assessing whether a dataset contains higher-order signal distinguishable from noise, and a posterior interval rule for pruning weakly supported hyperedges. Finally, we examine a separate, structural limitation of Taylor-based hypergraph reconstruction: higher-order interactions can produce spurious lower-order signatures that are indistinguishable from genuine lower-order edges. We find that the severity of this non-identifiability depends on cross-order support, showing that structural properties of the target hypergraph can be as important as data quality in determining what can be inferred from dynamics within this framework.

In phase I dose escalation studies for dual-agent combinations, at least one drug often has an established monotherapy dose. Consequently, substantial prior clinical safety data often exist for one or more monotherapies, allowing the study to focus on a subset of selected dose combinations rather than exhaustively evaluating all possible dose combinations for two agents. The Bayesian Optimal Interval (BOIN) design framework is widely recognized for its robust performance and ease of implementation; however, the BOIN for combination design, abbreviated as BOIN-C in this paper, was originally developed to evaluate full combinations and may not be directly applicable for the subset of selected combinations. In this paper, we propose three extensions to the BOIN-C design to address scenarios involving selected dose combinations: (a) BOIN-CS: a generalized BOIN-C design to accommodate any subset of dose combinations. (b) BOIN-CE: Exploration of new off-diagonal dose combinations when de-escalating. This option provides additional opportunities to treat patients with dose combinations that have not been administered. (c) BOIN-CB: Bayesian logistic regression model (BLRM)-guided BOIN design, which uses the BLRM model to break the tie when two dose combinations have an equal posterior probability of being selected. This can be useful when the dose-toxicity relationship is expected to be reasonably aligned with a logistic relationship. These study design options are motivated by practical considerations, and their operating characteristics are evaluated through extensive simulations under various scenarios, demonstrating satisfactory performance.

Fine particulate matter (PM$_{2.5}$) concentration data are positive, right-skewed series that arise naturally in environmental monitoring and are well described by the Birnbaum-Saunders (BS) distribution. In this paper, we propose a multivariate BS autoregressive moving average (MBSARMA) model with exogenous terms for the joint analysis of correlated positive asymmetric time series. The proposed model combines the multivariate log-linear BS framework with dynamic autoregressive moving average components on the conditional location parameter of each response. We estimate the model parameters by means of the Expectation-Maximisation (EM) algorithm. The performance of the proposed conditional likelihood estimators is evaluated by means of a Monte Carlo simulation study under several correlation levels and sample sizes. An application to weekly PM$_{2.5}$ pollution concentration data recorded at three monitoring stations in Santiago, Chile, obtained from the National Air Quality Information System of Chile (SINCA), is presented. The results show the good performance of the proposed methodology.

We study contextual dynamic pricing under a semiparametric demand model in which the purchase probability is $1-F(p-m(\mathbf{x}))$, where $m(\mathbf{x})$ captures mean utility as a function of product features and buyer covariates, and $F$ is an unknown market-noise distribution. Existing methods either incur suboptimal regret or rely on restrictive structural assumptions. We propose a stagewise greedy pricing algorithm that iteratively refines the estimate of $F$ via local polynomial regression while pricing greedily with current estimates. By exploiting feature diversity, the algorithm reuses endogenous samples collected during exploitation for nonparametric estimation, avoiding costly global random exploration used in prior work. We establish a general regret bound that applies to any estimator $\hat m$ of the utility function, and derive explicit rates for linear, nonparametric additive, and sparse linear classes of $m$. For the linear class, our regret scales as $T^{\max\{1/2,\,3/(2\beta+1)\}}$, where $\beta$ is the smoothness of $F$ and $T$ is the time horizon. This improves the best known rates for semiparametric contextual pricing and achieves the parametric $\sqrt{T}$ rate when $\beta \ge 5/2$. We further prove a matching lower bound, showing the optimality of our rate, and present numerical experiments that corroborate the theory and demonstrate the practical advantages of iterative refinement.

