Papers

Community detection is the problem of identifying dense communities in networks. Motivated by transitive behavior in social networks ("thy friend is my friend"), an emerging line of work considers spatially-embedded networks, which inherently produce graphs containing many triangles. In this paper, we consider the problem of exact label recovery in the Geometric Stochastic Block Model (GSBM), a model proposed by Baccelli and Sankararaman as the spatially-embedded analogue of the well-studied Stochastic Block Model. Under mild technical assumptions, we completely characterize the information-theoretic threshold for exact recovery, generalizing the earlier work of Gaudio, Niu, and Wei.

The growing scale of deep learning models has rendered standard hyperparameter (HP) optimization prohibitively expensive. A promising solution is the use of scale-aware hyperparameters, which can enable direct transfer of optimal HPs from small-scale grid searches to large models with minimal performance loss. To understand the principles governing such transfer strategy, we develop a general conceptual framework for reasoning about HP transfer across scale, characterizing transfer as fast when the suboptimality it induces vanishes asymptotically faster than the finite-scale performance gap. We show formally that fast transfer is equivalent to useful transfer for compute-optimal grid search, meaning that transfer is asymptotically more compute-efficient than direct tuning. While empirical work has found that the Maximal Update Parameterization ($\mu$P) exhibits fast transfer when scaling model width, the mechanisms remain poorly understood. We show that this property depends critically on problem structure by presenting synthetic settings where transfer either offers provable computational advantage or fails to outperform direct tuning even under $\mu$P. To explain the fast transfer observed in practice, we conjecture that decomposing the optimization trajectory reveals two contributions to loss reduction: (1) a width-stable component that determines the optimal HPs, and (2) a width-sensitive component that improves with width but weakly perturbs the HP optimum. We present empirical evidence for this hypothesis across various settings, including large language model pretraining.

We develop polynomial-time algorithms for near-optimal minimax mean estimation under $\ell_2$-squared loss in a Gaussian sequence model under convex constraints. The parameter space is an origin-symmetric, type-2 convex body $K \subset \mathbb{R}^n$, and we assume additional regularity conditions: specifically, we assume $K$ is well-balanced, i.e., there exist known radii $r, R > 0$ such that $r B_2 \subseteq K \subseteq R B_2$, as well as oracle access to the Minkowski gauge of $K$. Under additional conditions guaranteeing an efficient approximate quadratic-form-maximization oracle on $K$, our procedures achieve the minimax rate up to factors that depend polylogarithmically on the dimension, while remaining computationally efficient. We further extend our methodology to the linear regression and robust heavy-tailed settings, establishing polynomial-time near-optimal estimators when the constraint set satisfies the regularity conditions above. To the best of our knowledge, these results provide the first general framework for attaining statistically near-optimal performance under such broad geometric constraints while preserving computational tractability.

We study instrumental variable regression in data rich environments. The goal is to estimate a linear model from many noisy covariates and many noisy instruments. Our key assumption is that true covariates and true instruments are repetitive, though possibly different in nature; they each reflect a few underlying factors, however those underlying factors may be misaligned. We analyze a family of estimators based on two stage least squares with spectral regularization: canonical correlations between covariates and instruments are learned in the first stage, which are used as regressors in the second stage. As a theoretical contribution, we derive upper and lower bounds on estimation error, proving optimality of the method with noisy data. As a practical contribution, we provide guidance on which types of spectral regularization to use in different regimes.

We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling. To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound.

Ranked set sampling (RSS) is a cost-efficient study design that uses inexpensive baseline ranking to select a more informative subset of individuals for full measurement. While RSS is well known to improve precision over simple random sampling (SRS) for uncensored outcomes, survival analysis under RSS has largely been limited to estimation of the Kaplan-Meier survival curve under random censoring. Consequently, many standard tools routinely used with SRS data, including log-rank and weighted log-rank tests, restricted mean survival time summaries, and window-based mean life measures, are not yet fully developed for RSS settings, particularly when ranking is imperfect and censoring is present. This work develops a unified survival analysis framework for balanced RSS designs that preserves efficiency gains while providing the inferential tools expected in applied practice. We formalize Kaplan-Meier and Nelson-Aalen estimators for right-censored data under both perfect and concomitant-based imperfect ranking and establish their large-sample properties using martingale and empirical process methods adapted to the rank-wise RSS structure. Rank-aware Greenwood-type variance estimators are proposed, and efficiency relative to SRS is evaluated through simulation studies varying set size, number of cycles, censoring proportion, and ranking quality. The framework is further extended to log-rank and Fleming-Harrington weighted tests, as well as restricted and window mean life functionals with asymptotic variance formulas and two-sample comparisons. An implementation plan with real-data illustrations is provided to facilitate practical use.

