Papers

Detecting distributional changes in high dimension is difficult when neither the pre-change nor post-change density is parametrically specified. We introduce a representation-based approach that retains all degree-at-most-two density information while replacing density estimation by matrix mean estimation. For observations in $[-1,1]^d$, a symmetric feature matrix $H_2(X)\in\R^{(d+1)\times(d+1)}$ is constructed so that $M(f)=\E_f H_2(X)$ is an isometric encoding of the degree-two orthogonal projection of the density. We scan matrix CUSUMs after rank-$r$ truncation, exploiting the low rank of the projected jump rather than sparsity of individual coordinates. The resulting \LRD{} estimator has a tent-shaped population objective and a nonasymptotic operator-norm analysis whose leading stochastic term scales as $\sqrt{rd\log(nd)}$. For multiple changes, we give a seeded narrowest-over-threshold procedure and prove exact recovery by an induction that preserves an isolating interval for every undetected change. A cross-fitted scalar refinement learns the changing low-rank direction on one fold and localizes on the other, attaining $\widetilde O_{\Pp}(\kappa^{-2})$ error; a matching Le Cam lower bound shows optimality up to logarithms. A geometrically $\beta$-mixing extension follows from a dependent matrix Bernstein inequality. Experiments with ambient dimension up to $200$, a three-change $d=100$ sequence, and a $128$-feature human-activity benchmark show that the method remains computationally practical and accurately detects pure dependence changes that are invisible to mean CUSUMs.

Entropy functionals and their associated divergences underlie many statistical methods, including maximum entropy inference, minimum divergence estimation, and goodness-of-fit testing, yet choosing among Shannon, Rényi, Tsallis, and more general entropies is often a matter of convention rather than structural principle. We introduce a measure theoretic framework in which admissibility requires the entropy of an input measure to be bounded above by that of its reference measure whenever the former is absolutely continuous with respect to the latter. Under generalized mean-value composition, we characterize all such entropy functionals and obtain a four-level hierarchy determined successively by the mean generator, entropy scale, and additivity assumptions. A continuous strictly monotone generator $g$ is admissible exactly when $t\mapsto g(1/t)$ is strictly convex for increasing $g$, or strictly concave for decreasing $g$. This resolves a question posed by Rényi (Proc. 4th Berkeley Sympos. Math. Statist. Prob., 1961) concerning which generalized means may replace the arithmetic mean in his entropy axiomatization. The same criterion is equivalent to strict convexity of an associated Csiszár $f$-divergence generator and therefore yields data processing under Markov kernels with an exact equality condition. Within this hierarchy, product additivity singles out the Rényi family, while internal additivity, or product additivity together with arithmetic mean-value composition, singles out Shannon entropy. The characterization is constructive and yields new admissible entropy and divergence families, including integral-transform examples.

Forecast combining and forecast reconciliation for hierarchical and grouped time series have largely developed as separate research areas. This paper connects the two by developing a forecast combination framework for forecast reconciliation. For each bottom-level series, we construct a maximal linearly independent set of structured candidate forecasts from aggregation constraints, and show that combining these candidates and aggregating the resulting bottom-level forecasts is equivalent to standard unbiased linear reconciliation. Within this representation, we prove that mean-squared-error optimal combination weights exactly recover the widely used Minimum Trace (MinT) reconciliation. We further show that the optimal weight problem is separable across different bottom-level series, each yielding a Bates–Granger optimal forecast combination. This reveals MinT as a collection of optimal combinations over hierarchy-induced candidate forecasts. For finite-sample estimation, we propose a modular penalized framework that nests existing MinT variants and supports rich extensions including covariance shrinkage, weight penalization, and scalable series-wise separate estimation. Empirical results show that the framework is practically implementable, competitive with existing methods, and can improve accuracy while preserving coherence. Overall, the forecast combination perspective offers new interpretations of existing reconciliation approaches and provides a flexible basis for designing new methods.

