Papers

This paper develops a unified framework for identifying spatial and temporal boundaries of treatment effects in difference-in-differences designs. Starting from fundamental fluid dynamics equations (Navier-Stokes), we derive conditions under which treatment effects decay exponentially in space and time, enabling researchers to calculate explicit boundaries beyond which effects become undetectable. The framework encompasses both linear (pure diffusion) and nonlinear (advection-diffusion with chemical reactions) regimes, with testable scope conditions based on dimensionless numbers from physics (Péclet and Reynolds numbers). We demonstrate the framework's diagnostic capability using air pollution from coal-fired power plants. Analyzing 791 ground-based PM$_{2.5}$ monitors and 189,564 satellite-based NO$_2$ grid cells in the Western United States over 2019-2021, we find striking regional heterogeneity: within 100 km of coal plants, both pollutants show positive spatial decay (PM$_{2.5}$: $\kappa_s = 0.00200$, $d^* = 1,153$ km; NO$_2$: $\kappa_s = 0.00112$, $d^* = 2,062$ km), validating the framework. Beyond 100 km, negative decay parameters correctly signal that urban sources dominate and diffusion assumptions fail. Ground-level PM$_{2.5}$ decays approximately twice as fast as satellite column NO$_2$, consistent with atmospheric transport physics. The framework successfully diagnoses its own validity in four of eight analyzed regions, providing researchers with physics-based tools to assess whether their spatial difference-in-differences setting satisfies diffusion assumptions before applying the estimator. Our results demonstrate that rigorous boundary detection requires both theoretical derivation from first principles and empirical validation of underlying physical assumptions.

We forecast the full conditional distribution of macroeconomic outcomes by systematically integrating three key principles: using high-dimensional data with appropriate regularization, adopting rigorous out-of-sample validation procedures, and incorporating nonlinearities. By exploiting the rich information embedded in a large set of macroeconomic and financial predictors, we produce accurate predictions of the entire profile of macroeconomic risk in real time. Our findings show that regularization via shrinkage is essential to control model complexity, while introducing nonlinearities yields limited improvements in predictive accuracy. Out-of-sample validation plays a critical role in selecting model architecture and preventing overfitting.

Learning-to-Defer (L2D) enables hybrid decision-making by routing inputs either to a predictor or to external experts. While promising, L2D is highly vulnerable to adversarial perturbations, which can not only flip predictions but also manipulate deferral decisions. Prior robustness analyses focus solely on two-stage settings, leaving open the end-to-end (one-stage) case where predictor and allocation are trained jointly. We introduce the first framework for adversarial robustness in one-stage L2D, covering both classification and regression. Our approach formalizes attacks, proposes cost-sensitive adversarial surrogate losses, and establishes theoretical guarantees including $\mathcal{H}$, $(\mathcal{R }, \mathcal{F})$, and Bayes consistency. Experiments on benchmark datasets confirm that our methods improve robustness against untargeted and targeted attacks while preserving clean performance.

In real-world applications, the limited availability of labeled outcomes presents significant challenges for statistical inference due to high collection costs, technical barriers, and other constraints. In this work, we propose a method to construct efficient conformal prediction sets for new target outcomes by leveraging a source distribution that is distinct from the target but related through a distributional shift assumption and provides abundant labeled data. When the target data are fully unlabeled, our predictions rely solely on the source distribution, whereas partial target labels, when available, are integrated to improve efficiency. To address the challenges of data non-exchangeability and distribution non-identifiability, we identify the likelihood ratio by matching the covariate distributions of the source and target domains within a finite B-spline space. To accommodate complex error structures such as asymmetry and multimodality, our method constructs highest predictive density sets using a novel weight-adjusted conditional density estimator. This estimator models the source conditional density along a quantile process and transforms it, through appropriate weighting adjustments, to approximate the target conditional density. We establish the theoretical properties of the proposed method and evaluate its finite-sample performance through simulation studies and a real-data application to the MIMIC-III clinical database.

