Papers

The integrated density of states (IDS) is a fundamental spectral quantity for quantum Hamiltonians modeling condensed matter systems, describing how densely energy levels are distributed. It can be interpreted as a volume-averaged spectral distribution. Hence, there are two equivalent definitions of the IDS related by the Pastur-Shubin formula: an operator-theoretic trace formula and a limit of normalized eigenvalue counting functions on finite volumes. We study a discrete random Schrödinger operator with bounded random potentials of finite-range correlations and prove a quantitative concentration inequality ensuring, with explicit high probability, that the empirical IDS (normalized eigenvalue counting function) uniformly approximates the abstract IDS trace formula within a prescribed error, thereby implying confidence regions for the IDS.

We address the problem of uncertainty quantification for the deconvolution model \(Z = X + Y\), where \(X\) and \(Y\) are nonnegative random variables and the goal is to estimate the signal's distribution of \(X \sim F_0\) supported on~\([0,\infty)\), from observations where the noise distribution is known. Existing frequentist methods often produce confidence intervals for $F_0(x)$ that depend on unknown nuisance parameters, such as the density of \(X\) and its derivative, which are difficult to estimate in practice. This paper introduces a novel and computationally efficient nonparametric Bayesian approach, based on projecting the posterior, to overcome this limitation. Our method leverages the solution \(p\) to a specific Volterra integral equation as in \cite{74}, which relates the cumulative distribution function (CDF) of the signal, \(F_0\), to the distribution of the observables. We place a Dirichlet Process prior directly on the distribution of the observed data $Z$, yielding a simple, conjugate posterior. To ensure the resulting estimates for \(F_0\) are valid CDFs, we isotonize posterior draws taking the Greatest Convex Majorant of the primitive of the posterior draws and defining what we term the Isotonic Inverse Posterior. We show that this framework yields posterior credible sets for \(F_0\) that are not only computationally fast to generate but also possess asymptotically correct frequentist coverage after a straightforward recalibration technique for the so-called Bayes Chernoff distribution introduced in \cite{54}. Our approach thus does not require the estimation of nuisance parameters to deliver uncertainty quantification for the parameter of interest $F_0(x)$. The practical effectiveness and robustness of the method are demonstrated through a simulation study with various noise distributions for $Y$.

We introduce Box Thirding (B3), a flexible and efficient algorithm for Best Arm Identification (BAI) under fixed-budget constraints. It is designed for both anytime BAI and scenarios with large N, where the number of arms is too large for exhaustive evaluation within a limited budget T. The algorithm employs an iterative ternary comparison: in each iteration, three arms are compared–the best-performing arm is explored further, the median is deferred for future comparisons, and the weakest is discarded. Even without prior knowledge of T, B3 achieves an epsilon-best arm misidentification probability comparable to Successive Halving (SH), which requires T as a predefined parameter, applied to a randomly selected subset of c0 arms that fit within the budget. Empirical results show that B3 outperforms existing methods under limited-budget constraints in terms of simple regret, as demonstrated on the New Yorker Cartoon Caption Contest dataset.

This paper develops Kolmogorov-type maximal inequalities for sums of Negative Binomial random variables under both independence and dependence structures. For independent heterogeneous Negative Binomial variables we derive sharp Markov-type deviation inequalities and Kolmogorov-type bounds expressed in terms of Tweedie dispersion parameters, providing explicit control limits for NB2 generalized linear model monitoring. For dependent count data arising through a shared Gamma mixing variable, we establish a sub-exponential Bernstein-type refinement that exploits the Poisson-Gamma hierarchical structure to yield exponentially decaying tail probabilities – this refinement is new in the literature. Through moment-matched Monte Carlo experiments ($n=20$, 2{,}000 replications), we document a 55% reduction in mean maximum deviation under appropriate dependence structures, a stabilization effect we explain analytically. A concrete epidemiological application with NB2 parameters calibrated from COVID-19 surveillance data demonstrates practical utility. These results materially advance the applicability of classical maximal inequalities to overdispersed and dependent count data prevalent in public health, insurance, and ecological modeling.

