Papers

Refugee integration is a central policy challenge for host countries, and where governments initially place refugees shapes their integration trajectories. Yet placement officers often have limited information about where each case is most likely to succeed. Algorithmic refugee matching uses administrative data, machine learning, and constrained optimization to recommend employment-optimized placements in real time as cases arrive, with human placement officers retaining final authority. Between January 2020 and June 2023, the Swiss State Secretariat for Migration randomly assigned about 2,000 refugee cases to receive a canton recommendation either algorithmically optimized for employment or drawn to approximate existing procedures, with placement officers and refugees blinded to assignment. The two arms used identical but separate canton and origin-group quotas, so gains reflect better refugee-canton matching rather than reallocation toward stronger labor markets. The trial began just before the COVID-19 pandemic shifted labor-market conditions. For the pre-registered primary outcome – the share of months employed during the first three years – the pooled intention-to-treat (ITT) estimate across the 2020-2023 placement cohorts was +2.2 percentage points (about 10% of the 22.3% control mean; 95% CI [+0.05, +4.33]), rising to +3.9 pp (about 17%; [+1.11, +6.68]) for the post-COVID 2022-2023 cohorts. Effects grew over time: at 36 months, the pooled ITT on the employment rate was +5.2 pp (about 11%; 95% CI [+1.10, +9.25]) – comparable to the gains from hundreds of hours of intensive language training. Overall, the results provide rare field evidence that AI-based decision support can improve high-stakes public-sector allocation, offering a scalable, low-cost way to raise refugee employment.

Estimating conditional statistics and learning representations of a population of conditional distributions are central problems in many data-driven applications, including uncertainty quantification and dynamical systems analysis. Conditional mean operators (CMOs), a class of linear operators between function spaces, resolve these objectives by providing access to a broad class of conditional statistics. However, existing methods typically estimate each CMO independently or constrain it to prespecified function spaces, thereby preventing the exploitation of shared structure across related distributions. In this work, we posit that related CMOs share finite-dimensional input and output function spaces, and are specialized for each task with a linear operator mapping these spaces. Based on this hypothesis, we introduce MTL-CMO, a multi-task framework that jointly learns shared function spaces and task-specific operators across multiple datasets. We further introduce T-CMO, a transfer learning method that reuses the shared spaces to estimate, in closed form, the operator of a new conditional distribution. We establish statistical guarantees quantifying the benefits of jointly learning the shared function spaces. Our experiments demonstrate that learning shared function spaces improves uncertainty quantification across a broad range of conditional distributions and, when applied to Langevin and plasma dynamics, yields compact representations of complex dynamics that retain physically meaningful information and enable parameter identification.

This paper studies convex optimization when the gradient cannot be evaluated exactly, but only approximated by a hierarchy of algorithms whose compute grows like $\delta^{-\gamma}$ in the accuracy $\delta$. When $\gamma>2$, falling into the Harder-Than-Monte-Carlo (HTMC) regime, the price of accuracy outruns the variance reduction that Monte Carlo would buy and we show that minimizing a loss function costs no more, up to a factor depending only on $\gamma$, than a single evaluation of its gradient at the accuracy the problem demands. A randomized multilevel oracle replaces the deterministic approximation of accuracy $\delta$ by an unbiased estimator of it, whose variance $\sigma^2$ becomes a second, independently priced dial: the cost of one call drops from $\delta^{-\gamma}$ to $\delta^{2-\gamma}\sigma^{-2}$. Plain inexact gradient descent driven by that oracle reaches loss $\varepsilon$ at expected compute $\Theta(\varepsilon^{-\gamma})$ in the convex case, against $\Theta(\varepsilon^{-(\gamma+1)})$ for the same method run at a fixed accuracy: randomization buys a full power of $\varepsilon$. Under $\mu$-strong convexity the exponent halves, to $\varepsilon^{-\gamma/2}$, because the iterates settle at a noise floor and the bias budget relaxes accordingly. Both bounds are independent of the step size, and hence of the smoothness constant, and we show that the cost is a functional of the underlying gradient flow rather than of any discretization of it.