Variational Bayes (VB) is a popular and computationally efficient method to approximate the posterior distribution in Bayesian inference, especially when the exact posterior is analytically intractable and sampling-based approaches are computationally prohibitive. While VB often yields accurate point estimates, its uncertainty quantification (UQ) is known to be unreliable. For example, credible intervals derived from VB posteriors tend to exhibit undercoverage, failing to achieve nominal frequentist coverage probabilities. In this article, we address this challenge by proposing Trustworthy Variational Bayes (TVB), a method to recalibrate the UQ of broad classes of VB procedures. Our approach follows a bend-to-mend strategy: we intentionally misspecify the likelihood to correct VB's flawed UQ. In particular, we first relax VB by building on a recent fractional VB method, and then identify the optimal fraction parameter using conformal techniques such as sample splitting and bootstrapping. This yields recalibrated UQ for any given parameter of interest. On the theoretical front, we establish that the calibrated credible intervals achieve asymptotically correct frequentist coverage for a given parameter of interest; this, to the best of our knowledge, is the first such theoretical guarantee for VB. On the practical front, we introduce the "TVB table", which enables (1) massive parallelization and remains agnostic to the parameter of interest during its construction, and (2) efficient identification of the optimal fraction parameter for any specified parameter of interest. The proposed method is illustrated via Gaussian mixture models and Bayesian mixture linear regression models, and numerical experiments demonstrate that the TVB method outperforms standard VB and achieves normal frequentist coverage in finite samples. A real data application is also discussed.

Modern machine learning embeddings provide powerful compression of high-dimensional data, yet they typically destroy the geometric structure required for classical likelihood-based statistical inference. This paper develops a rigorous theory of likelihood-preserving embeddings: learned representations that can replace raw data in likelihood-based workflows – hypothesis testing, confidence interval construction, model selection – without altering inferential conclusions. We introduce the Likelihood-Ratio Distortion metric $\Delta_n$, which measures the maximum error in log-likelihood ratios induced by an embedding. Our main theoretical contribution is the Hinge Theorem, which establishes that controlling $\Delta_n$ is necessary and sufficient for preserving inference. Specifically, if the distortion satisfies $\Delta_n = o_p(1)$, then (i) all likelihood-ratio based tests and Bayes factors are asymptotically preserved, and (ii) surrogate maximum likelihood estimators are asymptotically equivalent to full-data MLEs. We prove an impossibility result showing that universal likelihood preservation requires essentially invertible embeddings, motivating the need for model-class-specific guarantees. We then provide a constructive framework using neural networks as approximate sufficient statistics, deriving explicit bounds connecting training loss to inferential guarantees. Experiments on Gaussian and Cauchy distributions validate the sharp phase transition predicted by exponential family theory, and applications to distributed clinical inference demonstrate practical utility.

We show that differentiable sorting and ranking operators are structurally incompatible with within-vector rank normalization. We formalize admissibility through monotone invariance (C1), batch independence (C2), and a rank-space stability condition (C3). Gap-sensitive relaxations such as SoftSort violate (C1) by a quantitative margin that depends on the temperature and input scale. Batchwise rank relaxations such as SinkhornSort violate (C2): the same sample can be assigned outputs arbitrarily close to 0 or 1 depending solely on batch context. Condition (C3) implies (C1) under the rank representation used here and should not be read as a third independent failure mode. We also characterize the admissible class: any admissible operator must factor through the rank representation via a Lipschitz function.