This paper introduces two new robust methods for estimation of parameters in a given parametric family. The first method is that of `minimum weighted L2', effectively minimising an estimate of the integrated (and possibly weighted) squared error. The second is `robust Kullback-Leibler', consisting of minimising a robust version of the empirical Kullback-Leibler distance, and can be viewed as a general robust modification of the maximum likelihood procedure. This second method is also related to recent local likelihood ideas for semiparametric density estimation. The methods are described, influence functions are found, as are formulae for asymptotic variances. In particular large-sample efficiencies are computed under the home turf conditions of the underlying parametric model. The methods and formulae are illustrated for the normal model.

The choice of sample size in the context of co-primary endpoints for a randomised trial is discussed. Current guidance can leave endpoints with unequal marginal power. A method is provided to achieve equal marginal power by using the flexibility provided in multiple testing procedures. A comparison is made to several choices of rule to determine the sample size, in terms of the study design and its operating characteristics.

Often, real-world entities are explained by multiple indicators, and ranking those is challenging, e.g., administrative units, sport teams, and machine learning models. Existing methodologies rely on single indicators or composite indices, leaving a gap for frameworks capable of handling multi-indicator scenarios. We present a ranking methodology motivated by the Bayesian Bradley-Terry (BT) model. We convert indicators to paired comparisons to utilize the BT framework. The prior covariance of the BT latent merit parameters is modeled as a function of a covariate to accommodate prior information. For parameter estimation, we use a Markov chain Monte Carlo (MCMC) algorithm that employs a hybrid Metropolis–Hastings scheme with a preconditioned Crank–Nicolson proposal and Gibbs sampling. We show that the proposed estimator converges. Furthermore, we propose a problem-specific stopping rule based on ranking stability. To demonstrate the methodology's practical utility, we apply it to rank the states and union territories (UTs) of India using multi-indicator data from the National Family Health Survey-5. By estimating and comparing rankings across various economic regimes, including low-income, mid-income, and outlier-excluded subsets, our application uncovers meaningful deviations between regional economic standing and overall developmental performance, illustrating the framework's broad value for resolving complex multidimensional ranking problems.

This paper presents methods for meta-analysis of $2 \times 2$ tables, both with and without allowing heterogeneity in the treatment effects. Meta-analysis is common in medical research, but most existing methods are unsuited for $2 \times 2$ tables with rare events. Usually the tables are modelled as pairs of binomial variables, but we will model them as Poisson pairs. The methods presented here are based on confidence distributions, and offer optimal inference for the treatment effect parameter. We also propose an optimal method for inference on the ratio between treatment effects, and illustrate our methods on a real dataset.

In orbital mechanics, Gauss's method for orbit determination (OD) is a popular, minimal assumption solution for obtaining the initial state estimate of a passing resident space object (RSO). Since much of the cislunar domain relies on three-body dynamics, a key assumption of Gauss's method is rendered incompatible, creating a need for a new, minimal assumption method for initial orbit determination (IOD). In this work, we present a framework for short and long term probabilistic target tracking in cislunar space which produces an initial state estimate with as few assumptions as possible. Specifically, we propose an IOD method involving the kinematic fitting of several series of noisy, consecutive ground-based observations. Once a probabilistic initial state estimate in the form of a particle cloud is formed, we apply the powerful Particle Gaussian Mixture (PGM) Filter to reduce the uncertainty of our state estimate over time. This combined IOD/OD framework is demonstrated for several classes of trajectories in cislunar space and compared to better-known filtering frameworks.

Conditional Value-at-Risk (CVaR) is a widely used risk-sensitive objective for learning under rare but high-impact losses, yet its statistical behavior under heavy-tailed data remains poorly understood. Unlike expectation-based risk, CVaR depends on an endogenous, data-dependent quantile, which couples tail averaging with threshold estimation and fundamentally alters both generalization and robustness properties. In this work, we develop a learning-theoretic analysis of CVaR-based empirical risk minimization under heavy-tailed and contaminated data. We establish sharp, high-probability generalization and excess risk bounds under minimal moment assumptions, covering fixed hypotheses, finite and infinite classes, and extending to $\beta$-mixing dependent data; we further show that these rates are minimax optimal. To capture the intrinsic quantile sensitivity of CVaR, we derive a uniform Bahadur-Kiefer type expansion that isolates a threshold-driven error term absent in mean-risk ERM and essential in heavy-tailed regimes. We complement these results with robustness guarantees by proposing a truncated median-of-means CVaR estimator that achieves optimal rates under adversarial contamination. Finally, we show that CVaR decisions themselves can be intrinsically unstable under heavy tails, establishing a fundamental limitation on decision robustness even when the population optimum is well separated. Together, our results provide a principled characterization of when CVaR learning generalizes and is robust, and when instability is unavoidable due to tail scarcity.