Power laws are often claimed for Bitcoin quantities from the slope of a log-log plot. We test one such claim with a formal protocol: whether the upper tail of the values of live unspent transaction outputs (UTXOs) follows a power law. We reconstruct the exact UTXO set at eight block heights from 2012 to 2025, validate it against an independent Bitcoin Core snapshot, and apply the Clauset-Shalizi-Newman protocol with three adaptations the data require: exact discrete support, several near-tied cutoffs retained and reselected in every bootstrap replicate, and resampling by transaction, because the outputs of one transaction are not independent. The protocol was fixed before inference and every later change is recorded. The pure power law is rejected at every height, with observed Kolmogorov-Smirnov distances five to forty times those the fitted law produces, and where the exponent is below 2 its extrapolation predicts, in expectation, several outputs each larger than the total coin supply. A lognormal, a stretched exponential and a power law with an exponential cutoff near a thousand bitcoin fit better in sample at every height, as families that contain the power law must, by margins that exceed those seen on simulated power-law tails at six of the eight heights, and they predict held-out transactions better at nearly every height. They rank differently from height to height, and the lognormal and the cutoff power law differ clearly at only two. The tail's shape and the share of supply it holds drifted slowly while the number of outputs grew seventy-fold. The rejection survives a null that reproduces the dependence measured within transactions, which halves the one exceptional effect size in the panel and changes no verdict. Bitcoin UTXO values are heavy-tailed but not power-law distributed.

We study the supervised classification problem for interacting particle systems (IPS) from trajectory observations. The IPS belongs to one of K classes, each characterized by a distinct interaction drift. Given a learning dataset, the goal is then to predict the class label of a newly observed system based on the trajectory of a single particle. This setting raises several statistical challenges. First, the particles within each system interact and are therefore dependent. Second, although the full particle system is Markovian, the trajectory of a single particle is not. To address these difficulties, we exploit the McKean-Vlasov limit of the IPS, which describes the dynamics of a typical particle as the number of particles tends to infinity. We propose a plug-in classification procedure based on estimating the interaction drift associated with each class from discretely observed trajectory data. Hence, for each class, we provide a new nonparametric estimator of the drifts by minimizing a ridge-regularized least-squares contrast over a B-spline basis. In particular, our theoretical findings reveal that the convergence rate of the resulting classifier is of order N -1/6+$\epsilon$ for any $\epsilon$ > 0, where N denotes the number of particles in each system. Numerical experiments illustrate the performance of the proposed method and show that it outperforms an end-to-end neural network baseline that does not exploit the underlying particle-system structure.

Virtual simulators are widely used for studying complex physical phenomena, from particle collisions to rocket propulsion. Such "computer experiments" can be highly time-intensive, and a Bayesian surrogate model can be used for efficient emulation with reliable uncertainty quantification. To train accurate surrogates with a limited sample size $n$, recent work has explored the incorporation of monotonicity (or isotonicity) information, which can often be elicited from physical systems. In practical applications with many input variables, however, existing Bayesian isotonic models can face statistical and computational limitations, which may result in worse performance compared to models that do not incorporate isotonicity. We propose a new transformed additive isotonic model (TAIM), which aims to tame this "curse-of-dimensionality". TAIM makes use of a flexible transformed additive isotonic modeling framework, which leverages a data-estimated link transformation and a monotone basis model with spike-and-slab priors on basis weights. Prediction-wise, TAIM achieves (up to log factors) a posterior contraction rate of $O(n^{-1/3})$ when the true black-box function is in a transformed additive isotonic form with mild smoothness conditions. Such a rate does not depend on the input dimension $d$ for terms involving $n$, which softens the effect of dimensionality on posterior predictions. Computation-wise, TAIM allows for efficient posterior inference via a carefully designed Gibbs sampler, where each sampling iteration requires only linear work in $d$. We further present an extension of TAIM that can model potential deviations from transformed additivity. Numerical experiments and two applications show the effectiveness of TAIM for isotonic surrogate modeling with many input variables.