We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices – a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved optimal covering numbers for axis-parallel boxes (Gnewuch, 2024) into the dyadic chaining framework, we achieve explicitly computable constants that improve upon all previously known bounds. For dimension $d \ge 3$, we prove that with high probability, \[ D_N^d \le \sqrt{\alpha A_d + \beta B \frac{\ln \log_2 N}{d}} \sqrt{\frac{d}{N}}, \] where $A_d$ is given by an explicit series and satisfies $A_3 \le 745$, a 14% improvement over the previous best constant of 868 (Fiedler et al., 2023). For $d=2$, we obtain the currently smallest known constant $A_2 \le 915$. Our analysis reveals a precise trade-off between the dimensional dependence and the logarithmic factor in $N$, highlighting how optimal covering estimates directly translate to tighter discrepancy bounds. These results immediately yield improved error guarantees for quasi-Monte Carlo integration, uncertainty quantification, and high-dimensional sampling, and provide a new benchmark for the probabilistic analysis of geometric discrepancy. Keywords: Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds.

This article addresses the fuzzy logistic regression model under conditions of multicollinearity, which causes instability and inflated variance in parameter estimation. In this model, both the response variable and parameters are represented as fuzzy triangular numbers. To overcome the multicollinearity problem, various Liu-type estimators were employed: Fuzzy Maximum Likelihood Estimators (FMLE), Fuzzy Logistic Ridge Estimators (FLRE), Fuzzy Logistic Liu Estimators (FLLE), Fuzzy Logistic Liu-type Estimators (FLLTE), and Fuzzy Logistic Liu-type Parameter Estimators (FLLTPE). Through simulations with various sample sizes and application to real fuzzy data on kidney failure, model performance was evaluated using mean square error (MSE) and goodness of fit criteria. Results demonstrated superior performance of FLLTPE and FLLTE compared to other estimators.

Bayesian variable selection (BVS) depends critically on the specification of a prior distribution over the model space, particularly for controlling sparsity and multiplicity. This paper examines the practical consequences of different model space priors for BVS in logistic regression, with an emphasis on streaming data settings. We review some popular and well-known Beta–Binomial priors alongside the recently proposed matryoshka doll (MD) prior. We introduce a simple approximation to the MD prior that yields independent inclusion indicators and is convenient for scalable inference. Using BIC-based approximations to marginal likelihoods, we compare the effect of different model space priors on posterior inclusion probabilities and coefficient estimation at intermediate and final stages of the data stream via simulation studies. Overall, the results indicate that no single model space prior uniformly dominates across scenarios, and that the recently proposed MD prior provides a useful additional option that occupies an intermediate position between commonly used Beta–Binomial priors with differing degrees of sparsity.

Transformers empirically perform precise probabilistic reasoning in carefully constructed “Bayesian wind tunnels” and in large-scale language models, yet the mechanisms by which gradient-based learning creates the required internal geometry remain opaque. We provide a complete first-order analysis of how cross-entropy training reshapes attention scores and value vectors in a transformer attention head. Our core result is an advantage-based routing law for attention scores, \[ \frac{\partial L}{\partial s_{ij}} = \alpha_{ij}\bigl(b_{ij}-\mathbb{E}_{\alpha_i}[b]\bigr), \qquad b_{ij} := u_i^\top v_j, \] coupled with a responsibility-weighted update for values, \[ \Delta v_j = -\eta\sum_i \alpha_{ij} u_i, \] where $u_i$ is the upstream gradient at position $i$ and $\alpha_{ij}$ are attention weights. These equations induce a positive feedback loop in which routing and content specialize together: queries route more strongly to values that are above-average for their error signal, and those values are pulled toward the queries that use them. We show that this coupled specialization behaves like a two-timescale EM procedure: attention weights implement an E-step (soft responsibilities), while values implement an M-step (responsibility-weighted prototype updates), with queries and keys adjusting the hypothesis frame. Through controlled simulations, including a sticky Markov-chain task where we compare a closed-form EM-style update to standard SGD, we demonstrate that the same gradient dynamics that minimize cross-entropy also sculpt the low-dimensional manifolds identified in our companion work as implementing Bayesian inference. This yields a unified picture in which optimization (gradient flow) gives rise to geometry (Bayesian manifolds), which in turn supports function (in-context probabilistic reasoning).