Conformal prediction gives exact finite-sample coverage guarantees under exchangeability, but deployed systems are judged by more than coverage alone. For a fixed calibrated rule reused over a finite operational window, stakeholders also care about deployment-facing quantities such as commitment frequency, deferral, and decisive error exposure. These are not determined by coverage: calibration choices with similar coverage can still induce materially different operational profiles. We study this characterization gap in a scoped setting: binary split conformal prediction under exchangeability with a fixed deployed rule. We introduce the Small-Sample Beta Correction (SSBC) which gives finite-sample coverage semantics for the deployed rule: it inverts the Beta/Beta–Binomial law governing calibration-conditional coverage to map a user request $(\alpha^\star,\delta)$ to the least conservative calibration grid point with calibration-conditional PAC semantics for the realized deployed rule. Calibrate-and-Audit then fixes the rule by calibration and uses an independent audit split to estimate the induced region–class label table, a reusable summary from which deployment-facing Key Performance Indicators (KPIs) follow by projection. Under this design, fixed operational rates admit exact finite-sample Binomial inference, while Beta–Binomial envelopes serve as practical predictive summaries for future windows. The induced partition also exposes regime boundaries, Pareto-relevant tradeoffs, and inverse-pricing questions for fixed downstream conventions. Simulations validate the SSBC semantics and compare audit-based summaries with leave-one-out planning proxies; molecular toxicity data provide an audit-based empirical example, and a solubility case study illustrates scenario planning once coverage semantics are fixed.

Tourism significantly affects the economies of many countries. Understanding the causal relationship between the length of overnight stay and traveller's expenditure is crucial for stakeholders to characterize spending profiles and to design marketing strategies. Causal mechanisms differ between personal and work-related travel because the decision-making processes have different drivers and constraints. We apply context-specific independence relations to model causal mechanisms in contexts specified by trip purpose and identify the causal effect of the length of stay on expenditure. Using the international visitor survey data on foreign travellers to Finland, we fit a hierarchical Bayesian model to estimate the posterior distribution of the counterfactual expenditure due to extending the length of stay by one night. We also perform a Bayesian sensitivity analysis of the estimated causal effect with respect to omitted variable bias.

Repetitive motion tasks are common in robotics, but performance can degrade over time due to environmental changes and robot wear and tear. Iterative learning control (ILC) improves performance by using information from previous iterations to compensate for expected errors in future iterations. This work incorporates the use of Quasi-Periodic Gaussian Processes (QPGPs) into a predictive ILC framework to model and forecast disturbances and drift across iterations. Using a recent structural equation formulation of QPGPs, the proposed approach enables efficient inference with complexity $\mathcal{O}(p^3)$ instead of $\mathcal{O}(i^2p^3)$, where $p$ denotes the number of points within an iteration and $i$ represents the total number of iterations, specially for larger $i$. This formulation also enables parameter estimation without loss of information, making continual GP learning computationally feasible within the control loop. By predicting next-iteration error profiles rather than relying only on past errors, the controller achieves faster convergence and maintains this under time-varying disturbances. We benchmark the method against both standard ILC and conventional Gaussian Process (GP)-based predictive ILC on three tasks, autonomous vehicle trajectory tracking, a three-link robotic manipulator, and a real-world Stretch robot experiment. Across all cases, the proposed approach converges faster and remains robust under injected and natural disturbances while reducing computational cost. This highlights its practicality across a range of repetitive dynamical systems.