Recently, multi-fidelity Bayesian optimization (MFBO) has been successfully applied to many engineering design optimization problems, where the cost of high-fidelity simulations and experiments can be prohibitive. However, challenges remain for constrained optimization problems using the MFBO framework, particularly in efficiently identifying the feasible region defined by the constraints. In this paper, we propose a constrained multi-fidelity Bayesian optimization (CMFBO) method with novel acquisition functions. Specifically, we design efficient acquisition functions that 1) have analytically closed-form expressions; 2) are straightforward to implement; and 3) do not require feasible initial samples, an important feature often missing in commonly used acquisition functions such as expected constrained improvement (ECI). We demonstrate the effectiveness of our algorithms on synthetic test problems using different combinations of acquisition functions. Then, we apply the proposed method to a data-driven inertial confinement fusion (ICF) design problem, and a high-current joint design problem using finite element simulations with computational contact mechanics.

This paper develops a finite-sample statistical theory for in-context learning (ICL), analyzed within a meta-learning framework that accommodates mixtures of diverse task types. We introduce a principled risk decomposition that separates the total ICL risk into two orthogonal components: Bayes Gap and Posterior Variance. The Bayes Gap quantifies how well the trained model approximates the Bayes-optimal in-context predictor. For a uniform-attention Transformer, we derive a non-asymptotic upper bound on this gap, which explicitly clarifies the dependence on the number of pretraining prompts and their context length. The Posterior Variance is a model-independent risk representing the intrinsic task uncertainty. Our key finding is that this term is determined solely by the difficulty of the true underlying task, while the uncertainty arising from the task mixture vanishes exponentially fast with only a few in-context examples. Together, these results provide a unified view of ICL: the Transformer selects the optimal meta-algorithm during pretraining and rapidly converges to the optimal algorithm for the true task at test time.

Self-supervised learning (SSL) has achieved remarkable success by learning meaningful representations without labeled data. However, a unified theoretical framework for understanding and comparing the efficiency of different SSL paradigms remains elusive. In this paper, we introduce a novel information-geometric framework to quantify representation efficiency. We define representation efficiency $\eta$ as the ratio between the effective intrinsic dimension of the learned representation space and its ambient dimension, where the effective dimension is derived from the spectral properties of the Fisher Information Matrix (FIM) on the statistical manifold induced by the encoder. Within this framework, we present a theoretical analysis of the Barlow Twins method. Under specific but natural assumptions, we prove that Barlow Twins achieves optimal representation efficiency ($\eta = 1$) by driving the cross-correlation matrix of representations towards the identity matrix, which in turn induces an isotropic FIM. This work provides a rigorous theoretical foundation for understanding the effectiveness of Barlow Twins and offers a new geometric perspective for analyzing SSL algorithms.

Derivative-free Bayesian inversion arises in science and engineering applications, particularly when forward model is costly or infeasible to differentiate through. Existing derivative-free methods collapse the posterior to a point estimate or return severely over-confident uncertainty on high-dimensional, nonlinear problems. We introduce Blade, which produces accurate and well-calibrated posteriors using an ensemble of interacting particles. Blade leverages diffusion models as data-driven priors, and only queries the forward model through forward evaluations (i.e., derivative-free). Theoretically, we show the convergence and stability of Blade under forward model approximation and prior score estimation error. Empirically, on nonlinear fluid dynamics, Blade produces well-calibrated posterior samples that existing derivative-free methods cannot, as measured by CRPS, the spread-skill ratio, and the rank histogram. Its accuracy and calibration improve consistently with more iterations and particles, backed by our convergence and stability analysis and empirical experiments.

Reasoning ability has become a defining capability of Large Language Models (LLMs), with Reinforcement Learning with Verifiable Rewards (RLVR) emerging as a key paradigm to enhance it. However, RLVR training often suffers from policy entropy collapse, where the policy becomes overly deterministic, hindering exploration and limiting reasoning performance. While entropy regularization is a common remedy, its effectiveness is highly sensitive to the fixed coefficient, making it unstable across tasks and models. In this work, we revisit entropy regularization in RLVR and argue that its potential has been largely underestimated. Our analysis shows that (i) tasks of varying difficulty demand distinct exploration intensities, and (ii) balanced exploration may require the policy entropy to be maintained within a moderate range below its initial level. Therefore, we propose Adaptive Entropy Regularization (AER)–a framework that dynamically balances exploration and exploitation via three components: difficulty-aware coefficient allocation, initial-anchored target entropy, and dynamic global coefficient adjustment. Experiments on multiple mathematical reasoning benchmarks show that AER consistently outperforms baselines, improving both reasoning accuracy and exploration capability.