AI reliability concerns whether an AI system performs its intended function dependably over a stated period and under stated operating conditions, with stated evidence. As these systems become more autonomous, that function includes more than a correct output. Retrieval, memory, tool use, permissions, human oversight, and interactions among systems must operate consistently and safely, and, for generative systems, so must the reasoning process that produces the output. Average benchmark accuracy measures capability; it does not quantify this broader reliability claim. This paper adapts established reliability engineering methods, from failure definitions and operational envelopes to FMEA, accelerated testing, field monitoring, and reliability growth, to AI systems. A four-level diagnostic framework classifies failures as component, operational-loop, agentic-conduct, or network and governance failures. Test, evaluation, verification, and validation (TEVV), sequential monitoring, and FRACAS create and refresh evidence. SMART provides statistical guidance for measurement, analysis, assessment, and test planning; the NIST AI Risk Management Framework provides organizational guidance for governance, evaluation, monitoring, and mitigation. Three cases illustrate the program: adversarial testing of a convolutional neural network, perception-error propagation, and autonomous-vehicle disengagements. Established reliability engineering provides a usable foundation; new measurements and safety guardrails are still needed as these systems are self-evolving.

Modern probabilistic machine learning models increasingly produce multivariate outputs with complex dependence structure, from multi-step time-series forecasts to sample path predictions. Understanding which input features drive the predictive uncertainty is important for risk-aware decisions, model diagnostics, and deciding whether the uncertainty should be mitigated or hedged against. This attribution problem requires a choice of how dependencies between output components are treated. Existing approaches reduce the output to a scalar through aggregation or projection before attribution, thereby obscuring whether features affect marginal uncertainty, dependence structure, or both, while component-wise analyses can miss dependence effects entirely. We close this gap by introducing a hierarchy of three entropy-based Shapley games that make this output-side choice explicit for any ordered multivariate outcome, ranging from per-component marginal entropy to fully joint entropy. The hierarchy isolates a cross-component attribution term that captures how each feature shifts the dependence between output components, a quantity invisible to component-wise methods. We establish a chain-rule decomposition of the joint attribution and characterize the cross-component term through conditional total correlation, providing both closed-form and sample-based estimators. Finally, we demonstrate how the framework captures differences in learned joint structure across probabilistic models from distributional regression to a zero-shot time series foundation model.

We study post-hoc refinement of frozen node classifiers: given only the graph $G$ and class distributions $Q$ predicted by a frozen model, can we improve accuracy without access to node features, model parameters, or gradients? APPNP answers this by propagating logits with a restart towards the initial predictions, minimizing the anchored Dirichlet energy. Instead, we consider the Potts energy, and decompose it into a Dirichlet term, which penalizes disagreement between neighbouring nodes, and a Gini term, which penalizes indecision within each node. This decomposition motivates Propagate, Then Sharpen (PtS), which alternates between propagation of class probabilities and node-wise, mass-preserving sharpening, with only one additional hyperparameter selected using labelled validation nodes. Across nine homophilic graphs, with a frozen MLP backbone, PtS improves mean test accuracy over independently tuned APPNP by $1.71$ percentage points on clean inputs and $3.90$ under severe Gaussian feature corruption. Gains over APPNP become smaller, but remain positive with frozen GCN and GraphSAGE backbones. Sharpening also removes most of the accuracy loss of deep propagation: on clean inputs without restart, accuracy falls by $2.2$ points between $2$ and $100$ propagation steps under PtS, compared with $33.8$ for APPNP.

Long contexts are central to modern transformer systems, but most expressivity results choose a different network for each fixed sequence length. We study whether one masked transformer can approximate causal token-to-token maps uniformly over sequences of arbitrary length sampling a fixed normalized horizon. To relate sampling resolutions, we model tokens by $\alpha$-Hölder sequences or, more generally, a common modulus of continuity. Our notion of continuity across resolutions characterizes the causal families admitting uniform approximation on these compact input classes by a single transformer with length-independent parameters. The result extends to the infinite-length mean-field limit, where tokens form continuous curves and masked attention becomes a causal time integral. For bounded regression with target maps satisfying a $\beta$-smooth stability condition defined using regular test functions, quantitative approximation yields a generalization bound: exact empirical risk minimization over suitably sized bounded-weight transformers gives root mean-square prediction error $O((\log\log N/\log N)^{\beta/(d+2)})$ from $N$ iid labeled sequences. The bound holds at fixed confidence on the same sampling distribution, with $d$ the token dimension and no maximum-length factor. Finally, experiments on physical time series support the Hölder-regular token model at observed scales, with dataset-dependent fitted exponents, whereas text input embeddings provide a contrasting case. Native and dense sampling, shuffled controls, and refinement checks delimit this empirical regularity regime.