We propose a new ensemble prediction method, Random Subset Averaging (RSA), tailored for settings with many covariates, particularly in the presence of strong correlations. RSA constructs candidate models via binomial random subset strategy and aggregates their predictions through a two-round weighting scheme, resulting in a structure analogous to a two-layer neural network. All tuning parameters are selected via cross-validation, requiring no prior knowledge of covariate relevance. We establish the asymptotic optimality of RSA under general conditions, allowing the first-round weights to be data-dependent, and demonstrate that RSA achieves a lower finite-sample risk bound under orthogonal design. Simulation studies demonstrate that RSA consistently delivers superior and stable predictive performance across a wide range of sample sizes, dimensional settings, sparsity levels and correlation structures, outperforming conventional model selection and ensemble learning methods. An empirical application to financial return forecasting further illustrates its practical utility.

Transformers often appear to perform Bayesian reasoning in context, but verifying this rigorously has been impossible: natural data lack analytic posteriors, and large models conflate reasoning with memorization. We address this by constructing Bayesian wind tunnels – controlled environments where the true posterior is known in closed form and memorization is provably impossible. In these settings, small transformers reproduce Bayesian posteriors with $10^{-3}$-$10^{-4}$ bit accuracy, while capacity-matched MLPs fail by orders of magnitude, establishing a clear architectural separation. Across two tasks – bijection elimination and Hidden Markov Model (HMM) state tracking – we find that transformers implement Bayesian inference through a consistent geometric mechanism: residual streams serve as the belief substrate, feed-forward networks perform the posterior update, and attention provides content-addressable routing. Geometric diagnostics reveal orthogonal key bases, progressive query-key alignment, and a low-dimensional value manifold parameterized by posterior entropy. During training this manifold unfurls while attention patterns remain stable, a frame-precision dissociation predicted by recent gradient analyses. Taken together, these results demonstrate that hierarchical attention realizes Bayesian inference by geometric design, explaining both the necessity of attention and the failure of flat architectures. Bayesian wind tunnels provide a foundation for mechanistically connecting small, verifiable systems to reasoning phenomena observed in large language models.

Post-hoc residual correction can reduce forecasting error, but always-on updates often fail to generalize: when residual signals are weak or noisy, expressive correctors can overfit and even increase the overall forecasting errors. We introduce CRC, a robust residual correction framework that couples a hybrid corrector with a validation-calibrated selective deployment rule. CRC proposes corrections via a conservative ridge "floor" plus conditional nonlinear refinement, but deploys them only when held-out evidence indicates net error reduction over a frozen baseline forecaster. CRC treats when to revise a forecast as a validation-calibrated deployment decision, not only how to fit residuals. Across standard long-sequence benchmarks and multiple backbones, CRC improves horizon-averaged MSE and MAE in most dataset-backbone pairs, with particularly large gains on complex datasets (e.g., over 20% relative MSE reduction on Traffic). We additionally report the non-degradation rate (NDR) as a stability diagnostic of harmful updates and ablate the validation-calibrated "firewall" that stabilizes MSE gains.

When sociologists and other social scientist ask whether the return to college differs by race and gender, they face a choice between two fundamentally different modes of inquiry. Traditional interaction models follow deductive logic: the researcher specifies which variables moderate effects and tests these hypotheses. Machine learning methods follow inductive logic: algorithms search across vast combinatorial spaces to discover patterns of heterogeneity. This article develops a framework for navigating between these approaches. We show that the choice between deduction and induction reflects a tradeoff between interpretability and flexibility, and we demonstrate through simulation when each approach excels. Our framework is particularly relevant for inequality research, where understanding how treatment effects vary across intersecting social subpopulation is substantively central.

