Papers

We continue the study of relatively smart learning, introduced by Dughmi and Pour (2026), which asks a supervised learner to compete, marginal by marginal, with every distribution-fixed error guarantee soundly certifiable from unlabeled data. They showed that the One-Inclusion Graph (OIG) learner is relatively smart with a quadratic sample-complexity blowup, and that no relatively smart learner can do better, leaving open whether ERM or another natural or tractable learner achieves comparable guarantees. They also left open whether the blowup can be restricted to unlabeled data. Our firs results shows that ERM—and in fact any proper consistent learner—is relatively smart for binary classification in the distribution-free setting. We show that a small certifiable error with $m$ samples implies a similarly small error on the uniform distribution over a random sample of size $O(m^2)$, yielding a cover of size at most $2^{m+1}$ on that sample. This suffices to control the error of proper consistent learners with $O(m^2)$ samples. We then show that semi-supervised relatively smart learning is information-theoretically possible with a quadratic blowup only in unlabeled sample complexity and no blowup in labeled sample complexity. The learner uses a natural generalization of OIG to a leave-most-out transductive problem, where labels of part of a finite pool are revealed and the remaining labels are predicted. Finally, this label efficiency comes at a cost in simplicity and tractability. If the hypothesis class is accessed only through an agnostic ERM oracle, any semi-supervised relatively smart learner with substantially sub-quadratic labeled-sample blowup requires super-polynomially many oracle calls. This holds even when the marginal is given explicitly, and thus also yields an intractability result for distribution-fixed learning that may be of independent interest.

Recent work has identified incremental learning in shallow networks trained on single-index and multi-index models. However, existing analyses often rely on simplifying settings, such as small initialization, correlation loss, or layer-wise training. These choices reduce neuron interactions and leave some feature learning dynamics under standard initialization unexplored. We study training dynamics for polynomial-width two-layer networks learning orthogonal multi-index targets under standard initialization using polynomially many samples. We first prove that incremental learning still occurs: the loss decreases sequentially according to the Hermite expansion of the target, with lower-order components learned before higher-order components recover the individual target directions. In this standard initialization regime, training also shows a competitive reallocation of parameter mass: after the total mass fits the target mean and stabilizes, mass shifts into the target subspace and then concentrates on aligned neurons. Our theoretical analysis uses slightly modified gradient flow, while vanilla gradient descent empirically exhibits the same qualitative dynamics. Technically, we introduce a symmetry-based finite-width approximation via symmetrized networks, rather than comparing directly with an infinite-width limit. This yields better control of approximation errors and may be of independent interest.

Constructing a confidence interval for the difference between two independent binomial proportions involves a nuisance direction that is not identified by the estimand. The one-sample Wilson score interval inverts a scalar score test, but has no direct bivariate analogue isolating the difference: inverting the joint normal approximation yields an elliptical region in the unit square, whereas the estimand \(p_1-p_2\) is one-dimensional. We define the Elliptically Optimal (EO) confidence interval as the range of \(p_1-p_2\) over this region and solve the resulting optimization problem in closed form, obtaining explicit bounds in six mutually exclusive and exhaustive cases. The solution admits a compact characterization: the EO interval is the score interval obtained by maximizing over the nuisance variance rather than estimating it. It is therefore the shortest interval obtained by projecting the elliptical region, and inherits its coverage guarantee. We derive the exact coverage excess, \(2[\Phi(z\mathcal{R})-\Phi(z)]\), where \(\mathcal{R}\) is the ratio of the least-favourable to the true standard deviation. The excess vanishes on an explicit line through the parameter space, is bounded by \(\alpha\), and is invariant under proportional scaling of the sample sizes. Exact enumeration of the binomial coverage shows that the Wald interval, whose variance estimator is downward biased by a factor \(1-1/n\) under balanced allocation, falls below nominal coverage almost everywhere. The EO interval never under-covers under the normal approximation and always yields admissible, non-degenerate bounds. Its price is over-coverage when both proportions are extreme, which we quantify exactly.