Federated Bayesian Optimization (FBO) enables distributed agents to collaboratively optimize expensive black-box objectives without sharing raw local observations. However, effective knowledge transfer remains challenging under communication constraints and task heterogeneity. We propose GUIDE-FBO, in which agents exchange compact distributions over the locations of their respective optima inferred from local Gaussian process (GP) posteriors, rather than raw observations, query points, or surrogate parameters. The server merges and reweights these distributional components before returning a subset to each agent. Each agent then constructs a Federated Interventional GP (FI-GP), which preserves the local posterior mean and spatially rescales its covariance for local decision making. For the upper confidence bound (UCB) instantiation, GUIDE-UCB, we prove that any bounded FI-GP uncertainty intervention preserves the leading-order cumulative regret rate of standard GP-UCB. When the transferred distributions place greater support near an optimum than in a suboptimal region, selecting the latter requires greater local posterior uncertainty. Experiments on 12 synthetic benchmarks and three real-world optimization tasks show that GUIDE-FBO remains effective across settings ranging from homogeneous to severely heterogeneous. Ablation results highlight the importance of spatially localized uncertainty intervention, while the communication analysis shows that GUIDE-FBO exchanges only compact distributional messages.

The Baringhaus–Henze–Epps–Pulley (BHEP) tests form a widely applicable class of affine invariant and consistent tests for multivariate normality. Under the null hypothesis, the test statistic converges to a weighted sum of independent chi-squared random variables. Although closed-form expressions for the first three cumulants of this limiting distribution are known for arbitrary dimension $d$ and smoothing parameter $\beta$, an explicit expression for the fourth cumulant has so far been available only in a special univariate case. We derive the fourth cumulant for arbitrary $d\geq1$ and $\beta>0$ and use the resulting first four moments to construct Johnson and Pearson approximations to the limiting null distribution. These yield essentially instantaneous analytic approximations to BHEP $p$-values, without numerical eigenvalue calculations. Using the recently obtained complete spectrum as a benchmark, we show that both four-moment approximations are highly accurate over a broad range of dimensions, smoothing parameters and upper-tail probabilities, and substantially improve on two- and three-parameter lognormal approximations. Monte Carlo results indicate good finite-sample calibration for many parameter combinations, although convergence to the limiting distribution may be slow in higher dimensions for extreme values of the smoothing parameter.

Hyperparameter transfer across model width can substantially reduce the cost of tuning large neural networks, but its behavior when the training horizon grows with width is not fully understood. Building on the framework of fast hyperparameter transfer (Ghosh et al., 2026), which formalizes when transfer is effective, we investigate conditions that ensure fast transfer in the growing-horizon regime. Specifically, we study learning-rate transfer in a shallow linear network with a single trainable hidden matrix, trained by full-batch gradient descent. Under additional spectral assumptions, our main results are threefold. (i) We prove fast learning-rate transfer as $n,T\to\infty$ whenever $T=o(\sqrt{n})$. (ii) We characterize the transfer rates through the finite-width perturbation scale, the first-order sensitivities of the loss and its learning-rate derivative to finite-width perturbations, and the local loss curvature. (iii) We derive limiting distributions for the optimal learning rate and optimized loss, governed by fluctuations associated with the extreme eigenvalues of the data Gram matrix. These results clarify how spectral structure and local loss sensitivities govern learning-rate transfer at growing horizons.

Price stability remains a pillar in monetary policy practices and carries a special importance within monetary unions. Mainstream economics tried to leverage price stability using price indices and several metrics to shed light on specific dynamics and optimal macroeconomic levels. The wide availability of data led researchers to consider the study of systems using Random Matrix Theory, based on inner correlation patterns. This aims to enhance the multivariate analysis by removing noisy patterns from the signal and improve data quality for further inferences. This work considers the collection of monthly inflation indices in the Eurozone as a system of prices to analyze its eigenvalues' statistical and asymptotic properties and uncover inner country-level insights. Results confirm the system cannot assumed to be randomly generated, and the data exhibit noise-dominated patterns, due to small and persistent variations at the country-level. The latter make the inter-country correlations more dynamic and the separation of the signal from the noise quiet difficult. Findings identified two countries as distorting inflation dynamics besides three other distinct, regional-based groups of countries. Variability sources might stem from economic episodes fueling inflation spikes in some countries, as well as methodological aspects used to ensure data quality and representativeness in the European Union. Despite being complex, the system demonstrates a certain stability, in terms of self-organization; while large monthly fluctuations cannot be considered as rare events, but part of the data-generating process.