Early detection of Alzheimer's disease (AD) is crucial because its neurodegenerative effects are irreversible, and neuropathologic and social-behavioral risk factors accumulate years before diagnosis. Identifying higher-risk individuals earlier enables prevention, timely care, and equitable resource allocation. We predict cognitive performance from social determinants of health (SDOH) using the NIH NIA-supported PREPARE Challenge Phase 2 dataset derived from the nationally representative Mex-Cog cohort of the 2003 and 2012 Mexican Health and Aging Study (MHAS). Data: The target is a validated composite cognitive score across seven domains-orientation, memory, attention, language, constructional praxis, and executive function-derived from the 2016 and 2021 MHAS waves. Predictors span demographic, socioeconomic, health, lifestyle, psychosocial, and healthcare access factors. Methodology: Missingness was addressed with a singular value decomposition (SVD)-based imputation pipeline treating continuous and categorical variables separately. This approach leverages latent feature correlations to recover missing values while balancing reliability and scalability. After evaluating multiple methods, XGBoost was chosen for its superior predictive performance. Results and Discussion: The framework outperformed existing methods and the data challenge leaderboard, demonstrating high accuracy, robustness, and interpretability. SHAP-based post hoc analysis identified top contributing SDOH factors and age-specific feature patterns. Notably, flooring material emerged as a strong predictor, reflecting socioeconomic and environmental disparities. Other influential factors, age, SES, lifestyle, social interaction, sleep, stress, and BMI, underscore the multifactorial nature of cognitive aging and the value of interpretable, data-driven SDOH modeling.

We present a theoretical framework that extends classical information theory to finite and structured systems by redefining redundancy as a fundamental property of information organization rather than inefficiency. In this framework, redundancy is expressed as a general family of informational divergences that unifies multiple classical measures, such as mutual information, chi-squared dependence, and spectral redundancy, under a single geometric principle. This reveals that these traditional quantities are not isolated heuristics but projections of a shared redundancy geometry. The theory further predicts that redundancy is bounded both above and below, giving rise to an optimal equilibrium that balances over-compression (loss of structure) and over-coupling (collapse). While classical communication theory favors minimal redundancy for transmission efficiency, finite and structured systems, such as those underlying real-world learning, achieve maximal stability and generalization near this equilibrium. Experiments with masked autoencoders are used to illustrate and verify this principle: the model exhibits a stable redundancy level where generalization peaks. Together, these results establish redundancy as a measurable and tunable quantity that bridges the asymptotic world of communication and the finite world of learning.

Understanding and modeling mortality patterns, especially differences in mortality rates between populations, is vital for demographic analysis and public health planning. We compare three statistical models within the age-period framework to examine differences in death counts. The models are based on the double Poisson, bivariate Poisson, and Skellam distributions, each of which provides unique strengths in capturing underlying mortality trends. Focusing on mortality data from 1960 to 2015, we analyze the two leading causes of death in Italy, which exhibit significant temporal and age-related variations. Our results reveal that the Skellam distribution offers superior accuracy and simplicity in capturing mortality differentials. These findings highlight the potential of the Skellam distribution for analyzing mortality gaps effectively.

Auditing information disclosure across every datapoint during the training of LLMs is challenging. We propose a principled, attack-agnostic approach that uses mutual information to measure what the final model reveals about a datapoint's training membership. We show that this final-model disclosure is upper bounded by the sum of per-iteration gradient disclosures and that, under a reasonable set of assumptions, these gradient disclosures increase with Gradient Uniqueness (GNQ), which measures how distinguishable a datapoint's gradient is relative to other gradients in the batch. While naively computing GNQ requires forming and inverting a $P\times P$ matrix for every datapoint (for a model with $P$ parameters), we introduce Batch-Space Ghost (BS-Ghost). This efficient algorithm performs all computations in a much smaller batch space and uses ghost kernels to compute GNQ "in-run" for every datapoint in the training corpus, with minimal computational and memory overhead. Our experiments show the following: (i) GNQ predicts MIA vulnerability without the need for shadow models. (ii) Beyond membership disclosure, GNQ predicts the success of reconstruction attacks. (iii) GNQ-guided removal and retraining identify datapoints that causally contribute to disclosure. (iv) GNQ outperforms counterfactual memorization in text extraction and common-knowledge discrimination without the need for additional model training. (v) For data attribution, GNQ-guided filtering reduces emergent misalignment in Qwen2.5-7B more than baselines. Further, GNQ attributes 1000 datapoints in 17 seconds—roughly $70\times$ faster than the baselines. (vi) GNQ explains how training choices affect training-set disclosure and captures how per-datapoint disclosure emerges during training.