Continuous and discrete flow matching are usually treated as separate constructions. This paper identifies a duality between them: projecting continuous convex-interpolant flows with one-hot targets through a position-wise argmax yields discrete convex-interpolant flows. The result requires source laws with appropriate coordinate symmetry and regularity, and it makes the continuous source distribution an explicit design choice for categorical generation. We derive the induced discrete interpolation behavior for Gaussian, bounded-uniform, and centered negative-exponential sources, showing that different source geometries lead to qualitatively different transition timing and vocabulary-size dependence. Small visual diagnostics and a short language-modeling pilot suggest that these source-design effects can also appear in learned transports and early generative quality.

In July 2026, Josh Kerr broke the 26-year-old mile world record, the latest in an unusual concentration of recent middle-distance records that has led to widespread speculation about the role of advancing shoe and other technologies. However, like almost every other recent record-breaking performance, Kerr's race featured a split-staggered start – an often overlooked, relatively novel start configuration that places a subset of athletes in a front group in the outside lanes, reducing the geometric burden associated with accelerating on tight turns while also avoiding the bulk of the extra distance associated with outside positioning in a traditional distance race. Here, these advantages are quantified using empirical lane-performance data, OMEGA measurements of realized race distance, video reconstruction, and a force-based model of curved running. Applying the resulting corrections to record-breaking performances from 2026, estimated advantages range from approximately 0.1 seconds to more than 1 second, depending on the event, race configuration, and model assumptions. Although small in absolute terms, these advantages demonstrate that start configuration represents a meaningful and previously overlooked source of performance advantage in modern middle-distance racing – large enough to make the difference between a world record and an otherwise great performance.

This study proposes an integrated framework for the processing and classification of invasive bird species vocalizations within natural soundscapes, characterized by high levels of environmental noise. We address the challenge of signal degradation by employing a Bayesian wavelet shrinkage methodology based on the Epanechnikov kernel prior, which offers a closed form decision rule and high computational efficiency for processing large bioacoustic datasets. The methodology was applied to recordings of three species obtained from the iNaturalist platform: Euphonia violacea, Leiothrix lutea, and Passer domesticus. After signal denoising, we extracted a comprehensive set of features, including Mel-Frequency Cepstral Coefficients (MFCCs) and spectral indices such as entropy and zero-crossing rate. Several supervised learning models: Random Forest, Multinomial Logistic Regression and Support Vector Machine (SVM) were evaluated across different feature dimensionalities. Our results demonstrate that the proposed wavelet based preprocessing significantly enhances classification performance, with the SVM model achieving the highest accuracy (up to 0.9398) under a 10-dimensional MFCC configuration. This research provides a robust statistical tool for automated ecological monitoring and the management of biological invasions.

We develop an exact almost-sure learning theory for smooth parametric models trained by regularly weighted empirical risk minimization on long-range dependent data. The training observations are generated from a fixed finite window of a stationary Gaussian sequence, and the sample weights are regularly varying. If the loss gradient at the population minimizer has Wiener-chaos rank $m$ and a nonzero low-frequency coefficient, then, in the long-memory interior regime, the finite-lag score reduces on the iterated-logarithm scale to a single weighted Hermite chaos. This yields an almost-sure Bahadur representation, an exact limsup law for the learned parameter, and, for $m\ge2$, the functional cluster set of the complete learning trajectory. The polynomial learning exponent is determined by the memory parameter and the chaos rank and is invariant under the admissible power weighting, whereas the sharp pathwise constant and cluster geometry depend on the weights. In the rank-one case, global optimization over the admissible power exponents shows that every optimizer is positive. Time-series prediction and classification examples illustrate the results.

Conformal prediction converts point predictions into set-valued predictions with coverage guarantees under exchangeability between calibration and deployment data. We study conformal calibration transfer, where this requirement fails because labeled calibration is available only in a source space, while prediction sets are needed in a target space linked to the source through unlabeled paired observations (e.g., paired modalities or sensor changes). We propose Transported Conformal Calibration (TCC): we transport labeled source calibration into the target space using the paired data, and then correct residual post-transport mismatch using only unlabeled target inputs. We instantiate this correction with two complementary methods: TCC-KS, which uses a label-free uncertainty surrogate to detect mismatch and adjust calibration conservatively, and weighted-TCC, which reweights transported calibration toward the target domain for improved efficiency when weights are stable. We provide finite-sample target-domain coverage guarantees that adapt to an observable measure of mismatch. Across CIFAR-100-C, Tiny-ImageNet-C, and SEN12MS, we show reliable target-domain coverage transfer without labeled target calibration data, with label-free diagnostics that predict when correction is needed.