High-dimensional functional panels consist of temporally ordered curves observed across many subjects and naturally exhibit heterogeneous structural changes. Under sparse subject-level break signals or opposite-signed shifts, traditional mean-aggregated CUSUM procedures may suffer noticeable power loss due to signal attenuation or cancellation induced by cross-sectional averaging. We propose a novel Energy–PE statistic, which combines subject-wise squared CUSUM energy aggregation with a generalized power-enhancement component. The energy aggregation preserves subject-level evidence under sign-heterogeneous changes, while the power-enhancement component improves sensitivity to sparse weak break signals. Under regularity conditions, we establish the asymptotic behavior of the proposed statistic. We further incorporate a latent group structure and an information-criterion-based clustering algorithm to estimate the unknown group number and membership for heterogeneous break points. Numerical studies and an intraday stock application demonstrate that Energy–PE controls size, improves power under sparse and sign-heterogeneous alternatives, and yields interpretable post-test summaries.

This paper studies trajectory-wise estimation of generalization error for primal–dual algorithms in non-smooth regression. Motivating examples include \(\ell_1\)-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle–Pock algorithm and related primal–dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.

Receiver operating characteristic (ROC) and the area under the ROC curve (AUC) are widely used to evaluate the discriminative ability of biomarkers. In many clinical settings, however, diagnostic accuracy varies substantially across patient characteristics, and failure to account for such heterogeneity can lead to misleading conclusions. We propose a new semiparametric framework based on generalized additive models to estimate covariate-specific and covariate-adjusted AUC while allowing for nonlinear and interaction effects of covariates on biomarker performance. Our method accommodates both binary and multicategory disease status. We also establish the asymptotic properties of the proposed estimators. Simulations demonstrate favorable finite-sample performance. We illustrate the method using data from the Alzheimer's Disease Neuroimaging Initiative, where substantial heterogeneity in biomarker discrimination across covariates is observed.

Timely urban-canopy information is essential for linking remote sensing with heat, mobility, and neighborhood planning. We developed an optical GeoAI workflow for Davis, California, using 2022 National Agriculture Imagery Program imagery (0.6 m RGB+NIR). DeepForest generated crown candidates; an NDVI threshold, non-maximum suppression, and box-prompted Segment Anything Model (ViT-B) produced a crown-anchored canopy surface. Analyses used the 25.92 km2 Census TIGER municipal boundary and a 100 m grid. The workflow retained 11,741 candidate crowns and mapped 7.71 km2 of canopy (29.8% of the city). On the identical extent, pixel precision was 0.804, pixel recall was 0.873, and 97.4% of candidate centers agreed with the 2022 USDA/CAL FIRE LiDAR-assisted canopy product (IoU 0.719; Dice 0.837; area recovery 108.5%). Approximately 49% of candidates occurred within 15 m of a road. Canopy was inversely associated with Landsat land-surface temperature (Spearman rho = -0.477; partial rho = -0.551 controlling for built probability), and spatial-lag modeling confirmed clear neighborhood structure. Two transparent attention surfaces combined canopy need with thermal and contextual indicators. The framework provides a reproducible, updateable screening layer that complements structural canopy products and municipal inventories while retaining assumptions, data provenance, and spatial diagnostics for planning interpretation.

We present the first application of simulation-based inference to resonant inelastic X-ray scattering spectroscopy. Using truncated marginal neural ratio estimation to efficiently restrict the prior and conditional flow matching as the joint density estimator, we infer full posteriors with a modest simulation budget for two Ni$^{2+}$ compounds—NiPS$_3$ as a representative covalent case and K$_2$NiF$_4$ as a more atomic one. We demonstrate that a vision transformer encoder whose tokenization matches the physical layout of the RIXS map yields better-covered and sharper posteriors than generic image encoders. Applying the validated method to experimental NiPS$_3$ and K$_2$NiF$_4$ data, we recover a joint posterior that reveals parameter correlations invisible to point estimators, and a posterior predictive distribution that closely matches the observed spectrum. The amortized posterior unlocks a class of analyses not previously available to the field such as nuisance-marginalized uncertainty quantification, multi-measurement posterior fusion and active experimental design.