Public attitudes toward artificial intelligence are heterogeneous, ordinally measured, and poorly captured by any single dependency graph. Existing ordinal structure learners assume a shared directed acyclic graph (DAG) across all respondents; recent heterogeneous ordinal graphical-model approaches focus on subgroup discovery rather than confirmatory cluster-specific DAG estimation; and latent profile analyses discard dependency structure entirely. We introduce a heterogeneous ordinal structure-learning framework combining monotone Gaussian score embedding, Bayesian nonparametric (BNP) complexity discovery via a truncated stick-breaking prior, and confirmatory fixed-K estimation with cluster-specific sparse DAG learning. The key methodological insight is a discovery-to-confirmation workflow: the nonparametric stage calibrates plausible archetype complexity, while inner-validated confirmatory refitting yields stable, interpretable structural estimates. On the 2024 Pew American Trends Panel AI attitudes survey, Wave 152 (W152) survey, (N = 4,788, 8 ordinal items), the confirmatory K*=5 model reduces holdout transformed-score mean squared error (MSE) by 25.8% over a single-graph baseline and by 4.6% over mixture-only clustering. A controlled tiered semi-synthetic benchmark calibrated to W152 structure validates recovery across difficulty regimes and transparently reveals failure modes under stress conditions.

Modern heterogeneity-robust difference-in-differences estimators derive their asymptotic properties under iid, cluster, or fixed-design frameworks that abstract from complex survey sampling, yet practitioners routinely apply them to nationally representative surveys with stratified cluster designs. We show that, under standard regularity conditions, the influence functions of each smooth IF-based or regression-based modern DiD estimator satisfy Binder's (1983) smoothness conditions, so the standard stratified-cluster variance formula applied to their values produces design-consistent standard errors. A Monte Carlo study with 66,000 replications shows where the design effect comes from. HC1 standard errors that treat observations as iid produce coverage as low as 34% under a baseline survey design and below 11% under informative sampling. Combining the survey-weighted point estimate with PSU-level clustering - the practitioner's cluster=psu heuristic - recovers near-nominal coverage across all scenarios. Adding strata and finite-population corrections yields incremental precision but is not required for valid coverage. Survey-weighted doubly robust estimation produces well-calibrated inference when parallel trends hold only conditionally. An NHANES illustration of the ACA dependent coverage provision shows that point estimates and standard errors change substantively - enough to reverse significance conclusions - when the survey design is accounted for. We provide diff-diff (https://github.com/igerber/diff-diff), an open-source Python package implementing design-based variance for fifteen modern DiD estimators.

Sampling from unnormalized multimodal distributions with limited density evaluations remains a fundamental challenge in machine learning and natural sciences. Successful approaches construct a bridge between a tractable reference and the target distribution. Parallel Tempering (PT) serves as the gold standard, while recent diffusion-based approaches offer a continuous alternative at the cost of neural training. In this work, we introduce Conditional Diffusion Sampling (CDS), a framework that combines these two paradigms. To this end, we derive Conditional Interpolants, a class of stochastic processes whose transport dynamics are governed by an exact, closed-form stochastic differential equation (SDE), requiring no neural approximation. Although these dynamics require sampling from a non-trivial initialization distribution, we show both theoretically and empirically that the cost of this initialization diminishes for sufficiently short diffusion times. CDS leverages this by a two-stage procedure: (1) PT is used to efficiently sample the initial distribution, and then (2) samples are transported via the transport SDE. This combination couples the robust global exploration of PT with efficient local transport. Experiments suggest that CDS has the potential to achieve a superior trade-off between sample quality and density evaluation cost compared to state-of-the-art samplers.

Skill scores, which measure the relative improvement of a forecasting method over a benchmark via consistent scoring functions and proper scoring rules, are a standard tool in forecast evaluation, yet their sampling uncertainty is rarely rigorously quantified. With modern forecasting applications being increasingly multivariate and involving evaluations across multiple horizons, variables, spatial locations, and forecasting methods, standard tools like the pairwise Diebold-Mariano forecast accuracy test or pointwise confidence intervals fail to account for the multiple comparison problem, leading to inflated Type I error rates and invalid joint inference. To address the lack of a coherent, statistically rigorous framework for quantifying uncertainty across these multi-dimensional evaluation problems, we introduce simultaneous confidence bands for expected scores and skill scores. Our framework provides a versatile tool for joint inference that is applicable to any forecast type from mean and quantile to full distributional forecasts. We develop a bootstrap implementation and show that our bands are valid under multivariate extensions of the classical Diebold-Mariano assumptions. We demonstrate the practical utility of the approach in two case studies by quantifying the benefits of time-varying parameter models for macroeconomic forecasting, and by comparing data-driven and physics-based models in probabilistic weather forecasting.

