Papers

Multi-judge evaluation is increasingly used to assess LLMs and reward models, and the prevailing heuristic is to curate: keep the most accurate judges and discard weaker ones. We show that this heuristic can reverse when the target is not point accuracy, but calibrated probabilistic evaluation from a labeled calibration set. Holding the aggregation and calibration procedures fixed, we compare accuracy-ranked top-$k$ judge selection with using the full judge panel. Across four labeled pairwise-evaluation benchmarks spanning LLM-as-judge and reward-model settings, the calibrated full panel consistently outperforms accuracy-based selection. On RewardBench2, retaining all judges achieves negative log-likelihood (NLL) of $0.006$ versus $0.013$ under top-5 selection, halving the calibration error. This advantage persists after judge-family deduplication and against stronger same-pipeline subset search. We explain this reversal with oracle analyses showing that the optimal calibrated risk under proper scoring rules cannot increase when additional judge signals are made available, and that even below-chance judges can be useful when their biases are learnable and their signals are non-redundant. The resulting operating principle is simple: in multi-judge evaluation with labeled calibration data, do not discard weak judges by accuracy alone; keep them when they are parseable, non-redundant, and calibratable.

Although spatial models for areal data are widely used in multilevel settings, the conditions under which spatial and nonspatial random effects yield equivalent posterior inference for regression coefficients have never been formally characterized. We address this question within a hierarchical Bayesian framework for Gaussian outcomes, using the Leroux conditional autoregressive (CAR) prior distribution as a representative specification. We derive a closed-form sample size threshold, $m^*$, below which spatial modeling materially affects inference on regression coefficients and above which a simpler nonspatial model yields effectively equivalent results, and show that the absolute relative difference in posterior variances converges to zero at rate $O(m^{-1})$. The threshold depends on three interpretable quantities: the spatial correlation parameter, the ratio of between-area to within-area variance, and the alignment between the covariate and dominant spatial patterns in the data. Because each can often be estimated prior to model fitting, $m^*$ can serve as a practical study design tool. Simulation studies confirm that $m^*$ accurately identifies this threshold across a range of settings. However, when the covariate does not vary within a given location, spatial modeling remains necessary regardless of within-area sample size. These results offer formal guidance for practitioners deciding whether the added complexity of spatial modeling is warranted.

Sampling from score-based diffusion models incurs bias due to both time discretisation and the approximation of the score function. A common strategy for reducing this bias is to apply corrector steps based on the unadjusted Langevin algorithm (ULA) at each noise level within a predictor-corrector framework. However, ULA is itself a biased sampler, as it discretises a continuous diffusion process. In this work, we consider adjusted Langevin correctors that employ Metropolis–Hastings (MH) or Barker's accept-reject steps to correct for this bias. Since the target density ratio typically required by MH-based algorithms is unavailable, we propose methods that instead utilise the score function to compute the correct acceptance probability. We introduce the first exact method for adjusting Langevin corrections in diffusion models, based on a two-coin Bernoulli factory algorithm. We also propose an efficient approximation based on Simpson's rule that achieves accuracy of order $5/2$ in the step size at near-zero marginal cost. We demonstrate that these procedures improve sample quality on both synthetic and image datasets, yielding consistent gains in Fréchet Inception Distance (FID) on the latter.

Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted submatrix in the mean matrix $\mathbf X$. Specifically, we aim to distinguish the null hypothesis $\mathbf X = 0$ from the alternative hypothesis in which $\mathbf X$ is non-zero only on a submatrix of size $s_1 \times s_2$ with elevated entries bounded below by $\mu>0$. We establish a minimax lower bound characterizing how large $\mu$ must be to ensure that the two hypotheses are distinguishable with high probability. Furthermore, we derive novel minimax-optimal tests achieving the lower bound, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. In contrast with previous work, which required restrictive assumptions on $s_1,s_2, d_1$ and $d_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters.

We introduce Dirichlet process mixtures of marked Poisson point processes, a Bayesian nonparametric model for clustering replicated marked point patterns. The model jointly infers latent cluster structure, the number of clusters, and continuous mark-specific intensity surfaces with uncertainty quantification, a combination existing methods provide only in part. Using a squared-link intensity whose likelihood is tractable without gridding or thinning, we develop, to our knowledge, the first variational Bayes method for clustering marked Poisson point processes. Its constrained Laplace approximation eliminates sign ambiguity and nodal lines by turning each coefficient update into a well-posed convex problem, attaining computational efficiency and inferential fidelity with theoretical guarantees of mode consistency, exponential-weight concentration, and uniform total-variation accuracy. Our method outperforms competitors in accuracy and speed on synthetic data, and an NBA shot-chart analysis uncovers offensive archetypes.