Recovering geographic variation in survival requires separating spatial risk from patients' clinical characteristics, a problem complicated by prognostic covariates that are themselves spatially structured. We develop a nonparametric two-stage method for this separation. A clinical survival tree fit to the covariates alone supplies leaf Nelson-Aalen cumulative hazard residuals, transferring the censored survival structure to a clinically adjusted scale without imposing a functional form on the clinical hazard, and a second tree fit to these residuals on the coordinates recovers the spatial structure. Both stages are kernel dipole-splitting survival trees, so the resulting spatial risk map is piecewise constant, with sharp, possibly curved boundaries. We establish when the residuals recover the spatial signal: under a multiplicative frailty and an exogeneity condition, they are free of the clinical covariates given location and stochastically ordered by the frailty. An expansion of the frailty Laplace exponent quantifies their approximate exponentiality and identifies the leading remainder. These results assume consistency of the first stage rather than a model class, so the construction extends to other survival estimators. On the LeukSurv leukemia data the method agrees with a Bayesian Gaussian random field frailty about where risk is elevated while resolving sharp adjacencies the smooth surface averages away, and an unadjusted spatial analysis misattributes clinical variation to location. Simulations with known zones, including a sweep through graded violations of exogeneity, locate the point at which the two contributions cease to be separately identifiable, with the smooth benchmark degrading in parallel as that point is approached.

Three-player Auction Bridge is a finite imperfect-information game in which dynamic partnerships create a distinctive interaction between scoring, strategic incentives, and payoff distribution. This paper develops a unified statistical and game-theoretic framework to evaluate a traditional scoring rule against a modified mechanism designed to improve strategic incentives. We first identify a structural defect in the original scheme: a bid-invariant slam bonus can make lower contracts strictly more attractive than higher ones. A bid-dependent correction removes this incentive distortion. We then compare the two schemes using higher-moment analysis, fairness measures, Nash equilibrium analysis, and General-Sum Counterfactual Regret Minimization (GS-CFR). The results show significant differences in the shape of the payoff distributions and reveal that, although both schemes remain highly balanced across physical seats, the modified scheme substantially increases inequality across strategic roles. The game is shown to be neither zero-sum nor constant-sum, motivating a general-sum rather than minimax analysis. Full-game GS-CFR further indicates a substantial increase in the bidder's payoff under the modified scheme, accompanied by a reduction in defender payoff and a shift from separating to partially pooling bidding behavior. These findings demonstrate that scoring-rule design can fundamentally reshape incentives, payoff distribution, and information transmission in imperfect-information games. The proposed framework provides a systematic approach to evaluating such mechanisms from both statistical and strategic perspectives.

Despite growing interest in using LLMs to add robustness or reduce data-collection costs in survey experiments, their efficacy in conjoint design—an increasingly popular method in political science—remains underexplored. This paper addresses that gap by investigating whether synthetic agents can reproduce the multi-dimensional human preference patterns that conjoint is designed to capture. It replicates published conjoint studies and compares the results generated by synthetic agents with original human data along three dimensions: representational correspondence, inferential correspondence, and procedural stability. Our analysis evaluates the alignment of choice distributions as well as the statistical and substantive similarity of estimates, and the results are uneven across these dimensions and studies replicated. This implies that the validity of synthetic participants should be considered claim-dependent and hierarchical. Reproducing a figure or obtaining strong sign agreement is evidence of similar aggregate outputs, but not enough to support replacing human respondents. Our results suggest that the discipline as a whole must first map this innovation's boundaries across various levels before considering synthetic agents a robust substitute for human samples.

The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.