For complex simulation problems, inferring parameters often precludes the use of classical likelihood-based techniques due to intractable likelihoods. Simulation-based inference (SBI) methods offer a likelihood-free approach to directly learn posterior distributions $p(\bftheta \mid \xobs)$ from simulator outputs. Recently, diffusion models have emerged as promising tools for SBI, addressing limitations of earlier neural methods such as neural likelihood/posterior estimation and normalizing flows. This review examines diffusion-based SBI from first principles to applications, emphasizing robustness in three non-ideal data scenarios common to scientific computing: model misspecification (simulator-reality mismatch), unstructured or infinite-dimensional observations, and missing data. We synthesize mathematical foundations and survey eight methods addressing these challenges, such as conditional diffusion for irregular data, guided diffusion for prior adaptation, sequential and factorized approaches for efficiency, and consistency models for fast sampling. Throughout, we maintain consistent notation and emphasize conditions required for accurate posteriors. We conclude with open problems and applications to geophysical uncertainty quantification, where these challenges are acute.

In this article, we study sequential change-point methods for discretely observed generalized Ornstein-Uhlenbeck processes with periodic drift. Two detection methods are proposed, and their respective performance is studied through numerical experiments for several choices of parameters.

We study the problem of active nonparametric sequential two-sample testing over multiple heterogeneous data sources. In each time slot, a decision-maker adaptively selects one of $K$ data sources and receives a paired sample generated from that source for testing. The goal is to decide as quickly as possible whether the pairs are generated from the same distribution or not. The gain achieved by such adaptive sampling (in terms of smaller expected stopping time or larger error exponents) has been well-characterized for parametric models via Chernoff's adaptive MLE selection rule [1]. However, analogous results are not known for the case of nonparametric problems, such as two-sample testing, where we place no restrictions on the distributions. Our main contribution is a general active nonparametric testing procedure that combines an adaptive source-selecting strategy within the testing-by-betting framework of [2] that works under minimal distributional assumptions. In each time slot, our scheme proceeds by selecting a source according to a probability that mixes exploitation, favoring sources with the largest empirical distinguishability, and exploration via a vanishing greedy strategy. The (paired) observations so collected are then used to update the "betting-wealth process", which is a stochastic process guaranteed to be a nonnegative martingale under the null. The procedure stops and rejects the null when the wealth process exceeds an appropriate threshold; an event that is unlikely under the null. We show that our test controls the type-I error at a prespecified level-$\alpha$ under the null, and establish its power-one property and a bound on its expected sample size under the alternative. Our results provide a precise characterization of the improvements achievable by a principled adaptive sampling strategy over its passive analog.

Graph Neural Networks (GNNs) are powerful tools for learning from graph-structured data, but their application to large graphs is hindered by computational costs. The need to process every neighbor for each node creates memory and computational bottlenecks. To address this, we introduce BLISS, a Bandit Layer Importance Sampling Strategy. It uses multi-armed bandits to dynamically select the most informative nodes at each layer, balancing exploration and exploitation to ensure comprehensive graph coverage. Unlike existing static sampling methods, BLISS adapts to evolving node importance, leading to more informed node selection and improved performance. It demonstrates versatility by integrating with both Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs), adapting its selection policy to their specific aggregation mechanisms. Experiments show that BLISS maintains or exceeds the accuracy of full-batch training.

Hyperparameter tuning can dramatically impact training stability and final performance of large-scale models. Recent works on neural network parameterisations, such as $\mu$P, have enabled transfer of optimal global hyperparameters across model sizes. These works propose an empirical practice of search for optimal global base hyperparameters at a small model size, and transfer to a large size. We extend these works in two key ways. To handle scaling along most important scaling axes, we propose the Complete$^{(d)}$ Parameterisation that unifies scaling in width and depth – using an adaptation of CompleteP – as well as in batch-size and training duration. Secondly, with our parameterisation, we investigate per-module hyperparameter optimisation and transfer. We characterise the empirical challenges of navigating the high-dimensional hyperparameter landscape, and propose practical guidelines for tackling this optimisation problem. We demonstrate that, with the right parameterisation, hyperparameter transfer holds even in the per-module hyperparameter regime. Our study covers an extensive range of optimisation hyperparameters of modern models: learning rates, AdamW parameters, weight decay, initialisation scales, and residual block multipliers. Our experiments demonstrate significant training speed improvements in Large Language Models with the transferred per-module hyperparameters.