Recently, Muon and related spectral optimizers have demonstrated strong empirical performance as scalable stochastic methods, often outperforming Adam. Yet their behaviour remains poorly understood. We analyze stochastic spectral optimizers, including Muon, on a high-dimensional matrix-valued least squares problem. We derive explicit deterministic dynamics that provide a tractable framework for studying learning behaviour with a focus on (stochastic) SignSVD, which Muon approximates, and (stochastic) SignSGD, the latter serving as a proxy for Adam. Our analysis shows that for large batch size, SignSVD performs a square-root preconditioning with respect to the data covariance spectrum, while for small batch size smaller eigenmodes behave like SGD, slowing down convergence. We contrast with SignSGD which for generic covariance performs no preconditioning and has no transition, leading to different optimal learning rates and convergence characteristics. The two methods match up to a constant factor with isotropic data, but behave differently with anisotropic data. An analysis of a power law covariance model with data exponent $\alpha$ and target exponent $\beta$ shows there are three phases in the $(\alpha,\beta)$ plane: one where SignSGD is uniformly favored, one where SignSVD is uniformly favored, and a third where the two methods exhibit a trade-off in performance.

When testing a number of statistical hypotheses using data from location families, it is often useful to control the false discovery rate (FDR) not just for hypotheses of the null values but also of other parameter values that are deemed practically insignificant. Here we consider FDR as a curve indexed by the location parameter and suggest a simple generalization of the Benjamini-Hochberg procedure that controls the FDR curve below any user-specified level. As a corollary of our main result, we show that the standard Benjamini-Hochberg procedure – designed to control the FDR at the null – also provides simultaneous control of the whole FDR curve for free. We further demonstrate the implications of our results and some practical considerations with a numerical example.

Neural operators provide fast surrogate models for time-dependent partial differential equations, but their standard autoregressive use usually assumes that the instantaneous field $u(t,\cdot)$ is a complete state. This assumption fails for delay equations, distributed-memory systems, and other non-Markovian dynamics: two trajectories may agree at time $t$ and nevertheless have different futures because their histories differ. We introduce the History-Space Fourier Neural Operator (HS-FNO), a neural operator for delay and memory-driven PDEs formulated on the lifted state $u_t(\theta,x)=u(t+\theta,x)$, $\theta\in[-\tau,0]$. The key computational step is to decompose one history-state update into a learned predictor for the newly exposed future slice and an exact shift-append transport for the portion of the history window already known from the previous state. This avoids learning deterministic history coordinates, reduces the learned output dimension, and enforces the natural discrete history update. We test HS-FNO on five benchmark families covering delayed reaction–diffusion, spatial epidemiology, nonlocal neural-field dynamics, delayed waves, and distributed-memory closures. Across ten random seeds, HS-FNO attains the lowest aggregate one-step, history-space, and rollout errors among the principal baselines. The largest gain occurs in autoregressive prediction, where aggregate rollout error decreases from $0.241$, $0.188$, and $0.185$ for current-state, lag-stack, and unconstrained history-to-history operators, respectively, to $0.094$. The same model uses fewer parameters than unconstrained history prediction. These results indicate that enforcing the discrete shift structure of history-state evolution is an effective inductive bias for non-Markovian PDE surrogate modeling.

The problem of predicting unobserved entries in a binary matrix, known as 1-bit matrix completion, has found diverse applications in fields such as recommendation systems. In this study, we develop an empirical Bayes method for 1-bit matrix completion motivated by the Efron–Morris estimator, a matrix generalization of the James–Stein estimator that shrinks singular values toward zero. The proposed method exploits the underlying low-rank structure of binary matrices, drawing parallels with multidimensional item response theory. Simulation studies and real-data applications demonstrate that the proposed method achieves competitive predictive accuracy and favorable predictive calibration.