Circulatory shock is one of the leading causes of mortality in intensive care units (ICUs), and its early detection is critical to enable timely treatment and improve clinical outcomes. This study aimed to develop and evaluate a two-stage cascade machine learning framework for the early detection and etiological classification of circulatory shock in critically ill patients. Using data from the Medical Information Mart for Intensive Care (MIMIC)-IV database, four patient groups were defined: septic shock, cardiogenic shock, hypovolemic shock, and a non?shock control group, comprising a total of 32,907 patients. Vital signs and laboratory data were collected during the first six hours after ICU admission. After data cleaning and missing-value imputation, the mean value of each variable was used for model development. Several machine learning algorithms were compared, including logistic regression, Random Forest, XGBoost, and multilayer perceptron (MLP) networks. Random Forest and XGBoost achieved the highest overall performance, with an AUROC of approximately 0.82-0.83 for shock detection, and a macro-averaged sensitivity of approximately 0.61 and precision of approximately 0.58 across all four classes. Classification performance was highest for the non?shock group, followed by septic and cardiogenic shock, while hypovolemic shock showed the lowest performance. These results indicate that machine learning models can identify early signs of hemodynamic deterioration associated with circulatory shock and may support clinical decision-making in the ICU. However, further improvements are needed for the classification of specific shock subtypes, particularly hypovolemic and cardiogenic shock, as well as for real-time clinical implementation.

We propose a novel joint sampling design for surveys involving two interrelated populations of parents and children in the presence of complex family structures, including shared parental responsibility and blended families. The design combines a graph-theoretic representation of co-parenting relationships with balanced sampling. This framework ensures that, in the first sampling phase, exactly one parent is selected from each co-responsibility pair while maintaining accurate inference for both the parent and child populations. A second sampling phase selects one child per sampled parent using unequal probabilities that produce consistent weighting and unbiased estimation. The proposed framework accommodates overlapping family relationships while balancing auxiliary information for both populations. Results based on the Luxembourg population registers show that the estimated totals are virtually identical to the known population totals for all balancing variables, and other auxiliary variables, demonstrating the excellent precision and efficiency of the proposed sampling design.

Tabular foundation models (TFMs) provide predictive distributions for regression, but their prediction regions can exhibit undercoverage or overcoverage even when point predictions are accurate. We introduce C-USIM (Conditionally-Uniformized Score Integration Method), a lightweight application of highest predictive density split conformal prediction that accommodates multimodal predictions. Given calibration and test outputs, it requires no additional training or model inference. It provides finite-sample marginal validity under our assumptions. We bound conditional-marginal coverage gaps using distribution-estimation error and score discreteness, and examine coverage heterogeneity through percentile rank-score plots. Experiments with TabPFN and TabICL show improved marginal coverage accuracy and lower average conditional and group coverage errors. Under a fixed data budget, allocating more observations to calibration can reduce marginal coverage error despite less accurate point predictions.

We present spsurv, an R package for semi-parametric time-to-event regression based on Bernstein-polynomial estimation of unknown baseline functions. The package provides a unified modelling interface for proportional hazards (PH), proportional odds (PO), and accelerated failure time (AFT) models for right-censored data, with either maximum likelihood or Bayesian estimation via Stan. Smooth baseline hazard, odds-function, or log-time structures are estimated without assuming a parametric baseline family, while retaining familiar hazard-ratio, odds-ratio, and time-ratio interpretations. We describe methodology, implementation, and syntax; evaluate finite-sample behaviour in a Monte Carlo study; and illustrate usage with oncology trials.