Multiparameter quantum metrology is ubiquitous in physics, yet it remains fundamentally challenging. As reflected by the uncertainty principle, distinct unknown parameters generally cannot be estimated simultaneously at their individual ultimate precision limits. The quantum Cramér–Rao bound (QCRB) and the Holevo Cramér–Rao bound (HCRB) represent the ultimate precision bounds in multiparameter quantum metrology. Here we introduce new formulations of these bounds based on quantum state purification. We show that, for any mixed state, their values are directly related to those of its purification, provided that nuisance parameters are introduced on the environmental system. Leveraging this connection, we develop a remarkably simple method for asymptotically attaining either the HCRB or twice the QCRB for arbitrary mixed states, using random purification channels and individual measurements.

Standard differential privacy imposes uniform privacy constraints across all features, overlooking the inherent distinction between sensitive and insensitive features in practice. In this paper, we introduce a relaxed definition of differential privacy that accounts for such heterogeneity, allowing certain features to be treated as insensitive even when correlated with sensitive ones. We propose a correlation-aware framework, $\textsf{CorrDP}$, which relaxes privacy for insensitive features while accounting for their correlations with sensitive features, with the correlations quantified using total variation distance. We design algorithms for differentially private empirical risk minimization (DP-ERM) under the $\textsf{CorrDP}$ framework, incorporating distance-dependent noise into gradients for improved theoretical utility guarantees. When the correlation distance is unknown, we estimate it from the dataset and show that it achieves a comparable privacy-utility guarantee. We perform experiments on synthetic and real-world datasets and show that $\textsf{CorrDP}$-based DP-ERM algorithms consistently outperform the standard DP framework in the presence of insensitive features.

Survival analysis on tabular data is a well-studied problem. However, existing deep learning methods are often highly task-specific, which can limit the transfer of new approaches from other domains and introduce constraints that may affect performance. We propose TabSurv, an approach that adapts modern tabular architectures to survival analysis using either the Weibull distribution or non-parametric survival prediction. TabSurv optimizes SurvHL, a novel histogram loss function supporting censored data. In addition to a baseline feed-forward network, we implement deep ensembles of MLPs for survival analysis within TabSurv. In contrast to prior work, the ensemble components are trained in parallel, optimizing survival distribution parameters before averaging, which promotes diversity across ensemble component predictions. We perform a comprehensive empirical evaluation of different proposed architectures on 10 diverse real-world survival datasets. Our results show that TabSurv consistently outperforms on average established classical and deep learning baselines, such as RSF, DeepSurv, DeepHit, SurvTRACE. Notably, deep ensembles with Weibull parametrization instead of non-parametric models achieve the highest average rank by C-index. Overall, our study clarifies how modern tabular neural networks can be adapted and trained to tackle survival analysis problems, offering a strong and reliable approach. The TabSurv implementation is publicly available.

We study the $(\varepsilon, \delta)$-PAC policy identification problem in finite-horizon episodic Markov Decision Processes. Existing approaches provide finite-time guarantees for approximate settings ($\varepsilon>0$) but suffer from high computational cost, rendering them hard to implement, and also suffer from suboptimal dependence on $\log(1/\delta)$. We propose a randomized and computationally efficient algorithm for best policy identification that combines posterior sampling with an online learning algorithm to guide exploration in the MDP. Our method achieves asymptotic optimality in sample complexity, also in terms of posterior contraction rate, and runs in $O(S^2AH)$ per episode, matching standard model-based approaches. Unlike prior algorithms such as MOCA and PEDEL, our guarantees remain meaningful in the asymptotic regime and avoid sub-optimal polynomial dependence on $\log(1/\delta)$. Our results provide both theoretical insights and practical tools for efficient policy identification in tabular MDPs.

We introduce the Multiplicative Quasi-Instrumental Variable (MQIV) model, a framework for causal inference with unmeasured confounding that leverages an instrument that may be imperfectly exogenous. We allow the candidate quasi-instrument to have a direct effect on the outcome not mediated by the treatment, thus violating the standard IV exclusion restriction. We establish nonparametric identification of the population average treatment effect on the treated (ATT) under a treatment model that is multiplicative with respect to the quasi-IV and the hidden confounder (Hernan and Robins, 2006). Such a multiplicative treatment model may arise naturally either when treatment occurs only if two independent instrument-driven and confounder-driven causal mechanisms are present; or alternatively, when an instrument's effect on treatment uptake is inherently heterogeneous and scales with a person's latent propensity, best capturing settings in which it is challenging for a given instrument to overcome a person's inherent lack of preference for the treatment in view. Importantly, as we establish, the MQIV model is simultaneously agnostic to treatment-effect heterogeneity with respect to hidden confounders and violation of the core IV exclusion restriction condition. Identification is achieved via a modified Wald ratio estimand, which corrects the bias due to the exclusion restriction violation, and we propose a new class of estimators that are multiply robust and semiparametric efficient. Finally, we evaluate the approach in extensive simulations and an application to evaluate the causal effect of having three or more children on mothers' labor-market engagement.