Simulation-based inference (SBI) enables parameter estimation for complex stochastic models with intractable likelihoods when model simulation is feasible. Neural posterior estimation (NPE) is a popular SBI approach that often achieves accurate inference with far fewer simulations than classical approaches. But in practice, neural approaches can be unreliable for two reasons: incompatible data summaries arising from model misspecification yield unreliable posteriors due to extrapolation, and prior-predictive draws can produce extreme summaries that lead to difficulties in obtaining an accurate posterior for the observed data of interest. Existing preconditioning schemes target well-specified settings, and their behaviour under misspecification remains unexplored. We study preconditioning under misspecification and propose preconditioned robust neural posterior estimation, which computes data-dependent weights that focus training near the observed summaries and fits a robust neural posterior approximation. We also introduce a forest-proximity preconditioning approach that uses tree-based proximity scores to down-weight outlying simulations and concentrate computation around the observed dataset. Across two synthetic examples and one real example with incompatible summaries and extreme prior-predictive behaviour, we demonstrate that preconditioning combined with robust NPE increases stability and improves accuracy, calibration, and posterior-predictive fit over standard baseline methods.

In oncology phase I trials, model-assisted designs have been increasingly adopted because they enable adaptive yet operationally simple dose adjustment based on accumulating safety data, leading to a paradigm shift in dose-escalation methodology. In practice, a single mid-trial dose insertion may be considered to examine safer doses and/or to collect more informative efficacy data. In this study, we investigate methods to improve dose assignment and the selection of the maximum tolerated dose (MTD) or the optimal biological dose (OBD) when a new dose level is added during an ongoing trial under a model-assisted framework, by assigning informative prior information to the inserted dose. We propose a hybrid design that uses a non-informative model-assisted design at trial initiation and, upon dose insertion, applies an informative-prior extension only to the newly added dose. In addition, to address potential skeleton misspecification, we propose two adaptive extensions: (i) an online-weighting approach that updates the skeleton over time, and (ii) a Bayesian-mixture approach that robustly combines multiple candidate skeletons. We evaluate the proposed methods through simulation studies.

Machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before. This dissertation focuses on three different projects rooted in mathematical theory for machine learning applications. The first project deals with supervised learning and manifold learning. In theory, one of the main problems in supervised learning is that of function approximation: that is, given some data set $\mathcal{D}=\{(x_j,f(x_j))\}_{j=1}^M$, can one build a model $F\approx f$? We introduce a method which aims to remedy several of the theoretical shortcomings of the current paradigm for supervised learning. The second project deals with transfer learning, which is the study of how an approximation process or model learned on one domain can be leveraged to improve the approximation on another domain. We study such liftings of functions when the data is assumed to be known only on a part of the whole domain. We are interested in determining subsets of the target data space on which the lifting can be defined, and how the local smoothness of the function and its lifting are related. The third project is concerned with the classification task in machine learning, particularly in the active learning paradigm. Classification has often been treated as an approximation problem as well, but we propose an alternative approach leveraging techniques originally introduced for signal separation problems. We introduce theory to unify signal separation with classification and a new algorithm which yields competitive accuracy to other recent active learning algorithms while providing results much faster.

In clinical practice, there is significant interest in integrating novel biomarkers with existing clinical data to construct interpretable and robust decision rules. Motivated by the need to improve decision-making for early disease detection, we propose a framework for developing an optimal biomarker-based clinical decision rule that is both clinically meaningful and practically feasible. Specifically, our procedure constructs a linear decision rule designed to achieve optimal performance among class of linear rules by maximizing the true positive rate while adhering to a pre-specified positive predictive value constraint. Additionally, our method can adaptively incorporate individual risk information from external source to enhance performance when such information is beneficial. We establish the asymptotic properties of our proposed estimator and compare to the standard approach used in practice through extensive simulation studies. Results indicate that our approach offers strong finite-sample performance. We also apply the proposed methods to develop biomarker-based screening rules for pancreatic ductal adenocarcinoma (PDAC) among new-onset diabetes (NOD) patients.

Transfer learning aims to improve inference in a target domain by leveraging information from related source domains, but its effectiveness critically depends on how cross-domain heterogeneity is modeled and controlled. When the conditional mechanism linking covariates and responses varies across domains, indiscriminate information pooling can lead to negative transfer, degrading performance relative to target-only estimation. We study a multi-source, single-target transfer learning problem under conditional distributional drift and propose a semiparametric domain-varying coefficient model (DVCM), in which domain-relatedness is encoded through an observable domain identifier. This framework generalizes classical varying-coefficient models to structured transfer learning and interpolates between invariant and fully heterogeneous regimes. Building on this model, we develop an adaptive transfer learning estimator that selectively borrows strength from informative source domains while provably safeguarding against negative transfer. Our estimator is computationally efficient and easy to implement; we also show that it is minimax rate-optimal and derive its asymptotic distribution, enabling valid uncertainty quantification and hypothesis testing despite data-adaptive pooling and shrinkage. Our results precisely characterize the interplay among domain heterogeneity, the smoothness of the underlying mean function, and the number of source domains and are corroborated by comprehensive numerical experiments and two real-data applications.