Quantum-inspired classical algorithms have dequantized several quantum machine learning routines by replacing quantum linear-algebra subroutines with classical counterparts. However, the sampler based on quantum singular value transformation (QSVT) for learning with optimized random features is not covered by existing dequantization frameworks, because the matrix to be inverted is not itself available through sampling access. In this work, we develop a classical algorithm to address this type of quantum-advantage candidate. Our method samples heavy indices, reduces the transformation to a small principal block, and outputs a sparse classical representation with operator-norm guarantees. Applying this method dequantizes the sampler for optimized random features, giving a classical sampler with prescribed accuracy and polynomially related runtime. These results show that the factorization underlying a quantum block encoding can itself provide sufficient classical structure even when sampling-and-query access to the composite matrix is unavailable.

Consistent data on the sizes of key populations, such as female sex workers (FSWs), are often scarce, particularly at the sub-national level. Accurate size estimates are critical to effectively allocate resources and achieve HIV targets. Since FSW population sizes may be spatially correlated across areas, models that account for spatial dependence can improve estimation. An important component of such models is the covariance function, which characterizes the spatial dependence structure of the underlying process. In this work, we study spatial covariance functions to estimate FSW population sizes in Sub-Saharan Africa (SSA). Many spatial models rely on parametric covariance functions. However, parametric estimation can suffer from model mis-specification, potentially leading to inefficient or biased predictions. We therefore develop a robust non-parametric approach for estimating the covariance function of a stationary isotropic process in $\mathbb{R}^d$. We focus on a class of covariance functions that are valid in all dimensions, which includes popular kernels such as the exponential and Matérn kernels. Leveraging the fact that such covariance functions can be represented as infinite mixtures of scaled Gaussian kernels, we propose two estimation methods: weighted least squares and nonparametric maximum likelihood estimation to estimate the mixing measure of scaled Gaussian kernels. We also develop computationally efficient methods to solve these optimization problems using non-negative least squares and second-order descent updates. We evaluate the proposed methods through simulations and apply them to estimate the FSW population sizes at the sub-national level in SSA.

We present a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close) in the presence of nuisance parameters, based only on a sample generator from the distribution of interest. We show that the statistic is near-pivotal in the sense of minimum average KL-divergence of its $p$-values versus uniform and we argue that it can be expected to have good power when the dimension of the statistic equals the dimension of the parameter. It is able to incorporate prior knowledge about group invariances such as translation and scale. It can discover the one-sample $t$-test almost exactly, outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range and achieves good calibration on partial biserial correlations, while showing higher power (and being much faster) on small-to-moderate samples than profile likelihood-ratio techniques.

We prove the gap-entropy conjecture for fixed-confidence best-arm identification with independent unit-variance Gaussian arms, means in $[0,1]$, and a unique optimal arm. For each suboptimal arm $i$, let $\Delta_i=\mu_*-\mu_i$ be its gap from the optimal mean, and write $H=\sum_{i\ne *}\Delta_i^{-2}$. Let $p_r$ be the fraction of $H$ contributed by arms with $2^{-(r+1)}<\Delta_i\le2^{-r}$, and let $\mathrm{Ent}(I)=\sum_{r:p_r>0} p_r\log(1/p_r)$. Among all algorithms that identify the optimal arm with probability at least $1-\delta$ on every Gaussian instance, the optimal expected number of samples on a given instance, averaged over all permutations of the arm labels, is within absolute constant factors of $H(\log(1/\delta)+\mathrm{Ent}(I))$. Moreover, there is an algorithm, independent of the instance, whose expected number of samples is bounded by a constant multiple of this quantity plus $g^{-2}\log\log(e^e/g)$, where $g=\min_{i\ne *}\Delta_i$ is the gap to the closest competitor.