We propose a method for transfer learning in nonparametric regression using a random forest (RF) with distance covariance-based feature weights, assuming the unknown source and target regression functions are sparsely different. Our method obtains residuals from a source domain-trained Centered RF (CRF) in the target domain, then fits another CRF to these residuals with feature splitting probabilities proportional to feature-residual sample distance covariance. We derive an upper bound on the mean square error rate of the procedure as a function of sample sizes and difference dimension, theoretically demonstrating transfer learning benefits in random forests. A major difficulty for transfer learning in random forests is the lack of explicit regularization in the method. Our results explain why shallower trees with preferential selection of features lead to both lower bias and lower variance for fitting a low-dimensional function. We show that in the residual random forest, this implicit regularization is enabled by sample distance covariance. In simulations, we show that the results obtained for the CRFs also hold numerically for the standard RF (SRF) method with data-driven feature split selection. Beyond transfer learning, our results also show the benefit of distance-covariance-based weights on the performance of RF when some features dominate. Our method shows significant gains in predicting the mortality of ICU patients in smaller-bed target hospitals using a large multi-hospital dataset of electronic health records for 200,000 ICU patients.

Measuring dataset similarity is fundamental in machine learning, particularly for transfer learning and domain adaptation. In the context of supervised learning, most existing approaches quantify similarity of two data sets based on their input feature distributions, neglecting label information and feature-response alignment. To address this, we propose the Cross-Learning Score (CLS), which measures dataset similarity through bidirectional generalization performance of decision rules. We establish its theoretical foundation by linking CLS to cosine similarity between decision boundaries under canonical linear models, providing a geometric interpretation. A robust ensemble-based estimator is developed that is easy to implement and bypasses high-dimensional density estimation entirely. For transfer learning applications, we introduce a "transferable zones" framework that categorizes source datasets into positive, ambiguous, and negative transfer regions. To accommodate deep learning, we extend CLS to encoder-head architectures, aligning with modern representation-based pipelines. Extensive experiments on synthetic and real-world datasets validate the effectiveness of CLS for similarity measurement and transfer assessment.

Conflicting clinical trial results on omega-3 highly unsaturated fatty acids (n-3 HUFA) have prompted uncertainty about their cardioprotective effects. While the VITAL trial found no overall cardiovascular benefit from n-3 HUFA supplementation, its substantial African American (AfAm) enrollment provided a unique opportunity to explore racial differences in response to n-3 HUFA supplementation. The current observational study aimed to simulate randomized clinical trial (RCT) conditions by matching 3,766 AfAm and 15,553 non-Hispanic White (NHW) individuals from the VITAL trial utilizing propensity score matching to address the limitations related to differences in confounding variables between the two groups. Within matched groups (3,766 AfAm and 3,766 NHW), n-3 HUFA supplementation's impact on myocardial infarction (MI), stroke, and cardiovascular disease (CVD) mortality was assessed. A weighted decision tree analysis revealed belonging to the n-3 supplementation group as the most significant predictor of MI among AfAm but not NHW. Further logistic regression using the LASSO method and bootstrap estimation of standard errors indicated n-3 supplementation significantly lowered MI risk in AfAm (OR 0.17, 95% CI [0.048, 0.60]), with no such effect in NHW. This study underscores the critical need for future RCT to explore racial disparities in MI risk associated with n-3 HUFA supplementation and highlights potential causal differences between supplementation health outcomes in AfAm versus NHW populations.