We establish local laws for sample covariance matrices $K = N^{-1}\sum_{i=1}^N \g_i\g_i^*$ where the random vectors $\g_1, \ldots, \g_N \in \R^n$ are independent with common covariance $\Sigma$. Previous work has largely focused on the separable model $\g = \Sigma^{1/2}\w$ with $\w$ having independent entries, but this structure is rarely present in statistical applications involving dependent or nonlinearly transformed data. Under a concentration assumption for quadratic forms $\g^*A\g$, we prove an optimal averaged local law showing that the Stieltjes transform of $K$ converges to its deterministic limit uniformly down to the optimal scale $\eta \geq N^{-1+\eps}$. Under an additional structural assumption on the cumulant tensors of $\g$ – which interpolates between the highly structured case of independent entries and generic dependence – we establish the full anisotropic local law, providing entrywise control of the resolvent $(K-zI)^{-1}$ in arbitrary directions. We discuss several classes of non-separable examples satisfying our assumptions, including conditionally mean-zero distributions, the random features model $\g = \sigma(X\w)$ arising in machine learning, and Gaussian measures with nonlinear tilting. The proofs introduce a tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.

Multivariate mode estimation arises in many statistical problems such as inverse problems, multimodal sampling, and density-based clustering, but becomes challenging in moderate to high dimensions, especially when the underlying density is not directly evaluable. We introduce GERVE (Gibbs-measure Entropy-Regularized Variational Estimation), a sample-based method for estimating multivariate modes by approximating Gibbs distributions directly from samples, without estimating or evaluating the density. GERVE uses Gaussian-mixture variational annealing and natural-gradient optimization, producing a mixture concentrated in high-density regions whose component responsibilities also provide a clustering of the observations. We prove theoretical guarantees in two regimes: as the Gibbs temperature goes to zero, the optimal variational mixture concentrates around the global modes of the population density; at fixed positive temperature, we prove existence, consistency, and asymptotic normality of empirical maximizers and propose a bootstrap procedure for uncertainty quantification. Simulations and a real-data experiment show that GERVE accurately recovers modes and produces meaningful clusters.

Computational models of complex physical systems often rely on simplifying assumptions which inevitably introduce model error, with consequent predictive errors. Given data on model observables, the estimation of parameterized model-error representations, along with other model parameters, would be ideally done while separating the contributions of each of the two sets of parameters, in order to ensure meaningful stand-alone model predictions. This work builds an embedded model error framework using a weight-space representation of Gaussian processes (GPs) to flexibly capture model-error spatiotemporal correlations and enable inference with GP-embedding in non-linear models. To disambiguate model and model-error/bias parameters, we extend an existing orthogonal GP method to the embedded model-error setting and derive appropriate orthogonality constraints. To address the increased dimensionality introduced by the GP representation, we employ the likelihood-informed subspace method. The construction is demonstrated on linear and non-linear examples, where it effectively corrects model predictions to match data trends. Extrapolation beyond the training data recovers the prior predictive distribution, and the orthogonality constraints lead to meaningful stand-alone model predictions and nearly uncorrelated posteriors between model and model-error parameters.

The glmnet package in R is widely used for lasso estimation because of its computational efficiency. Despite its popularity, glmnet occasionally yields solutions that deviate substantially from the true ones because of the inappropriate default configuration of the algorithm. The accuracy of the obtained solutions can be improved by appropriately tuning the configuration. However, such improvements typically increase computational time, resulting in a tradeoff between accuracy and computational efficiency. Therefore, a systematic approach is required to determine the appropriate configuration. To address this need, we propose a unified data-driven framework specifically designed to optimize the configuration by balancing solution path accuracy and computational cost. Specifically, we generate a large-scale training dataset by measuring the accuracy and computation time of glmnet. Using this dataset, we construct neural networks to predict accuracy and computation time from data characteristics and configuration. For a new dataset, the proposed framework uses the trained networks to explore the configuration space and derive a Pareto front that represents the tradeoff between accuracy and computational cost. This front enables automatic selection of the configuration that maximizes accuracy under a user-specified time constraint. The proposed method is implemented in the R package glmnetconf, available at https://github.com/Shuhei-Muroya/glmnetconf.git.