Sparse index tracking is often evaluated through rolling reconstruction: a sparse portfolio is fitted on an in-sample window, held over the next period, and rebuilt when the window rolls forward. This can achieve low realised tracking error, but it treats rebalancing primarily as repeated construction and can generate large turnover and frequent substitutions in the selected constituents. We propose a new workflow that separates sparse-tracker construction from sparse-tracker maintenance. A hybrid optimisation-plus-sampling framework provides the metrics operating at the decision level for both layers. The initial tracker is built from a calibrated shrinkage model and uncertainty-aware posterior support screening. Subsequent rebalance dates are handled in the self-financing change variable $\Delta w$. The default action is to preserve the existing tracker; local repairs are implemented only when realised tracking deterioration and posterior directional evidence jointly suggest intervention. In a 2020-2025 S&P 500-style case study, we show that the proposed tracker occupies a distinct low-turnover operating region. Moreover, we demonstrate that the proposed $\Delta w$ maintenance layer can be attached to externally constructed trackers, where it gives consistent improvements over simply holding the initial tracker. Additional diagnostics, sensitivity experiments, and computational details are reported in the companion Supplementary Material. Replication code and logs of several experiments are available at \href{https://github.com/droxanas/Low-Turnover-Rebalancing-for-Sparse-Index-Tracking/}{https://github.com/droxanas/Low-Turnover-Rebalancing-for-Sparse-Index-Tracking/}.

We introduce a nonparametric model for inferring time-evolving, unobserved probability distributions from discrete-time data consisting of unlabelled partitions. The latent process is a two-parameter Poisson-Dirichlet diffusion, and observations arise via exchangeable sampling. Applications include social and genetic data where only aggregate clustering summaries are observed. To address the intractable likelihood, we develop a tractable inferential framework that avoids label enumeration and direct simulation of the latent state. We exploit a duality between the diffusion and a pure-death process on partitions, together with coagulation operators that encode the effect of new data. These yield closed-form, recursive updates for forward and backward inference. We compute exact posterior distributions of the latent state at arbitrary times and predictive distributions of future or interpolated partitions. This enables online and offline inference and forecasting with full uncertainty quantification, bypassing MCMC and sequential Monte Carlo. Compared to particle filtering, our method achieves higher accuracy, lower variance, and substantial computational gains. We illustrate the methodology with synthetic experiments and a social network application, recovering interpretable patterns in time-varying heterozygosity.

This paper introduces new methods for constructing prediction intervals using quantile-based techniques. The procedures are developed for both classical (homoscedastic) autoregressive models and modern quantile autoregressive models. They combine quantile estimation with multiplier bootstrap schemes to approximate the sampling variability of coefficient estimates, together with bootstrap replications of future observations. We consider both percentile-based and predictive-root-based constructions. Theoretical results establish the validity and pertinence of the proposed methods. Simulation experiments evaluate their finite-sample performance and show that the proposed methods yield improved coverage properties and computational efficiency relative to existing approaches in the literature. The empirical usefulness of the methods is illustrated through applications to U.S. unemployment rate data and retail gasoline prices.

Entropically regularized optimal transport between probability measures supported on compact subsets of Euclidean space admits a representation as an information projection under moment inequality constraints. Exploiting this structure, I develop a sieve-based approximation of the Fenchel dual, yielding a sequence of finite-dimensional convex programs whose sample analogues provide tractable estimators of the regularized optimal value and associated dual optimizers. Under minimal assumptions–compact support and continuity of the cost function–I establish almost sure consistency of these estimators. I further derive finite-sample bounds for the estimation error of the optimal value, featuring only logarithmic dependence on sieve complexity, and obtain asymptotic stochastic bounds characterized by suprema of centered Gaussian processes. The results furnish general statistical guarantees for sieve-based estimation of entropic optimal transport and apply to settings not covered by existing theory for the empirical Sinkhorn divergence and other sieve-based methods.