Monte Carlo permutation tests are a cornerstone of valid, model-free statistical inference. A widely held practical intuition is that increasing the number of sampled permutations improves test performance, in particular that statistical power tends to increase with the Monte Carlo budget. In this paper, we show that these intuitions are false in general. Leveraging the saw-toothed structure of power arising from distributional discreteness, we provide a simple structural explanation for why power can decrease as the number of sampled permutations increases, and we prove that such decreases occur infinitely often as the Monte Carlo budget grows.

We study strong universal Bayes-consistency in the realizable setting for learning with general metric losses, extending classical characterizations beyond $0$-$1$ classification (Bousquet et al., 2020; Hanneke et al., 2021) and real-valued regression (Attias et al., 2024). Given an instance space $(X,\rho)$, a label space $(Y,\ell)$ with possibly unbounded loss, and a hypothesis class $H \subseteq Y^{X}$, we resolve the realizable case of an open problem presented in Tsir Cohen and Kontorovich (2022). Specifically, we find the necessary and sufficient conditions on the hypothesis class $H$ under which there exists a distribution-free learning rule whose risk converges almost surely to the best-in-class risk (which is zero) for every realizable data-generating distribution. Our main contribution is this sharp characterization in terms of a combinatorial obstruction: Similarly to Attias et al. (2024), we introduce the notion of an infinite non-decreasing $(\gamma_k)$-Littlestone tree, where $\gamma_k \to \infty$. This extends the Littlestone tree structure used in Bousquet et al. (2020) to the metric loss setting.

When direct measurement of a clinically relevant primary endpoint in a clinical trial is infeasible, a surrogate endpoint may be used instead to infer treatment effects. Trial-level surrogates predict the average treatment effect on the primary endpoint and may be evaluated within the meta-analytic framework. However, traditional methods are ill-suited to the complex high-dimensional data now increasingly collected in modern trials, such as omics data. Although methods for high-dimensional surrogate evaluation exist, they have largely been developed for single-trial settings and therefore cannot assess surrogate generalisability. Here, we propose RISE-Meta, an approach for evaluating trial-level surrogate markers in the multi-trial, high-dimensional setting. In the first stage, an existing nonparametric method is applied to individual participant data to derive study-level surrogacy metrics for each candidate marker. Next, random-effects meta-analysis combines these metrics across studies, and equivalence testing provides operational criteria for surrogate validity. Finally, a subset of candidates is combined into a composite signature through a weighting scheme to improve surrogacy relative to any individual candidate. We evaluate RISE-Meta in both simulation studies and real data applications. In an application to high-dimensional data, we analyse gene expression as trial-level surrogate markers for the antibody response to seasonal influenza vaccination, while in a low-dimensional application we compare RISE-Meta to a reference meta-analytic approach and observe strong agreement between the two.

The Brier score conflates two distinct properties of probabilistic predictions: reliability (calibration error) and resolution (discriminatory power). We introduce the Manokhin Probability Matrix, a BCG-style two-dimensional diagnostic framework that separates them. Classifiers are placed on a 2x2 grid by Spiegelhalter Z-statistic and AUC-ROC expected rank, then assigned to one of four archetypes: Eagle (good on both axes), Bull (strong discrimination, poor calibration), Sloth (well-calibrated, weak discriminator), and Mole (poor on both). Each archetype carries a distinct prescription. We populate the matrix from a large-scale empirical study spanning 21 classifiers, 5 post-hoc calibrators, and 30 real-world binary classification tasks from the TabArena-v0.1 suite. The assignment is unambiguous. CatBoost, TabICL, EBM, TabPFN, GBC, and Random Forest are Eagles. XGBoost, LightGBM, and HGB are Bulls; Venn-Abers calibration cuts log-loss by 6.5 to 12.6% on Bulls but degrades Eagles by 2.1%. SVM, LR, LDA, and the empirical base-rate predictor are Sloths. MLP, KNN, Naive Bayes, and ExtraTrees are Moles. A theoretical asymmetry follows: no order-preserving post-hoc calibrator can add discriminatory power (Proposition 1), so calibration is the fixable part and discrimination is the hard part. The practical rule is direct: do not optimise aggregate Brier score without first decomposing it; optimise discrimination first, then fix calibration post-hoc. Code and raw experimental data are available at https://github.com/valeman/classifier_calibration.