We study the maximum likelihood estimation of the coefficient {\beta} in well-specified Poisson regression. Using tools from empirical process theory and random conic geometry, we show that the probability of existence of the maximum likelihood estimator (MLE) exhibits a sharp phase transition at the threshold n > d. We then determine a minimum threshold exponential in the norm of {\beta} on the sample size n to guarantee with high probability an excess risk of the asymptotic order d/n. We reveal the existence of an intermediate regime in Poisson regression, when n is larger than d but smaller than this exponential threshold, where the MLE exists but does not achieve the optimal rate d/n. We close the gap between the two regimes up to a d^{1+\epsilon} term with {\epsilon} \in (0, 1) by providing an upper bound on the distance between the MLE and {\beta} whenever n > d^{1+\epsilon}. Along the way, we provide two generalizations of well-known PAC-Bayes inequalities regarding sub-Gamma random vectors and sub-Gamma random matrices that are of independent interest and that we use extensively to prove the main results of the present paper.

Adapting pretrained models to downstream tasks with limited data has become a central paradigm in modern deep learning. Yet, despite its widespread practical success, how fine-tuning leverages information from pretraining remains poorly understood theoretically. We study fine-tuning from pretrained weights through the lens of sparse linear regression and two-layer diagonal linear networks. In our setting, pretraining provides information through the support (and signs) of the initialization predictor, which may contain coordinates relevant to the downstream task. We show how pretrained information reshapes the implicit bias and training dynamics, and can thereby reduce the sample complexity of recovering the target parameters and support. In particular, for a clean initialization with correctly inherited signs, we show that the required sample size is comparable to that of a weighted Lasso estimator that explicitly exploits the pretrained support through a suitably chosen regularizer. Our results thus show how information encoded in pretrained weights can be implicitly exploited by gradient-based fine-tuning, reducing the amount of data needed to recover a downstream task.

In this paper, we formulate a general differencing framework for variance estimation across a range of settings. We demonstrate that conventional difference-based noise variance estimators cannot achieve the desired bias-correcting power in nonparametric regression with repeated measurements. A new high-order bias-corrected differencing scheme, adapted to repeated measurements, is proposed by interlacing inter-group and intra-group differencing. The theoretical properties of the new sequences and estimators are studied. Our proposals are particularly efficient in finite samples and under high signal-to-noise ratio scenarios, where asymptotic convergence has not yet fully taken effect, due to their strong bias-correcting power.

We develop an information-theoretic framework for generalization in next-token prediction under temporally dependent data. We consider independent trajectories generated by finite-memory Markov processes and distinguish algorithmic dependence, quantified by mutual information, from temporal dependence, characterized by mixing. For cross-entropy loss, we derive an expected generalization bound using the Donsker–Varadhan variational representation and a McDiarmid-type concentration inequality for Markov chains. A refinement captures the joint effect of context length and temporal mixing through the mixing properties of the history-state process. We then extend the bound through a rate–distortion formulation, replacing mutual information with the minimum information rate required to represent the learned model within a prescribed distortion in the generalization gap, yielding informative guarantees for deterministic algorithms over continuous hypothesis spaces. For margin-based prediction, we derive explicit bounds for linear and self-attention next-token predictors via noisy low-dimensional compression, revealing the roles of context length, model complexity, sample size, margin, and temporal mixing. Experiments on TinyStories and ETTh2 show that longer contexts can reduce both training and test losses, but typically reduce training loss more, enlarging the generalization gap. A complementary ETTh2 analysis identifies an effective predictive-memory scale near 24 hours, with no statistically supported improvement beyond this scale, offering a plausible explanation for test-performance saturation at larger contexts.

Clustering problems increasingly involve complex objects observed over space, such as distributions, matrices, functions, images, or multivariate data, for which a scientifically meaningful dissimilarity between objects is often easier to specify and computationally more tractable than an object-response likelihood model. We propose DISCCO, a Bayesian framework for clustering spatially indexed complex objects using only a pairwise distance matrix and a spatial adjacency graph, with broad applicability and minimal user modeling requirements. The model combines a hierarchical distance-based likelihood accounting for within-cluster compactness and between-cluster separation with a random spatial graph partition prior, ensuring that posterior clusters are spatially contiguous. A key model feature is a set of node-specific frailty parameters that induce dependence among overlapping within-cluster distances and provide posterior summaries of object-level centrality or peripherality within each inferred cluster. We develop a partially collapsed Markov chain Monte Carlo algorithm for posterior inference. Simulations with distribution- and matrix-valued responses show that the proposed spatial distance-clustering framework improves region recovery relative to existing distance-clustering methods, while the frailty layer provides interpretable centrality summaries. Real applications to Houston Census Block Group racial-composition distributions and Western US county-level cancer mortality matrices illustrate how the method recovers interpretable contiguous clusters and frailty-based centrality maps.