Recovering the linear relationships that govern a system from noisy measurements is a basic task across the physical and engineering sciences. Because every measured variable may carry an unknown amount of noise, classical regression must commit in advance to a set of structural assumptions: ordinary least squares requires a declared input-output partition with input variables being noise-free, total least squares assumes equal noise variance across all variables, and generalized total least squares additionally requires the noisy-variable partition and variances to be known beforehand. Kalman~\cite{Kalman:1982} showed that any procedure returning a unique linear model from inexact data must rest on such unverifiable a priori assumptions – “prejudices” – that cannot be checked against the data itself, and that removing them leaves the identification problem fundamentally indeterminate. Whether these prejudices can instead be resolved directly from the data has remain unresolved. Here we show that an iterative generalized-eigenvalue algorithm, QZ-IPCA, recovers the noisy-variable partition, noise variances, number of linear relations, and regression coefficients of a multivariate linear system simultaneously, using only the raw data. Across all possible exhaustive noise configurations of a five-variable benchmark network, QZ-IPCA correctly identifies model structure and recovers coefficients with error below 6.4%. It outperforms ordinary least squares even when given the best partition, and succeeds in rank identification precisely where standard total least squares falls once noise variances differ across variables. These results show that the assumptions conventionally required for multivariate regression are not necessary, recasting model identification as a problem solvable from data geometry alone.

Motivated by Wasserstein measures of dependence, we study the largest possible 2-Wasserstein distance between a joint distribution and the product of its prescribed marginals. For two uniform marginals, Catalano and Lavenant conjectured that the monotone and antimonotone couplings maximize the distance from the independent coupling. We prove the Gaussian analogue for an arbitrary number $n\geq 2$ of one-dimensional standard Gaussian marginals. More generally, for every probability measure $\mu$ on $\mathbb R$ with finite second moment, we characterize the laws on $\mathbb{R}^n$ with all marginals equal to $\mu$ that are farthest from the $n$-dimensional standard Gaussian.

Algorithmic stability refers to the property of an algorithm being insensitive to perturbations of the input data, where the type of perturbation may vary depending on the setting. In this work, we develop a general framework to quantify the extent to which any ensembling strategy defined via averaging can yield stability guarantees for any type of data perturbation. Our main theoretical result is a guarantee on the stability of this ensembled algorithm, given in terms of the norm of a certain covariance operator that describes the ensembling process. We show how our general framework yields interpretable and intuitive insights in several examples of perturbations of practical interest, and provides much sharper guarantees than those obtained from privacy considerations.

Most work on survival prediction focuses on estimating survival probabilities rather than predicting individual event times. Recent conformal methods have made it possible to construct prediction intervals for survival times with right-censored outcomes, but existing approaches are restricted to settings with covariates only measured at baseline and do not address dynamic prediction with longitudinal data. We study prediction intervals for survival times in a dynamic prediction framework with longitudinal covariates. Our approach uses Penalized Regression Calibration (PRC) as a working dynamic prediction model, combining linear mixed models for the longitudinal histories with a Cox model for post-landmark survival, and then applies a conformal calibration step to obtain prediction intervals. We compare naive intervals obtained by direct inversion of the survival function estimated by PRC to our dynamic conformal method. A Monte Carlo simulation study evaluates empirical coverage and interval length across sample sizes, censoring levels, landmark times, and non-proportional hazards (NPH) scenarios. We illustrate the proposed methodology by computing dynamic prediction intervals for the time until a dementia diagnosis in the ADNI dataset. The results show that naive inversion is often unreliable, whereas the proposed dynamic conformal method yields more stable predictive performance.

Shapley values are widely used for post-hoc feature attribution, but most estimators return point quantities and do not quantify uncertainty, and popular implementations sample out-of-coalition features from their marginal distribution, which misattributes importance when features are dependent. This paper studies the conditional formulation, in which out-of-coalition features are integrated out under their true conditional distribution. The target is a global, loss-based importance that pairs a conditional value function with a SAGE-style loss aggregation. We propose a one-step estimator with K-fold cross-fitting and a U-statistic correction of the squared loss that removes the Monte Carlo bias of the naive plug-in; it is $\sqrt{n}$-consistent and asymptotically normal under double-robust rate conditions, and the resulting Wald interval attains nominal coverage. A Pinsker-type bound quantifies the bias from misspecifying the working copula class, while vine copulas keep conditional sampling tractable. In a Gaussian design study with n= 500, the empirical coverage of the 95% interval lies between 0.91 and 0.96 across all features, the test holds its Type-I rate at 0.05, and it reaches power one for moderate signals. Applied to the UCI Concrete and California Housing data, the method identifies the conditionally informative features with Bonferroni-controlled significance.