Empirical research in economics increasingly relies on restricted-access data held by multiple firms or agencies, making it impossible to construct the estimator of interest on the pooled sample. At the same time, heavy-tailed distributions are pervasive in economics and finance outcomes such as prices, expenditures and loan sizes. We study sparse, robust estimation in the restricted-access setting. The infeasible pooled benchmark is convoluted rank regression (CRR), a smooth rank-based estimator designed for heavy-tailed outcomes. Because the CRR criterion is a non-additive U-statistic, existing communication-efficient methods built for additive empirical losses do not directly apply. We propose distributed convoluted rank regression (DCRR), a surrogate criterion built from a single local CRR loss and an aggregated gradient correction, and show that it shares the same population minimizer as the pooled CRR objective. Building on this surrogate, we develop a two-stage sparse procedure: an iterative $l_1$- penalized stage followed by a folded-concave refinement. For the resulting estimator, we establish non-asymptotic error bounds, a distributed strong oracle property, and a distributed criterion for consistent model selection. Simulations and an application to used-car prices show that DCRR closely approximates pooled CRR and improves on naive divide-and-conquer, particularly under heavy-tailed errors.

Urban air quality forecasting is challenging because pollutant concentrations are nonlinear, nonstationary, spatiotemporally dependent, and often affected by anomalous observations caused by traffic congestion, industrial emissions, and seasonal meteorological variability. This study proposes a Graph Convolutional Support Vector Regression (GraphSVR) framework for robust spatiotemporal forecasting of urban air pollution. The model combines graph convolutional learning to capture inter-station spatial dependence with support vector regression to model nonlinear temporal dynamics while reducing sensitivity to outlier observations. The proposed framework is evaluated using air quality records from 37 monitoring stations in Delhi and 18 stations in Mumbai, representing inland and coastal metropolitan environments in India. Forecasting performance is assessed across multiple horizons and compared with established temporal and spatiotemporal benchmarks. The results show that GraphSVR consistently improves predictive accuracy and maintains stable performance across seasons and outlier-prone pollution episodes. Statistical test further confirms the reliability of the proposed approach across the datasets. Furthermore, the conformal prediction approach is integrated with GraphSVR to generate calibrated prediction intervals, enhancing its practical value for uncertainty-aware air quality monitoring and public health decision-making.

We propose Conformal Seasonal Pools (CSP), a training-free probabilistic time-series forecaster that mixes same-season empirical draws with signed residual draws around a seasonal naive forecast. In an audited rolling-origin benchmark on the six time-series datasets where DeepNPTS was originally evaluated (electricity, exchange_rate, solar_energy, taxi, traffic, wikipedia), CSP-Adaptive significantly outperforms DeepNPTS on every metric we report – CRPS (per-window paired Wilcoxon $p \approx 4 \times 10^{-10}$), normalized mean quantile loss ($p \approx 7 \times 10^{-10}$), and empirical 95% coverage ($p \approx 8 \times 10^{-45}$, mean 0.89 vs 0.66) – while running over 500x faster on CPU. Coverage is the most decision-critical of these: a 0.95 nominal interval that contains the truth in only ~66% of cases fails the basic calibration desideratum and would not survive deployment in safety- or decision-critical settings. The failure mode is also more severe than aggregate coverage suggests: in the worst 10% of windows, DeepNPTS's prediction interval covers none of the H forecast horizons – the entire multi-step trajectory misses the truth at every step simultaneously. This poses serious risk in safety- and decision-critical applications such as healthcare, finance, energy operations, and autonomous systems, where prediction intervals that systematically miss the truth across the entire planning horizon translate directly into misclassified patients, regulatory capital failures, grid imbalances, and safety-case violations. CSP achieves all of this with no learned parameters and no training. We argue training-free conformal samplers should be mandatory baselines when evaluating learned non-parametric forecasters.