Fine-tuning pre-trained models on specialized tasks with scarce data is central to modern deep learning. Despite its empirical success, theoretical understanding of fine-tuning remains limited. We introduce a Gaussian multi-index setting to study fine-tuning from pre-trained weights, where the teacher network has $m+1$ features, $m$ of which are learned during pre-training and one of which must be learned during fine-tuning. For two-layer ReLU networks, we show that two-timescale training, i.e., updating the outer weights infinitely faster than the hidden ones, learns the new task-specific feature while preserving the pre-trained ones in the model representation. Moreover, only $\mathcal{O}(d)$ fine-tuning samples are required for this recovery, independently of the number of pre-trained features. In contrast, with random initialization, the same number of samples is insufficient to recover the target parameters. Our results therefore demonstrate that pre-training can induce an implicit bias with a clear statistical advantage over random initialization, enabling feature learning from scarce fine-tuning data.

We study approximate sampling: given $N$ independent samples from a proposal distribution $\mu$, the goal is to select one whose distribution is close to a target $\pi$ specified only up to a normalizing constant. Block and Polyanskiy (2023) provide finite budget error bounds for approximate rejection sampling (RS) as a function of the acceptance threshold $M$. The threshold $M$ giving the smallest bound, however, depends on properties of $(\pi,\mu)$ that are typically unavailable from the observed sample. This raises a natural question: Can one attain the best RS guarantee without taking $M$ as input? We answer affirmatively by proposing a parameter-free sampling algorithm called uniform race (UR), based on importance weights, which are ratios of target to proposal probabilities (or densities). It divides each observed weight by an independent uniform random variable to form a score and returns the candidate with the largest score. For every budget $N$, its total variation error satisfies the RS upper bound for every fixed threshold $M$ simultaneously, thereby achieving the best such bound in hindsight. We also characterize its output distribution conditional on the largest score, identifying when it is exactly the target $\pi$. Uniform race has no larger total variation error than a natural budget-calibrated RS derived from Rohatgi et al. (2025) and sampling importance resampling (SIR). In particular, we exhibit instances where UR's error is exponentially smaller in $N$ than that of either baseline. Furthermore, we establish conditions under which attaining this RS guarantee for every $(\pi,\mu)$ uniquely determines the selection probabilities as those of UR. Finally, test-time scaling experiments on LLM math-reasoning tasks corroborate the theoretical comparisons and demonstrate that UR remains competitive in ground-truth accuracy without requiring threshold selection.

Flow matching (FM) has recently emerged as a promising framework for generative modeling due to its conceptual simplicity and strong empirical performance. In FM, samples are transported along a vector field parameterized by a neural network, inducing a probability path that evolves from a simple noise distribution to the target data distribution, governed by an ordinary differential equation (ODE). However, existing FM approaches predominantly rely on probability paths derived from optimal transport (OT) between Gaussian distributions, which may be suboptimal for capturing complex data with inhomogeneous structures such as heavy tail or sharp contrast. In this work, we generalize FM to the broader class of location-scale families for handling data inhomogeneity and introduce a novel class of probability paths defined as geodesics on the manifold of probability distributions. We name this approach probabilistic geodesic flow matching to distinguish it from prior geodesic (Riemannian) FM methods defined in input space. We argue that Euclidean OT-based paths are not necessarily optimal in probability space and may limit modeling flexibility. Through synthetic benchmarks and scientific datasets at different scales, we demonstrate that the proposed method more effectively captures complex distributions, leading to improved or comparable performance compared with SOTA geometry-motivated generative models.