Continuous measurement of quantum systems gives rise to stochastic dynamics of the conditional quantum state, described by diffusive stochastic master equations. In this paper, we study parameter estimation for such equations when the Hamiltonian and measurement operators depend on unknown parameters. Based on multiple independent observed trajectories with a known initial state, we construct a contrast function using the deterministic averaged state and define a maximum contrast estimator for the unknown parameter. We prove strong consistency and asymptotic normality of the parameter estimator in a fixed-time, many-trajectory asymptotic regime. A key point is that the covariance matrix appearing in the asymptotic normality is given in a form that naturally leads to a consistent covariance estimator. This covariance estimator is computable from the observed data together with the deterministic averaged dynamics, so the asymptotic normality result can be used to construct standard errors and assess uncertainty for the parameter estimator.

Models for droughts draw on a wide range of meteorological and hydrological inputs. Stakeholders classify droughts according to different purposes and priorities, and accordingly rely on different measures to explain and predict the onset of drought. While many meteorological inputs are available as gridded data products, hydrological streamflow measurements are often only available as point-referenced gauge measurements. This leads to an issue of misalignment for studies relying on both areal meteorological and point-referenced hydrological data. Such gauge data can also have notable spatial and/or temporal missingness. Many areas remain ungauged, and where gauges exist, equipment malfunctions cause temporal gaps. In this study, we document the value of an existing gridded streamflow measure for the conterminous United States, one which was specifically designed to match the spatio-temporal support of publicly available meteorological and ordinal drought measurements. We use this homogenized database to assess the relative importance of this streamflow measure in explaining US drought variability. This assessment first requires that we address autocorrelation and variability in the variance of observed droughts, two extensions to existing statistical methodology. After suitably controlling for both, our results show that the streamflow measure is often the most statistically important explanatory variable from among a wide set of meteorological variables. We explore the spatial variation and some drivers of this result.

Attias, Hanneke and Ramaswami (NeurIPS 2025) asked whether randomization provably reduces the oracle calls needed for online learning when the class is accessible only through an oracle. We study the instance they singled out: transductive online learning of thresholds on an unknown total order of T instances, with a consistency-type ERM oracle that returns a full concept consistent with a queried labeled set (or reports non-realizability). Our main result is a separation for a fixed natural oracle. When the oracle is the minimal-prefix rule (or the maximal-prefix rule), every deterministic learner makes M mistakes and Q calls with $M+Q\ge T-\varepsilon$ on some instance ($\varepsilon\in\{0,1\}$, according to whether the empty prefix is a concept), and the constant is exact; hence $O(\log T)$ mistakes cost $T-\varepsilon-O(\log T)$ calls, whereas that paper's randomized learner achieves $O(\log T)$ expected calls and mistakes under the same rule. The randomized order is optimal: on an explicit hard distribution under the minimal-prefix rule, every learner has expected mistakes at least $((T+1-\varepsilon)\,128^{-\mathbb{E}[Q]}-1)/2$, so $\Omega(\log T)$ expected calls are necessary for polylogarithmic mistakes. The separation is governed by the oracle's selection rule, not by the class alone: for a legal feasible-median ERM rule a deterministic learner achieves $O(\log T)$ calls and mistakes, while a global-median rule again forces linear total cost. The same linear bound holds when the oracle's answers are chosen adversarially and then frozen into a memoryless oracle. We add partial tradeoff results for fixed query budgets (the middle regime is open) and an interface contrast: with only a weak consistency oracle, returning a realizability bit, both deterministic and randomized learners need $\Theta(T)$ calls.

Tuberculosis (TB) remains a critical public health concern in Brazil, characterized by pronounced spatial heterogeneity and fluctuating temporal volatility. In this paper, we study monthly TB notifications across 61 microregions of Sao Paulo state from 2001 to 2024. To do this, we introduce a negative binomial spatial integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) model featuring jointly dynamic conditional means and time-varying dispersion. To capture inter-regional spillovers, we incorporate both discrete adjacency structures and a novel continuous distance-based formulation leveraging the Matern correlation function. Parameter estimation via conditional maximum likelihood employs a two-step profile-likelihood iterative scheme, demonstrating solid finite-sample performance in simulation studies. Applied to the Sao Paulo TB surveillance data, the framework substantially outperforms standard Poisson and fixed-dispersion spatiotemporal baselines in empirical fit and uncertainty quantification, maintaining nominal 95% predictive coverage across both dense metropolitan centers and rural microregions. Our results reveal marked spatial heterogeneity in baseline incidence, dynamic overdispersion driven by localized outbreaks, and short-range spatial interaction decay. By accurately modeling spatiotemporal volatility, the proposed methodology provides a robust statistical tool to support public health surveillance, policy-making, and resource allocation.