We study a sequential prediction problem in which an adversary is allowed to inject arbitrarily many adversarial instances in a stream of i.i.d. instances, but at each round, the learner may also abstain from making a prediction without incurring any penalty if the instance was indeed corrupted. This semi-adversarial setting naturally sits between the classical stochastic case with i.i.d. instances for which function classes with finite VC dimension are learnable; and the adversarial case with arbitrary instances, known to be significantly more restrictive. For this problem, Goel et al. (2023) showed that, if the learner knows the distribution $\mu$ of clean samples in advance, learning can be achieved for all VC classes without restrictions on adversary corruptions. This is, however, a strong assumption in both theory and practice: a natural question is whether similar learning guarantees can be achieved without prior distributional knowledge, as is standard in classical learning frameworks (e.g., PAC learning or asymptotic consistency) and other non-i.i.d. models (e.g., smoothed online learning). We therefore focus on the distribution-free setting where $\mu$ is unknown and propose an algorithm AbstainBoost based on a boosting procedure of weak learners, which guarantees sublinear error for general VC classes in distribution-free abstention learning for oblivious adversaries. These algorithms also enjoy similar guarantees for adaptive adversaries, for structured function classes including linear classifiers. These results are complemented with corresponding lower bounds, which reveal an interesting polynomial trade-off between misclassification error and number of erroneous abstentions.

We introduce the Integrated Tsallis Combination (ITC), a hybrid impurity measure for decision tree learning that combines normalized Tsallis entropy with an exponential polarization component. While many existing measures sacrifice theoretical soundness for computational efficiency or vice versa, ITC provides a mathematically principled framework that balances both aspects. The core innovation lies in the complementarity between Tsallis entropy's information-theoretic foundations and the polarization component's sensitivity to distributional asymmetry. We establish key theoretical properties-concavity under explicit parameter conditions, proper boundary conditions, and connections to classical measures-and provide a rigorous justification for the hybridization strategy. Through an extensive comparative evaluation on seven benchmark datasets comparing 23 impurity measures with five-fold repetition, we show that simple parametric measures (Tsallis $\alpha=0.5$) achieve the highest average accuracy ($91.17\%$), while ITC variants yield competitive results ($88.38-89.16\%$) with strong theoretical guarantees. Statistical analysis (Friedman test: $\chi^2=3.89$, $p=0.692$) reveals no significant global differences among top performers, indicating practical equivalence for many applications. ITC's value resides in its solid theoretical grounding-proven concavity under suitable conditions, flexible parameterization ($\alpha$, $\beta$, $\gamma$), and computational efficiency $O(K)$-making it a rigorous, generalizable alternative when theoretical guarantees are paramount. We provide guidelines for measure selection based on application priorities and release an open-source implementation to foster reproducibility and further research.

Content safety teams need metrics that reflect what users actually experience, not only what is reported. We study prevalence: the fraction of user views (impressions) that went to content violating a given policy on a given day. Accurate prevalence measurement is challenging because violations are often rare and human labeling is costly, making frequent, platform-representative studies slow. We present a design-based measurement system that (i) draws daily probability samples from the impression stream using ML-assisted weights to concentrate label budget on high-exposure and high-risk content while preserving unbiasedness, (ii) labels sampled items with a multimodal LLM governed by policy prompts and gold-set validation, and (iii) produces design-consistent prevalence estimates with confidence intervals and dashboard drilldowns. A key design goal is one global sample with many pivots: the same daily sample supports prevalence by surface, viewer geography, content age, and other segments through post-stratified estimation. We describe the statistical estimators, variance and confidence interval construction, label-quality monitoring, and an engineering workflow that makes the system configurable across policies.