Constructing honest confidence intervals often depends on reliable bias control and accurate calibration based on asymptotic normality, including standard-normal and folded-normal calibration. Substantial progress has been made in correcting or controlling smoothing bias, including robust bias correction and bias-aware inference. We first show that, even after smoothing bias has been well corrected or controlled, asymptotic-normality-based calibration may still be a binding source of finite-sample undercoverage. Thus, the resulting intervals may struggle to achieve the minimax shrinkage rate and uniformly small undercoverage error simultaneously. Instead of using distributional approximation, we calibrate the radius directly by combining an empirical Bernstein bound, a data-driven variance proxy, Lepski-type bandwidth selection, and a bias-aware fixed-length-radius criterion. The formal theory covers nonparametric regression and density estimation, with regression results ranging from local-polynomial to sieve estimators. The resulting empirical Bernstein confidence intervals are safe and sharp. Uniformly over functions with $S$-th order local smoothness, both one-sided and two-sided intervals attain nominal coverage up to $o(n^{-2S/(2S+1)})$, or exponential remainders under bounded or sub-Gaussian conditions, while their widths shrink at the minimax rate $n^{-S/(2S+1)}$ (or up to a $\sqrt{\log n}$-level factor). The calibration principle is modular and can also be combined with other existing bias-control strategies, like robust bias correction. Thus, the contribution of this paper is not a bias-control device but a new angle of calibration. Compared with asymptotic-normality-based calibration, empirical Bernstein calibration safely and conveniently converts the specified smoothness into both coverage accuracy and interval-length efficiency. Simulations support the theory.

Transformers are effective at inferring the latent task from context via two inference modes: recognizing a task seen during training, and adapting to a novel one. Recent interpretability studies have identified from middle-layer representations task-specific directions, or task vectors, that steer model behavior. However, a lack of rigorous foundations hinders connecting internal representations to external model behavior: existing work fails to explain how task-vector geometry is shaped by the training distribution, and what geometry enables out-of-distribution (OOD) generalization. In this paper, we study these questions in a controlled synthetic setting by training small transformers from scratch on latent-task sequence distributions, which allows a principled mathematical characterization. We show that two inference modes can coexist within a single model. In-distribution behavior is governed by Bayesian task retrieval, implemented internally through convex combinations of learned task vectors. OOD behavior, by contrast, arises through extrapolative task learning, whose representations occupy a subspace nearly orthogonal to the task-vector subspace. Taken together, our results suggest that task-vector geometry, training distributions, and generalization behaviors are closely related.

Estimating unknown parameters subject to prior constraints is important in statistical inference, particularly in fields such as reliability analysis, survival studies, and engineering, where prior structural information about the parameters is often available. Incorporating such prior information makes the analysis more realistic and usually yields better estimates than methods that ignore such information. In this article, we consider the problem of estimating the positive power of the scale parameter of a two-shifted exponential population under a prior ordering constraint on scale parameters. We derive sufficient conditions under which equivariant estimators are shown to dominate others under scale-invariant strictly convex loss functions. In addition, we derived various estimators that dominate the best affine equivariant estimators (BAEE). Moreover, we derive a smooth estimator which dominates the BAEE using an integrated approach, and we further show that it is a generalized Bayes estimator under a non-informative prior. We also provide an improved estimator based on the Pitman closeness criterion. An extensive simulation study has been done for computational purposes. Finally, we provided real examples to implement the results.

Multi Armed Bandit (MAB) algorithms are a cornerstone of reinforcement learning and have been studied both theoretically and numerically. One of the most commonly used implementation uses a softmax mapping to prescribe the optimal policy and served as the foundation for downstream algorithms, including REINFORCE. Distinct from vanilla approaches, we consider here the L2 regularized softmax policy gradient where a quadratic term is subtracted from the mean reward. Previous studies exploiting convexity failed to identify a suitable theoretical framework to analyze its convergence when the regularization parameter vanishes. We prove here theoretical convergence results and confirm empirically that this regime makes the L2 regularization numerically advantageous on standard benchmarks.

We consider a novel algorithm, for the completion of partially observed low-rank tensors, as a generalization of matrix completion. The proposed low-rank tensor completion (TC) method builds on the conventional nuclear norm (NN) minimization-based low-rank TC paradigm, by leveraging the alternating direction method of multipliers (ADMM) optimization framework. To that extend the original NN minimization problem is reformulated into multiple subproblems, which are then solved iteratively via closed-form proximal operators, making use of over-relaxation and an adaptive penalty parameter update scheme, to further speed up convergence and improve the overall performance of the method. Simulation results demonstrate the superior performance of the new method in terms of normalized mean square error (NMSE), compared to the conventional state-of-the-art (SotA) techniques, including NN minimization approaches, as well as a mixture of the latter with a matrix factorization approach, while its convergence can be significantly improved by initializing the algorithm with the solution of the SotA.