Generalized zero-shot learning (GZSL) has emerged as an important paradigm for visual recognition systems that must generalize to classes that were not observed during training. Traditional GZSL techniques are limited by their applicability to a relatively small number of such unseen classes, scalability beyond which is challenging due to its well-known misclassification bias towards classes observed during training. In this work, we investigate the GZSL paradigm through the lens of zero-shot handwritten word recognition over extremely large vocabularies. We propose a statistical approach to rectifying this bias, which views any classical GZSL feature learner as a black box mechanism whose intrinsic bias in identifying the training status (seen vs. unseen) of a typical data point we aim to correct, similar to an out of distribution inferential problem. Our method leverages a simple two-stage hierarchical architecture, combining a classical GZSL blackbox in the first stage and an ensemble of lightweight Monte Carlo bias-correctors in the second. Once debiased, the classification of test data is undertaken only restricted to its predicted training status via well-founded statistical methods (eg nearest neighbour, logistic regression and random forests). We achieve relative accuracy improvements of over 20% in the classification of unseen words compared to established techniques. A key outcome is that word recognition over large scale vocabularies is amenable to a much lower dimensional representation (~15 dimensions). Our approach is underpinned by mathematical analysis that captures the essence of the statistical approach to bias correction. Our approach to bias rectification can be combined in a turn-key fashion with any classical GZSL learner as a blackbox, thereby suggesting a wide scope of applicability of this method for a wide variety of GZSL implementations in different domains.

Parametric bootstrap and fiducial generalized methods are proposed to construct simultaneous confidence intervals (SCIs) for all pairwise mean differences of several two-parameter exponential distributions, based on doubly censored samples. Both methods have been shown to have correct asymptotic coverage probabilities. Simulation studies are conducted to compare the performance of the two proposed methods. The usefulness of our proposed procedure is illustrated with two examples.

Multidimensional item response theory (MIRT) is a statistical test theory that precisely estimates multiple latent skills of learners from the responses in a test. Both compensatory and non-compensatory models have been proposed for MIRT: the former assumes that each skill can complement other skills, whereas the latter assumes they cannot. This non-compensatory assumption is convincing in many tests that measure multiple skills; therefore, applying non-compensatory models to such data is crucial for achieving unbiased and accurate estimation. In contrast to tests, latent skills will change over time in daily learning. To monitor the growth of skills, dynamical extensions of MIRT models have been investigated. However, most of them assumed compensatory models, and a model that can reproduce continuous latent states of skills under the non-compensatory assumption has not been proposed thus far. To enable accurate skill tracing under the non-compensatory assumption, we propose a dynamical extension of non-compensatory MIRT models by combining a linear dynamical system and a non-compensatory model. This results in a complicated posterior of skills, which we approximate with a Gaussian distribution by minimizing the Kullback-Leibler divergence between the approximated posterior and the true posterior. The learning algorithm for the model parameters is derived through Monte Carlo expectation maximization. Simulation studies verify that the proposed method is able to reproduce latent skills accurately, whereas the dynamical compensatory model suffers from significant underestimation errors. Furthermore, experiments on an actual data set demonstrate that our dynamical non-compensatory model can infer practical skill tracing and clarify differences in skill tracing between non-compensatory and compensatory models.

Tests for cointegration are typically applied to a single window spanning the entire sample, assuming that the long-run relationship holds throughout. When it holds over only a part of the sample, such tests lose power, because the stationary episode is diluted by periods without cointegration. We propose three statistics for testing whether two or more series cointegrate only over a part of the sample, each an infimum of the Engle-Granger statistic over recursive, backward-expanding, or doubly-flexible windows. We derive their limiting distributions and establish which alternatives each is consistent against. Only the doubly-flexible statistic has power against both break directions. Inspired by the seminal work of James G. MacKinnon, critical values are obtained by simulation and summarized through response surface regressions. We apply the tests to global mean sea level and global mean surface temperature anomalies. All three reject the null of no cointegration at the 5% level, locating it in a sub-period of the 1880–2019 record that coincides with documented discontinuities in sea surface temperature data collection.