The global clustering coefficient serves as a powerful metric for the structural analysis and comparison of complex networks. Random geometric graphs offer a realistic framework for representing the spatial constraints and geometry often found in real-world network datasets. In this paper, we establish a central limit theorem for the global clustering coefficient of random geometric graphs. Our main result identifies the centering and scaling sequences required for convergence in law to the standard normal distribution. Our approach varies by regime: in the dense case, we employ the Lyapunov CLT; in the intermediate case, we utilize the asymptotic theory of $U$-statistics with sample-size-dependent kernels; and in the sparse regime, we use the method of moments to derive the asymptotic distribution. Notably, the convergence rates for non-uniform and uniform random geometric graphs diverge in the dense regime, yet they coincide in the sparse regime. In addition, we find that the global clustering coefficient for both uniform and non-uniform RGGs is asymptotically equal to $3/4$

Data collection is a critical component of modern statistical and machine learning pipelines, particularly when data must be gathered from multiple heterogeneous sources to study a target population of interest. In many use cases, such as medical studies or political polling, different sources incur different sampling costs. Observations often have associated group identities - for example, health markers, demographics, or political affiliations - and the relative composition of these groups may differ substantially, both among the source populations and between sources and target population. In this work, we study multi-source data collection under a fixed budget, focusing on the estimation of population means and group-conditional means. We show that naive data collection strategies (e.g. attempting to "match" the target distribution) or relying on standard estimators (e.g. sample mean) can be highly suboptimal. Instead, we develop a sampling plan which maximizes the effective sample size - the total sample size divided by $D_{\chi^2}(q\mid\mid\overline{p}) + 1$, where $q$ is the target distribution, $\overline{p}$ is the aggregated source distribution, and $D_{\chi^2}$ is the $\chi^2$-divergence. We pair this sampling plan with a classical post-stratification estimator and upper bound its risk. We provide matching lower bounds, establishing that our approach achieves the budgeted minimax optimal risk. Our techniques also extend to prediction problems when minimizing the excess risk, providing a principled approach to multi-source learning with costly and heterogeneous data sources.

Stochastic gradient descent (SGD) is a cornerstone algorithm for high-dimensional optimization, renowned for its empirical successes. Recent theoretical advances have provided a deep understanding of how SGD enables feature learning in high-dimensional nonlinear models, most notably the single-index model with i.i.d. data. In this work, we study the sequential learning problem for single-index models, also known as generalized linear bandits or ridge bandits, where SGD is a simple and natural solution, yet its learning dynamics remain largely unexplored. We show that, similar to the optimal interactive learner, SGD undergoes a distinct “burn-in” phase before entering the “learning” phase in this setting. Moreover, with an appropriately chosen learning rate schedule, a single SGD procedure simultaneously achieves near-optimal (or best-known) sample complexity and regret guarantees across both phases, for a broad class of link functions. Our results demonstrate that SGD remains highly competitive for learning single-index models under adaptive data.

We study the estimation of time-homogeneous drift functions in multivariate stochastic differential equations with known diffusion coefficient, from multiple trajectories observed at high frequency over a fixed time horizon. We formulate drift estimation as a denoising problem conditional on previous observations, and propose an estimator of the drift function which is a by-product of training a conditional diffusion model capable of simulating new trajectories dynamically. Across different drift classes, the proposed estimator was found to match classical methods in low dimensions and remained consistently competitive in higher dimensions, with gains that cannot be attributed to architectural design choices alone.

Generative Flow Networks (GFlowNets) are a flexible family of amortized samplers trained to generate discrete and compositional objects with probability proportional to a reward function. However, learning efficiency is constrained by the model's ability to rapidly explore diverse high-probability regions during training. To mitigate this issue, recent works have focused on incentivizing the exploration of unvisited and valuable states via curiosity-driven search and self-supervised random network distillation, which tend to waste samples on already well-approximated regions of the state space. In this context, we propose Adaptive Complementary Exploration (ACE), a principled algorithm for the effective exploration of novel and high-probability regions when learning GFlowNets. To achieve this, ACE introduces an exploration GFlowNet explicitly trained to search for high-reward states in regions underexplored by the canonical GFlowNet, which learns to sample from the target distribution. Through extensive experiments, we show that ACE significantly improves upon prior work in terms of approximation accuracy to the target distribution and discovery rate of diverse high-reward states.