Exposure control lets an adversary rank authentic public posts to strengthen private-attribute inference without altering content. AccretionLink defines confidentiality and integrity games for this attack, models bounded selection odds through partial identification, and constructs dependence-aware time-uniform e-processes. On 52 held-out synthetic profiles, odds-four selection reduced aggregate negative log likelihood at every horizon. At eight posts the advantage was 0.01595 nats (95% CI [0.00890, 0.02336]), three of four target effects survived Holm adjustment, and label-blind model-guided selection caused 6/109 high-confidence false reversals. On 142 PAN15 test profiles, exploratory selection produced a 0.01227-nat advantage but no reversal. A separate TF-IDF selector retained a 0.01470-nat advantage against the unchanged G5 target, while matched identity shuffling did not reproduce it. Pixel 10 encoded all 1,622 held-out posts once with a fallback-free Tensor G5 graph. A P-256 checkpoint authenticated the selected-replay, actual-model, native-report, and operation digests; local KeyInfo identified the signing key as StrongBox-backed.

We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $\rho_n^*$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion $\rho_n^* = n^{-1} - Cn^{-4/3} + o_n(n^{-4/3})$ for an explicit constant $C$ related to the Airy function. The minimax risk is attained by Bernoulli randomization with a nonlinear shrinkage estimator. Our results show that standard procedures such as complete randomization with difference in means are only minimax optimal up to first order in $n.$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.

Joint variability in temperature and precipitation is central in characterizing seasonal climate structure and associated environmental processes, yet many regional analyses rely on marginal or univariate summaries. We analyze seasonal temperature-precipitation patterns across the Southern United States using two complementary multivariate statistical approaches. First, functional multivariate analysis of variance (FMANOVA) is employed to test the equality of state-level bivariate mean functions, with the permutation-based Wilks' lambda and Pillai's trace statistics. Second, Gaussian mixture models are applied to station-level seasonal summaries to identify latent climate regimes based on the joint distribution of temperature and precipitation. The FMANOVA results indicate statistically significant differences in bivariate mean trajectories between states in both winter and summer, with seasonal contrasts reflecting differing contributions of temperature and precipitation. Clustering analysis indicates more clearly defined and spatially coherent winter regimes than summer regimes, with summer regimes exhibiting greater variability and a stronger role for precipitation.

Machine learning (ML) has become an indispensable part of modern engineering design workflows. A crucial step in training an ML model is the selection of the loss function which can be systematically formulated via various techniques such as maximum likelihood estimation (MLE) and cross-validation . While MLE is one of the most popular, effective, and intuitive mechanisms for training ML models, it is brittle: if the assumptions underpinning it are not met, the trained ML model may generalize poorly. This brittleness affects even Gaussian processes (GPs) which are widely used in engineering design and are often (incorrectly) presumed to be very robust to overfitting. In this paper, we fundamentally evaluate the brittleness of MLE in the context of training GPs for probabilistic regression or classification tasks. We compare theoretically grounded metrics against MLE and propose practical solutions. Our extensive studies demonstrate the effectiveness of our solutions in downstream design tasks such as Bayesian optimization and provide a blueprint for practitioners to build accurate and robust GPs that can even outperform tabular foundation models in terms of prediction accuracy, uncertainty quantification, and inference cost. Our contributions are publicly available via GitHub at https://github.com/Bostanabad-Research-Group/GP-vs-TabPFN-vs-GPyTorch.

In the era of modern data science, the rapid proliferation of high-dimensional and functional datasets has fostered increasing interest in the investigation of paradigm shifts and structural breaks. Unlike classical univariate time series, structural changes in functional data need not occur simultaneously across the entire domain; instead, they may emerge locally, producing heterogeneous distortions across the underlying functional structure. The patterns of instability often exhibit sparsity, where it is not known a priori which specific parameters are undergoing a transition. However, in functional contexts, these shifts are often "localised". The difficulty lies in the high dimensionality of the parameter space, where the signal-to-noise ratio may be low for individual components, necessitating the aggregation of information across dimensions to detect a global change. This paper addresses the problem of detecting structural shifts in a functional time series from a Bayesian perspective. We have developed various novel methodologies that capture the inherent structural distortion in a sequence of random functions, both individually and simultaneously. The formulation of the problem is based on the state-space representation of a functional time series. Efficient Blocked Gibbs Sampling algorithms have been proposed to identify these locations accurately. Further, we demonstrate the effectiveness of our methods on several financial and temperature datasets.