Diffusion models have demonstrated remarkable performance in generating high-dimensional samples across domains such as vision, language, and the sciences. Although continuous-state diffusion models have been extensively studied both empirically and theoretically, discrete-state diffusion models, essential for applications involving text, sequences, and combinatorial structures, remain significantly less understood from a theoretical standpoint. In particular, all existing analyses of discrete-state models assume score estimation error bounds without studying sample complexity results. In this work, we present a principled theoretical framework for discrete-state diffusion, providing the first sample complexity bound of $\widetilde{\mathcal{O}}(\epsilon^{-2})$. Our structured decomposition of the score estimation error into statistical, approximation, optimization, and clipping components offers critical insights into how discrete-state models can be trained efficiently. This analysis addresses a fundamental gap in the literature and establishes the theoretical tractability and practical relevance of discrete-state diffusion models.

We propose small area estimators of general indicators in off-census years, which avoid the use of deprecated census microdata, but are nearly optimal in census years. The procedure is based on replacing the obsolete census file with a larger unit-level survey that adequately covers the areas of interest and contains the values of useful auxiliary variables. However, the minimal data requirement of the proposed method is a single survey with microdata on the target variable and suitable auxiliary variables for the period of interest. We also develop an estimator of the mean squared error (MSE) that accounts for the uncertainty introduced by the large survey used to replace the census of auxiliary information. Our empirical results indicate that the proposed predictors perform clearly better than the alternative predictors when census data are outdated, and are very close to optimal ones when census data are correct. They also illustrate that the proposed total MSE estimator corrects for the bias of purely model-based MSE estimators that do not account for the large survey uncertainty.

We present two innovative functional partial quantile regression algorithms designed to accurately and efficiently estimate the regression coefficient function within the function-on-function linear quantile regression model. Our algorithms utilize functional partial quantile regression decomposition to effectively project the infinite-dimensional response and predictor variables onto a finite-dimensional space. Within this framework, the partial quantile regression components are approximated using a basis expansion approach. Consequently, we approximate the infinite-dimensional function-on-function linear quantile regression model using a multivariate quantile regression model constructed from these partial quantile regression components. To evaluate the efficacy of our proposed techniques, we conduct a series of Monte Carlo experiments and analyze an empirical dataset, demonstrating superior performance compared to existing methods in finite-sample scenarios. Our techniques have been implemented in the ffpqr package in R.

This study proposes an energy-based framework for exploring local variations around observed territorial configurations in Official Statistics. The observed distribution serves as the initial reference, without assuming equilibrium. A register-derived graph represents structural similarities between municipalities, while socio-economic composite indices are synthesised through Principal Component Analysis into a univariate external field. A Continuous Ising model and a Langevin-based procedure provide complementary stochastic exploration strategies. Simulated Annealing progressively limits exploration, favouring lower-energy configurations under each specification. Recorded energy trajectories and municipal deviations characterise the resulting ensembles. Conformal Prediction provides calibrated intervals whose widths and empirical coverage describe their relationship with fixed observed municipal values under repeated subsampling. The application to resident foreign population shares in North-East Italy yields simulated municipal averages close to the observed values, but substantially different calibrated interval widths between the two procedures. Comparisons across territorial groups reveal heterogeneous associations with socio-economic profiles, without a uniform ordering of interval widths between central and peripheral municipalities. The framework connects territorial interactions, socio-economic information, and stochastic variation within an exploratory analysis. Its contribution is to examine how nearby simulated alternatives relate to the observed configuration and to formulate hypotheses about territorial heterogeneity. Further applications and sensitivity analyses would assess the robustness of these interpretations.

Systematic reviews are crucial for synthesizing scientific evidence but remain labor-intensive, especially when extracting detailed methodological information. Large language models (LLMs) offer potential for automating methodological assessments, promising to transform evidence synthesis. Here, using causal mediation analysis as a representative methodological domain, we benchmarked state-of-the-art LLMs against expert human reviewers across 180 full-text scientific articles. Model performance closely correlated with human judgments (accuracy correlation 0.71; F1 correlation 0.97), achieving near-human accuracy on straightforward, explicitly stated methodological criteria. However, accuracy sharply declined on complex, inference-intensive assessments, lagging expert reviewers by up to 15%. Errors commonly resulted from superficial linguistic cues – for instance, models frequently misinterpreted keywords like "longitudinal" or "sensitivity" as automatic evidence of rigorous methodological approache, leading to systematic misclassifications. Longer documents yielded lower model accuracy, whereas publication year showed no significant effect. Our findings highlight an important pattern for practitioners using LLMs for methods review and synthesis from full texts: current LLMs excel at identifying explicit methodological features but require human oversight for nuanced interpretations. Integrating automated information extraction with targeted expert review thus provides a promising approach to enhance efficiency and methodological rigor in evidence synthesis across diverse scientific fields.