In Bayesian phylogenetics, our goal is to estimate the posterior distribution over phylogenetic trees. Markov chain Monte Carlo methods are widely used to approximate the phylogenetic posterior distributions. For large-scale sequence data, repeated evaluation of the likelihood function incurs a high computational cost. In this article, we propose a machine-learning algorithm with over 35 topological and branch-length features to predict the changes in the likelihood function caused by tree moves (\eg,~eSPR, stNNI) used in standard MCMC approaches. This algorithm is then used to design a delayed acceptance MCMC kernel, which utilized the predicted surrogate function for preliminary rejection, to accelerate tree space searches. Furthermore, we integrate our proposed MCMC kernel into the sequential Monte Carlo sampler framework. We validate the proposed delayed-acceptance sequential Monte Carlo approach (DA-SMC) on simulation and real data sets. Our delayed acceptance kernel can maintain robust estimation while reduces the number of likelihood evaluations significantly, yielding substantial computational time savings. We develop a Python package that is available at https://github.com/wentYu/DAphyloSMC.

Latent representations learned by neural networks often exhibit semantic structure, where concept similarity is reflected by geometric proximity in embedding space. However, comparing such spaces across models remains difficult: changes in architecture, pretraining data, objective, or random seed can yield embeddings with similar content but incompatible geometry. This latent space alignment problem is central to interpretability, transfer and multimodal learning, federated systems, and semantic communication; however, progress remains limited by the lack of large-scale, model-diverse, and metadata-rich benchmarks. To address this gap, we introduce SEMASIA, a large-scale collection of latent representations extracted from approximately 1,700 pretrained vision models across eight standard image-classification benchmarks. SEMASIA pairs embeddings with structured metadata describing architectures, training regimes, pretraining sources, and model scale. We demonstrate three applications of the resource. First, we analyze the conceptual organization of individual latent spaces, showing consistent prototype-like clustering and hierarchical semantic neighborhoods across models and datasets. Second, we benchmark supervised alignment mappings between latent spaces using reconstruction error and downstream task performance. Third, we perform a large-scale regression analysis of how pretraining-data complexity, specialization, transfer learning, augmentation, and model scale relate to geometric and probing properties of embeddings. By coupling representational scale with standardized metadata, SEMASIA provides a reproducible foundation for studying latent geometry, evaluating alignment methods, and developing next-generation heterogeneous and interoperable AI systems.

Multi-source transfer learning can improve target-domain estimation by leveraging related source data, but its benefits depend on unknown source-to-target biases. This raises a fundamental question: can a bias-agnostic estimator perform as well as an oracle that knows the true bias configuration? To study this, we introduce the intrinsic cost of adaptation, defined as the smallest worst-case ratio between the risk of any bias-agnostic estimator and the oracle risk. An intrinsic cost of one means oracle performance is achievable without knowing the biases, whereas a larger cost quantifies the unavoidable price of adaptation. Focusing on parametric estimation, we show that multi-source transfer behaves fundamentally differently from the single-source setting: adaptation is not always possible, even with only two sources. For a fixed number of sources, we characterize the intrinsic cost of adaptation and identify a phase transition separating regimes where oracle performance is achievable from those where it is not. As the number of sources grows, we further show that the adaptation cost increases. When adaptation over the full bias configuration space is impossible, additional structure can substantially reduce the cost. We study settings with ordered biases, clustered source parameters, and sufficiently separated non-informative sources, and propose estimators tailored to each regime, with supporting theoretical and empirical results. Overall, our results delineate the statistical limits of multi-source transfer, clarifying when oracle performance is attainable, when structural assumptions help, and when adaptation is fundamentally impossible.

Causal mediation analysis has been extended to estimate path-specific effects with multiple intermediate variables, isolating treatment effects through a mediator of interest while excluding pathways through its ancestors. Such analyses address bias from recanting witnesses, i.e., treatment-induced mediator-outcome confounders. However, existing methods typically rely on stringent assumptions precluding general unmeasured confounding, which are often violated in practice. In this paper, we relax these restrictions by leveraging observed covariates as proxy variables to accommodate unmeasured confounding among the treatment, recanting witness, mediator, and outcome. Using proximal confounding bridge functions, we develop four nonparametric identification strategies for the path-specific effect. We further derive the efficient influence function and propose a quadruply robust, locally efficient estimator. To handle high-dimensional nuisance parameters, we propose a proximal debiased machine learning approach. We theoretically guarantee that our estimator achieves $\sqrt{n}$-consistency and asymptotic normality even when machine learning estimators for nuisance functions converge at slower rates. Our approaches are validated via semiparametric and nonparametric simulations and an application to the CDC WONDER Natality study, estimating the path-specific effect of prenatal care on preterm birth through preeclampsia, independent of maternal smoking during pregnancy.