Neurodegenexrative diseases such as Alzheimer's disease and Parkinson's disease are diagnosed most reliably only after substantial, often irreversible, neuronal loss has already occurred, creating an urgent need for quantitative tools that can detect subtle, early, and individual-specific brain changes from neuroimaging data. This review surveys a broad and rapidly evolving toolkit of data-driven techniques for translational neuroscience and personalized neuro-health, organized around four complementary methodological pillars. Throughout, we emphasize how these methodologically diverse approaches converge on a common translational goal: personalized, mechanistically grounded, and clinically actionable models of individual brain health, and we close by discussing the principal open statistical, computational, and clinical challenges that remain.

We study the graph causal optimal transport problem, a generalisation of the classical optimal transport problem in which the allowed couplings satisfy causal restrictions prescribed by a directed graph. We characterise fully the directed acyclic graphs for which the associated graph causal Wasserstein discrepancy is a metric and show that the induced topology agrees with other natural adapted topologies. We characterise the gluing properties of graph causal couplings, prove denseness of Monge couplings, and obtain a dynamic programming principle which allows us to deduce when the graph causal Wasserstein and the adapted Wasserstein distances are equal. Our results link fundamental properties of graph causal optimal transport to structural properties of its underlying graph. Complementing Cheridito and Eckstein (2025), who first introduced such distances and established Lipschitz continuity for the average treatment effect in structural causal models, we obtain Lipschitz continuity of the value function in stochastic team problems.

Binary bipartite data often combine row and column heterogeneity with hierarchical interaction, as when students answer exercises organized by prerequisites or legislators vote on nested policy areas. Fixed-dimensional Euclidean volume grows only polynomially with radius, whereas branching hierarchies expand exponentially. Hyperbolic space matches this growth, and its rooted Gromov product measures shared ancestry. We propose the HypErbolic Latent Position model for bipartite Interaction data (HELPI), which places both object types in hyperbolic space while separating geometric interaction from additive propensities. With unrestricted main effects, a hyperbolic-distance predictor is likelihood-equivalent to a rooted Gromov-product predictor. Radial depth is absorbed by additive terms, leaving a root-invariant projected hierarchy signal as the identified geometric target. We establish identification and posterior contraction results and develop augmented variational procedures for binary outcomes. Simulations support recovery of hierarchical interaction and clarify weak-branch regimes. In Junyi Academy data, curriculum anchors yield an interpretable geometry with temporal prediction close to item-response benchmarks, while distinguishing attempted exercises from unconditional mastery. In U.S. House roll calls, flexible HELPI variants improve on additive and two-parameter logistic benchmarks and recover party and policy structure without using party labels during fitting.

Nonprobability samples (NPS) are attractive because they are less costly to collect, can provide substantially larger sample sizes, and may reach populations that traditional probability surveys do not. As response rates for traditional surveys fall, interest in NPS has grown rapidly within the field of survey statistics. These methods are especially relevant for small area estimation (SAE), where there is ever-present demand for estimates at fine geographic scales and detailed demographic domains. Despite rapid methodological development, there remains limited understanding of which approaches perform best under different conditions. In this paper, we review recent developments in NPS methodology, including the concept of data defect correlation (DDC) as a measure of data quality and as a tool for categorizing the various NPS methods. We then present a comprehensive simulation study that evaluates a range of NPS approaches under varying levels of DDC and extend several existing methods to the SAE setting.

Local polynomial smoothing is commonly used in non-parametric regression, but local linear derivative estimation still has a bias of order $O(h^2)$. This paper proposes an iterative data sharpening method to reduce the bias of derivative estimates while retaining the simplicity of local linear fitting. The method is based on two expectation operators: $L_0$, acting on the regression function, and $L_1$, acting on the first-order derivative. By repeatedly applying the residual operator $R=I-L_0$, a series of sharpened derivative estimates can be constructed. After $l$ sharpening steps, the bias order can be reduced from $O(h^2)$ to $O(h^{2l+2})$. For the Gaussian kernel, all sharpening coefficients equal 1, giving a simple closed-form single-bandwidth expression. Simulation experiments on three smooth test functions show that this method can significantly reduce the estimation bias while revealing a bias-variance trade-off.