Sequential data - ranging from financial time series to natural language - has driven the growing adoption of autoregressive models. However, these algorithms rely on the presence of underlying patterns in the data, and their identification often depends heavily on human expertise. Misinterpreting these patterns can lead to model misspecification, resulting in increased generalization error and degraded performance. The recently proposed evolving pattern (EvoRate) metric addresses this by using the mutual information between the next data point and its past to guide regression order estimation and feature selection. Building on this idea, we introduce a general framework based on predictive information, defined as the mutual information between the past and the future, $I(X_{past}; X_{future})$. This quantity naturally defines an information-theoretic learning curve, which quantifies the amount of predictive information available as the observation window grows. Using this formalism, we show that the presence or absence of temporal patterns fundamentally constrains the learnability of sequential models: even an optimal predictor cannot outperform the intrinsic information limit imposed by the data. We validate our framework through experiments on synthetic data, demonstrating its ability to assess model adequacy, quantify the inherent complexity of a dataset, and reveal interpretable structure in sequential data.

We provide a unified algorithmic framework for ensemble sampling in nonlinear contextual bandits and develop corresponding regret bounds for two most common nonlinear contextual bandit settings: Generalized Linear Model Ensemble Sampling (GLM-ES) for generalized linear contextual bandits and Neural Ensemble Sampling (Neural-ES) for neural contextual bandits. Both methods maintain multiple estimators for the reward model parameters via maximum likelihood estimation on randomly perturbed data. We prove high-probability frequentist regret bounds of $\widetilde{\mathcal{O}}(d^{3/2} \sqrt{T} + d^{4})$ for GLM-ES and $\widetilde{\mathcal{O}}(\widetilde{d}^{3/2} \sqrt{T})$ for Neural-ES, where $d$ is the dimension of feature vectors, $\widetilde{d}$ is the effective dimension of a neural tangent kernel (NTK) matrix and $T$ is the number of rounds. The regret bound of GLM-ES matches the state-of-the-art result of randomized exploration algorithms in generalized linear bandit setting. In the theoretical analysis, we introduce techniques that address challenges specific to nonlinear models. Practically, we remove fixed-time horizon assumption by developing anytime versions of our algorithms, suitable when $T$ is unknown. Finally, we empirically evaluate GLM-ES, Neural-ES and their anytime variants, demonstrating strong performance. Overall, our results establish ensemble sampling as a provable and practical randomized exploration approach for nonlinear contextual bandits.

Missing data in online reinforcement learning (RL) poses challenges compared to missing data in standard tabular data or in offline policy learning. The need to impute and act at each time step means that imputation cannot be put off until enough data exist to produce stable imputation models. It also means future data collection and learning depend on previous imputations. This paper proposes fully online imputation ensembles. We find that maintaining multiple imputation pathways may help balance the need to capture uncertainty under missingness and the need for efficiency in online settings. We consider multiple approaches for incorporating these pathways into learning and action selection. Using a Grid World experiment with various types of missingness, we provide preliminary evidence that multiple imputation pathways may be a useful framework for constructing simple and efficient online missing data RL methods.

Quantized neural network training optimizes a discrete, non-differentiable objective. The straight-through estimator (STE) enables backpropagation through surrogate gradients and is widely used. While previous studies have primarily focused on the properties of surrogate gradients and their convergence, the influence of quantization hyperparameters, such as bit width and quantization range, on learning dynamics remains largely unexplored. We theoretically show that in the high-dimensional limit, STE dynamics converge to a deterministic ordinary differential equation. This reveals that STE training exhibits a plateau followed by a sharp drop in generalization error, with plateau length depending on the quantization range. A fixed-point analysis quantifies the asymptotic deviation from the unquantized linear model. We also extend analytical techniques for stochastic gradient descent to nonlinear transformations of weights and inputs.