A parametric cluster model is a statistical model providing geometric insights onto the points defining a cluster. The spherical cluster model (SC) approximates a finite point set $P\subset \mathbb{R}^d$ by a sphere $S(c,r)$ as follows. Taking $r$ as a fraction $\eta\in(0,1)$ (hyper-parameter) of the std deviation of distances between the center $c$ and the data points, the cost of the SC model is the sum over all data points lying outside the sphere $S$ of their power distance with respect to $S$. The center $c$ of the SC model is the point minimizing this cost. Note that $\eta=0$ yields the celebrated center of mass used in KMeans clustering. We make three contributions. First, we show fitting a spherical cluster yields a strictly convex but not smooth combinatorial optimization problem. Second, we present an exact solver using the Clarke gradient on a suitable stratified cell complex defined from an arrangement of hyper-spheres. Finally, we present experiments on a variety of datasets ranging in dimension from $d=9$ to $d=10,000$, with two main observations. First, the exact algorithm is orders of magnitude faster than BFGS based heuristics for datasets of small/intermediate dimension and small values of $\eta$, and for high dimensional datasets (say $d>100$) whatever the value of $\eta$. Second, the center of the SC model behave as a parameterized high-dimensional median. The SC model is of direct interest for high dimensional multivariate data analysis, and the application to the design of mixtures of SC will be reported in a companion paper.

Policy alignment to preference data typically assumes a known link function between observed preferences and latent rewards (e.g., Bradley-Terry model / logistic link). Misspecification of this link can bias inferred rewards and misalign learned policies. We study policy alignment under an unknown and unrestricted link function. We formulate an $f$-divergence-constrained reward maximization problem and show that realizability in a policy class induces a semiparametric single-index binary choice model, where a scalar policy-induced index captures all dependence on demonstrations and the remaining preference distribution is unrestricted. Rather than impose identifiability of structural parameters of such a model and estimate them, as in econometrics, we develop methods that directly learn policies, with the reward function implicit, analyzing error to the optimal policy and allowing for unidentifiable and nonparametric indices. We prove link-agnostic convergence guarantees in terms of generic function complexity measures and validate the methods and theory empirically. Code is available at https://github.com/causalml/spo/.

In multicenter biomedical research, integrating data from multiple decentralized sites provides more robust and generalizable findings due to its larger sample size and the ability to account for the between-site heterogeneity. However, sharing individual-level data across sites is often difficult due to patient privacy concerns and regulatory restrictions. To overcome this challenge, many distributed algorithms, that fit a global model by only communicating aggregated information across sites, have been proposed. A major challenge in applying existing distributed algorithms to real-world data is that their validity often relies on the assumption that data across sites are independently and identically distributed, which is frequently violated in practice. In biomedical applications, data distributions across clinical sites can be heterogeneous. Additionally, the set of covariates available at each site may vary due to different data collection protocols. We propose a distributed inference framework for data integration in the presence of both distribution heterogeneity and data structural heterogeneity. By modeling heterogeneous and structurally missing data using density-tilted generalized method of moments, we developed a general aggregated data-based distributed algorithm that is communication-efficient and heterogeneity-aware. We establish the asymptotic properties of our estimator and demonstrate the validity of our method via simulation studies.

In biomedical research, to obtain more accurate prediction results from a target study, leveraging information from multiple similar source studies is proved to be useful. However, in many biomedical applications based on real-world data, populations under consideration in different studies, e.g., clinical sites, can be heterogeneous, leading to challenges in properly borrowing information towards the target study. The state of art methods are typically based on study-level matching to identify source studies that are similar to the target study, whilst samples from source studies that significantly differ from the target study will all be dropped at the study level, which can lead to substantial loss of information. We consider a general situation where all studies are sampled from a super-population composed of distinct subpopulations, and propose a novel framework of targeted learning via subpopulation matching. In contrast to the existing study-level matching methods, measuring similarities between subpopulations can effectively decompose both within- and between-study heterogeneity, allowing incorporation of information from all source studies without dropping any samples as in the existing methods. We devise the proposed framework as a two-step procedure, where a finite mixture model is first fitted jointly across all studies to provide subject-wise probabilistic subpopulation information, followed by a step of within-subpopulation information transferring from source studies to the target study for each identified subpopulation. We establish the non-asymptotic properties of our estimator and demonstrate the ability of our method to improve prediction at the target study via simulation studies.