We study offline inference for the optimal value in reinforcement learning under finite state and action spaces. Two new nuisances are derived as fixed points of a self-induced Bellman equation, in which we approximate the maximum Bellman operator by its softmax correspondence. We propose a debiased estimator through the Neyman orthogonality and establish its asymptotic normality under diverging horizons even when the behavior policy changes with time, as long as the nuisances have the statistical rates that can be achieved by many machine learning methods. We provide a concrete estimating procedure for these nuisances and show they can lead to valid inference. Synthetic experiments validate the numerical performance of our inference method, and we implement it in real-life decision-making problems, including bike repositioning and AI agentic tool use.

Machine learning-based credit scoring is increasingly central to Peer-to-Peer (P2P) lending, yet its resilience to adversarial manipulation, where applicants strategically alter self-reported inputs to secure favourable decisions, remains poorly understood. Most adversarial-robustness evidence comes from image and text domains and evaluates a single attack against a matching defence, offering little guidance on how defences generalise across attack types in tabular credit data. We address this with a systematic train-test robustness benchmark on a large Lending Club subset, spanning three model families (logistic regression, a feed-forward neural network, and a transformer for tabular data) and four attacks confined to applicant-mutable features: Fast Gradient Sign Method (FGSM), Projected Gradient Descent (PGD), Salt-and-Pepper (S&P) noise, and DeepFool, plus a mixed-attack regime. Across a full grid evaluated with stratified cross-validation, adversarial training sharply improves robustness against the attack it is trained on and transfers well within the gradient-based family, but transfers weakly to non-gradient corruption, so single-attack defences overstate real-world resilience. Mixed training delivers the most balanced robustness across heterogeneous attacks while preserving clean-test performance, supporting multi-attack stress testing in credit-model governance.

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

Preference alignment for flow and diffusion models now spans online reinforcement learning and offline preference optimization, but the relation between these methods remains unclear. In particular, existing forward-process alignment methods require fresh samples from the current model, while offline methods based on fixed preference pairs rely primarily on positive-only fine-tuning or DPO-style likelihood-ratio surrogates. We organize these approaches through a divergence-based framework and introduce FlowCPO, an offline forward-KL objective that uses both preferred and dispreferred samples without online rollouts. For linear interpolation, we show under explicit regularity conditions that the forward-KL objective is bounded by a contrastive flow matching loss, yielding a tractable surrogate on fixed data. We further show that this loss is nonnegative, whereas the signed regression loss of simplified FlowDPO can be unbounded below. In the in-domain setting, FlowCPO achieves higher mean GenEval and OCR scores than the evaluated baselines, reaching 0.84 and 0.87 versus 0.81 and 0.74 for FlowDPO at CFG 3.0. In the out-of-domain setting, the results are mixed, with the best GenEval result but lower reward scores than RFT on several metrics.

Federated clients that perform several local optimization steps can return parameter displacements with widely different magnitudes. The quadratic regularization of FedProx grows linearly with displacement and therefore offers limited control over the contrast between ordinary and unusually large client movements. We here introduce HiFedProx, which replaces the quadratic penalty with a scale-matched power-type regularizer indexed by $p\geq2$. All powers have the same regularization-gradient magnitude at a reference displacement $R$, while every $p>2$ gives a weaker response below $R$ and a stronger response above it. An exact affine reference calculation shows that increasing $p$ compresses relative displacement disparities, although very large powers approach fixed-radius behavior and increase local curvature. HiFedProx combines this geometry with finite-budget stochastic client optimization and same-minibatch Armijo backtracking. In paired five-seed experiments on a frozen 60-writer FEMNIST subset, a common-parameter study over $p\in\{2,3,4,5,6,7,8\}$ shows similar clean-training performance but substantial gains under composite stress. The lowest moderate- and severe-stress losses occur at $p=7$ and $p=6$, improving over $p=2$ by $11.44\%$ and $23.16\%$, respectively. Although displacement-tail ratios continue to decrease through $p=8$, predictive performance peaks in an intermediate range and Armijo trial cost increases with $p$. These results indicate that the exponent should be calibrated rather than maximized. In our experiments, $p=5$–$7$ provides the most useful range.