We study the task of noiseless linear regression under Gaussian covariates in the presence of additive oblivious contamination. Specifically, we are given i.i.d.\ samples from a distribution $(x, y)$ on $\mathbb{R}^d \times \mathbb{R}$ with $x \sim \mathcal{N}(0,\mathbf{I}_d)$ and $y = x^\top \beta + z$, where $z$ is drawn independently of $x$ from an unknown distribution $E$. Moreover, $z$ satisfies $\mathbb{P}_E[z = 0] = \alpha>0$. The goal is to accurately recover the regressor $\beta$ to small $\ell_2$-error. Ignoring computational considerations, this problem is known to be solvable using $O(d/\alpha)$ samples. On the other hand, the best known polynomial-time algorithms require $\Omega(d/\alpha^2)$ samples. Here we provide formal evidence that the quadratic dependence in $1/\alpha$ is inherent for efficient algorithms. Specifically, we show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $\tilde{\Omega}(d^{1/2}/\alpha^2)$.

A probabilistic clustering algorithm is proposed for the analysis of forensic DNA mixtures in which individual cells are isolated and short tandem repeats are amplified using the polymerase chain reaction to generate single cell electropherograms. The task of the algorithm is to use the peak height information in the electropherograms to group the cells according to their contributors. Using a recently developed experimental set of individual cell electropherograms, a large set of simulations shows that the proposed clustering algorithm has excellent performance in correctly grouping single cells, and for assigning likelihood ratios for persons of interest (of known genotype).

This paper investigates extreme value theory for processes obtained by applying transformations to stationary Gaussian processes, also called subordinated Gaussian processes. The main contributions are as follows. First, we refine the method of \cite{sly2008nonstandard} to allow the covariance of the underlying Gaussian process to decay more slowly than any polynomial rate, nearly matching Berman's condition. Second, we extend the theory to a multivariate setting, where both the subordinated process and the underlying Gaussian process may be vector-valued, and the transformation is finite-dimensional. In particular, we establish the weak convergence of a point process constructed from the subordinated Gaussian process, from which a multivariate extreme value limit theorem follows. A key observation that facilitates our analysis, and may be of independent interest, is the following: any bivariate random vector derived from transformations of two jointly Gaussian vectors with a non-unity canonical correlation always remains extremally independent. This observation also motivates us to introduce and discuss a notion we call $m$-extremal-dependence, which extends the classical concept of $m$-dependence. Moreover, we relax the restriction to finite-dimensional transforms, extending the results to infinite-dimensional settings via an approximation argument. As an illustration, we establish a limit theorem for a multivariate moving maxima process driven by regularly varying innovations that arise from subordinated Gaussian processes with potentially long memory.

Federated learning has attracted significant attention as a privacy-preserving framework for training personalised models on multi-source heterogeneous data. However, most existing approaches are unable to handle scenarios where subgroup structures coexist alongside within-group heterogeneity. In this paper, we propose a federated learning algorithm that addresses general heterogeneity through adaptive clustering. Specifically, our method partitions tasks into subgroups to address substantial between-group differences while enabling efficient information sharing among similar tasks within each group. Furthermore, we integrate the Huber loss and Iterative Hard Thresholding (IHT) to tackle the challenges of high dimensionality and heavy-tailed distributions. Theoretically, we establish convergence guarantees, derive non-asymptotic error bounds, and provide recovery guarantees for the latent cluster structure. Extensive simulation studies and real-data applications further demonstrate the effectiveness and adaptability of our approach.

We derive a novel PAC-Bayesian generalization bound for reinforcement learning that explicitly accounts for Markov dependencies in the data, through the chain's mixing time. This contributes to overcoming challenges in obtaining generalization guarantees for reinforcement learning, where the sequential nature of data breaks the independence assumptions underlying classical bounds. The new bound provides non-vacuous certificates for modern off-policy algorithms such as Soft Actor-Critic. We demonstrate the practical utility of the bound through PB-SAC, a novel algorithm that optimizes the bound during training to guide exploration. Experiments across several continuous control tasks show that the proposed approach provides meaningful confidence certificates while maintaining competitive performance.