We study the problem of learning directed acyclic graphs (DAGs) with nonnegative edge weights from observational data. We propose the Nonnegative and Concomitant (NoCo) DAG estimator, which jointly recovers the weighted graph structure and the exogenous noise variances in the linear structural equation model for the observations. Different from prior art, this noise-adaptive formulation blends a smoothed concomitant lasso criterion with a simpler log-determinant acyclicity constraint that exploits nonnegativity and yields a more benign optimization landscape. Specifically, nonnegative weights allow us to impose acyclicity directly on the adjacency matrix without elementwise squaring of its entries, thus avoiding the well-documented degeneracy of the Karush-Kuhn-Tucker conditions. Computationally, we develop a method of multipliers' algorithm that accommodates both homoscedastic and heteroscedastic noise profiles. Within each iteration, we use block successive convex approximation to minimize the augmented Lagrangian, alternating between proximal gradient steps for the adjacency matrix and closed-form updates for the noise scales. Simulated experiments demonstrate NoCo's improved structural and edge-weight recovery relative to competing methods in a variety of settings, highlighting the benefits of exploiting nonnegativity along with noise adaptivity.

Off-policy evaluation is a foundational component of offline reinforcement learning, aiming to assess and optimize policy performance using pre-collected datasets. However, such datasets often suffer from pronounced challenges, including distribution shift, $Q$-value overestimation, and low sample utilization efficiency. To address these issues, this paper introduces a weighted Bellman residual minimization framework that incorporates density ratio weighting by effectively integrating expert demonstrations with behavioral data. The proposed weighting scheme departs from the conventional completeness assumption commonly imposed in the theoretical analysis of deep reinforcement learning. We establish a sharp convergence rate for density ratio estimation and derive the convergence rate for the excess risk of resulting deep $Q^*$ estimator. Extensive empirical evaluations demonstrate that, compared to existing methods, our method achieves significant improvements in numerical performance and policy generalization, providing specific guidance for the rational utilization of expert demonstrations.

In target trial emulation, the treatment strategies that patients follow are inferred from the treatments they actually receive in routine care. However, in most settings, the outcome may preclude the observation of planned treatment, giving rise to immortal time bias through misclassification of treatment strategy, while markers of treatment response may influence subsequent treatment decisions, giving rise to time-varying confounding. A key step toward unbiased treatment effect estimation is to partition follow-up into sufficiently short time intervals to unfold the feedback relationships involving treatment and represent the resulting causal relations with a directed acyclic graph. In this study, we present the possible within-interval causal orderings induced by this partitioning, discuss their causal implications, and assess their plausibility across clinical settings. For each causal ordering, we derive the corresponding g-formula. Using ancestral multi-world networks and simulations, we show that the standard cloning-censoring-weighting estimator is invalid when treatment affects the outcome within a time interval, and we propose a modified version of the method that restores its validity in this setting. Finally, we analyze the consequences of choosing time intervals that are either excessively wide or excessively narrow, thereby formally establishing the need for time partitioning and providing practical guidance for selecting an appropriate partition based on the clinical setting.

We propose a Bayesian nonparametric factor model that infers the number of factors, induces row-wise sparsity, and merges redundant dictionary elements via exact clustering. A Dirichlet process prior is placed on the columns of an overcomplete loading matrix and fully marginalized to an exact Pólya urn, avoiding stick-breaking and auxiliary variables. A spike-and-slab base measure allows entire factors to be exactly zero. The model uniquely combines exact zeros, exchangeability over columns, and exact merging within a single marginalized Dirichlet process, unlike CUSP, MGP, or the beta process. An exact Gibbs sampler with canonical relabeling and parallel C/MPI implementation is developed. We prove posterior contraction at rate $\sqrt{M s_0 \log n / n}$ for the covariance matrix, and in the fixed-dictionary setting obtain the minimax optimal rate $\sqrt{s_0 \log n / n}$ plus underfitting consistency; the overfitting direction is an open conjecture. The spike-and-slab is essential: without it the effective dimension scales as $pM$, yielding a slower rate. Simulations show the method is the only fully adaptive approach to recover the true rank, achieving the smallest covariance, loading, and signal-reconstruction errors, beating an oracle baseline. On van 't Veer breast cancer data ($n=97$, $p=1213$), the posterior concentrates on eight interpretable programmes; seven pass coherence and two pass Bonferroni-corrected Hallmark enrichment.