Area-level small area estimation (SAE) models, such as the Fay–Herriot (FH) model, borrow strength across domains through covariates and random effects, but they can struggle when the relationship between the covariates and the outcome is spatially heterogeneous, that is, when it changes across the spatial domain of interest. We propose a spatially-clustered FH (SC-FH) framework that simultaneously (i) estimates cluster-specific regression coefficients and random effects variances and (ii) generates spatially coherent partitions of the geographical domain. Estimation maximizes a penalized likelihood that augments the FH likelihood with a Potts-type spatial cohesion term over the areal adjacency graph, through an efficient strategy that alternates between sequential label updates and closed-form FH updates within clusters. In simulation experiments run on the real geography of the application, the method recovers the latent regimes almost exactly whenever they are separated in the covariate–response space and improves prediction accuracy over the standard FH benchmark, with the spatial penalty acting as a stabilizer of both classification and estimation. An empirical application to the average standard output of farms in the Po Valley (Northern Italy) identifies two spatially compact production regimes with significantly different cluster-wise coefficients, and shows that the clusterwise predictor improves on the direct estimates while avoiding the over-shrinkage of the pooled model.

Clinically actionable, patient-specific hemodynamic assessment, specifically wall shear stress, vortex structure and pressure distributions, is critical for determining risky or unfavorable evolution in Abdominal Aortic Aneurysms (AAA). While Physics-Informed Deep Operator Networks (PI-DeepONets) show promising results in complementing established 5 tools such as Computational Fluid Dynamics (CFD), a persistent architectural challenge remains for complex 3D flows. In this direction, we propose a Modified Multi-Input Multi-Output PI-DeepONets (M3PI-DeepONet) designed for predicting unsteady flows in an idealized AAA geometry. Central to our model is the Aggregated Injection strategy, where latent representations from multiple input branches are fused prior to trunk injection, allowing the coordinate basis to adapt to multiple physical constraints. To the best of our knowledge, this is the first architecture to combine the layer-wise gating mechanism with a multi-branch operator-network topology, yielding an input-adaptive trunk basis. Additionally, we integrate the 3D Navier-Stokes equations as governing physical laws, so the model is trained based on physics-informed residuals, initial and boundary conditions, and only 0.3% of the labeled internal data together with the selected branch-conditioning signals. The M3PI-DeepONet simultaneously predicts unsteady 3D flow velocity and pressure fields with an average relative L2 velocity error below 4% and pressure error around 5% while achieving a conservative retained-cycle inference speedup of approximately 36x compared to reference CFD simulations once the branch inputs used for conditioning are available. This work advances the application of deep learning in cardiovascular disease modeling, marking step toward real-time, non-invasive clinical diagnostics.

We study online probabilistic forecasting of binary outcomes chosen by an adaptive adversary. Given an online learning algorithm for a weak hypothesis class $H$, we would like to efficiently obtain two incomparable guarantees that existing online boosting techniques provide separately. Online gradient boosting competes in Brier score with the best predictor induced by the span of $H$ on every sequence, but promises nothing when the span does not contain an accurate predictor. Online weak-to-strong boosting drives classification error to zero under a weak-learning condition, but promises little when that condition fails. We give a simple defensive forecasting algorithm, the Defensive Booster, that obtains both guarantees. On every adaptive sequence, its Brier score is competitive with the best prediction induced by the span of $H$ at the same rate as online gradient boosting; simultaneously, whenever the realized transcript satisfies the smooth weak-learning condition, its Brier score and randomized classification error satisfy the same rate guarantee as online classification boosting. This is achieved by operationalizing the "dual view" of boosting: When the algorithm's randomized classification error is persistently high, its mistake weights form a smooth reweighting on which every weak hypothesis has low edge, yielding an ex-post hard-core certificate that the weak-learning condition fails. We also develop a strongly adaptive variant, which satisfies both guarantees on every time interval. The Defensive Booster is very efficient: it accesses just one weak-class learner, whereas the prior online boosting methods we compare against maintain large weak-learner ensembles. Experiments on synthetic and real data streams demonstrate its strong predictive performance (sometimes substantially improving over all prior baselines) coupled with orders-of-magnitude faster runtime.

The per-instance Jaccard score, or intersection over union (IoU), is standard in multi-label classification and binary segmentation. With $s$ labels, its loss matrix has $2^s$ outcomes and reports. Under the convention $\mathrm{Jac}(\varnothing,\varnothing)=1$, we prove that the Jaccard score, shifted-loss, and ordinary loss matrices are nonsingular and that the loss columns have affine dimension $2^s-1$. The proof combines a finite MinHash Gram representation with Boolean Möbius inversion. For exact calibration, we prove $2^{s-1} \leq \mathrm{CCdim}(L^{\mathrm{Jac}}) \leq 2^s-1$. The lower bound uses a factorially weighted distribution with $2^{s-1}+1$ supported outcomes and Bayes-optimal reports. Consequently, every exactly calibrated convex surrogate requires exponentially many prediction coordinates. We also give two polynomial-dimensional approximation guarantees with explicit regret transfers. A new $F_1$-to-Jaccard transfer turns an existing $(s^2+1)$-dimensional $F_1$ surrogate into a polynomial-time rule with asymptotic Jaccard regret at most $3-2\sqrt{2}$. For any $\alpha>0$ and $0<\rho<1$, a MinHash square-loss surrogate attains Jaccard-regret floor $\alpha$ uniformly over arbitrary conditional label distributions. With probability at least $1-\rho$, the direct construction has dimension $O((s^2+s\log(1/\rho))/\alpha^2)$, while a signed variant has dimension $O((s+\log(1/\rho))/\alpha^2)$. Thus zero-regret calibration requires exponential dimension, whereas every fixed additive regret tolerance admits polynomial prediction dimension.

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the unmasking growth complexity ({UGC\xspace}). Its local increments directly control Kullback–Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {UGC\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to certified-optimal samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {UGC\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {UGC\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetilde{\Omega}(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

We revisit the problem of learning predictors robust to adversarial examples at test-time. We prove that VC classes are adversarially robustly learnable with sample complexity linear in the VC dimension $d$, providing an exponential improvement over the previous upper bound of Montasser, Hanneke, and Srebro (2019). Remarkably, this result is achieved with a simple improper algorithm that combines the classic heuristic bagging (bootstrap aggregation) of Breiman (1996) with robust empirical risk minimization (RERM). Our algorithm computes RERMs on $O(d^\star)$ independent bootstrap samples and outputs their majority vote, where $d^\star$ denotes the dual VC dimension. We complement this result with a lower bound showing that this is unavoidable: in general, any learner in this oracle model requires $\Omega(d^\star)$ calls to an RERM oracle, even when given arbitrarily many training examples.

Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cramér-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.

Covariate-adaptive randomization procedures are widely used in clinical trials to improve covariate balance. In modern applications, experimenters often have access to many covariates, motivating the need for a theory of covariate-adaptive randomization procedures with a diverging number of covariates. This paper studies two classes of covariate-adaptive randomization procedures, referred to as imbalance-efficient covariate-adaptive randomization and imbalance-robust covariate-adaptive randomization, when the feature dimension diverges. We derive convergence rates for the imbalance of the covariates used in randomization. For both procedures, the imbalance is of a smaller order than that under complete randomization when the feature dimension is $o(n)$, whereas it is of the same order when the feature dimension is $\Omega(n)$. For imbalance-robust covariate-adaptive randomization, we further establish the asymptotic properties of the imbalance of additional covariates and use these results to derive the asymptotic distribution of the difference-in-means estimator for the average treatment effect and construct asymptotically valid confidence intervals. Furthermore, we provide extensive numerical and empirical studies to illustrate the practical relevance of our theoretical results.