Stein Variational Gradient Descent (SVGD) is a deterministic interacting-particle method for sampling from a target probability measure given access to its score function. In the mean-field and continuous-time limit, it is known that the flow converges weakly toward the target, but no quantitative rate is known for the last iterate. In this paper, we establish quantitative local convergence in strong norms for this dynamics, when the interaction kernel is of Riesz type on the $d$-dimensional torus. Specifically, assuming that the initial density and the target are smooth and close in $L^2$-norm, we obtain explicit polynomial convergence rates in $L^2$-norm that depend on the dimension and on the regularity parameters of the kernel, the initialization and the target. We further show that these rates are sharp in certain regimes, and support the theory with numerical experiments. In the edge case of kernels with a Coulomb singularity, we recover the global exponential convergence result established in prior work. Our analysis is inspired by recent results on Wasserstein gradient flows of kernel mean discrepancies.

We study the $\textit{single-index bandit}$ problem, where rewards depend on an unknown one-dimensional projection of high-dimensional contexts through an unknown reward function. This model extends linear and generalized linear bandits to a nonparametric setting, and is particularly relevant when the reward function is not known in advance. While optimal regret guarantees are known for monotone reward functions, the general non-monotone case remains poorly understood, with the best known bound being $\tilde{\mathcal{O}}(T^{3/4})$ (under standard boundedness and Lipschitz assumptions on the reward function [Kang et al., 2025]). We close this gap by establishing the optimal regret for general single-index bandits. We propose a simple two-phase algorithm, namely, Zoomed Single Index Bandit with Upper Confidence Bound ($\texttt{ZoomSIB-UCB}$), that first estimates the projection direction via a normalized Stein estimator, and then reduces the problem to a one-dimensional bandit using discretization and finally use UCB. This approach achieves a regret of $\tilde{\mathcal{O}}(T^{2/3})$, and improves significantly upon prior work without any additional assumptions. We also prove a matching minimax lower bound of $\tilde{\Omega}(T^{2/3})$, showing that the upper bound is essentially tight. Our upper and lower bounds together provide a sharp characterization of the regret in single-index bandits. Moreover, the empirical results further demonstrate the effectiveness and robustness of our approach.

Generative models are powerful tools for sampling from a learned distribution $\mathcal{P}(Y \mid X)$, and inverse-design methods invert this map to find an input $x$ that produces a desired point output $y^*$. However, many design goals are naturally distributional rather than pointwise, incorporating the inherent uncertainty of $Y$ and targeting a specific form for it, a task not addressed by standard inverse design. To address this issue we introduce Conditional Distribution Matching (CDM), a new inverse-design problem class in generative modeling: given a joint distribution $\mathcal{P}(X, Y)$ and a target distribution $\mathcal{G}(Y)$, find an input $x^*$ whose induced conditional distribution $\mathcal{P}(Y \mid X = x^*)$ matches $\mathcal{G}$. We formally define two variants: Conditional Distribution Matching Sampling (CDMS) and Conditional Distribution Matching Optimization (CDMO). To solve these problems, we propose MLGD-F (Matching-Loss Guided Diffusion with a Fast inner sampler), a plug-and-play inference-time algorithm that combines a pretrained score-based diffusion model with a pretrained fast conditional sampler, requiring no additional training or fine-tuning. By leveraging single-step conditional sampling, MLGD-F enables tractable gradient computation, making the estimation of $\mathcal{P}(Y \mid X)$ both memory-efficient and computationally lightweight. We validate MLGD-F on synthetic benchmarks, structured image transformations, and generative editing optimization, demonstrating reliable recovery of inputs whose conditional distributions match diverse user-specified targets, including discrete mixtures and continuous low-rank supports.

Link prediction (inferring missing or future connections between nodes in a graph) is a fundamental problem in network science with widespread applications in, e.g., biological systems, recommender systems, finance and cybersecurity. The ability to accurately predict links has significant real-world applications, such as detecting fraudulent financial transactions or identifying drug-target interactions in biomedicine. Despite a rich literature, link prediction is still challenging, especially for graphs enriched with information on edges (direction) and nodes (attributes). In fact, research on link prediction, especially the one based on Graph Deep Learning (GDL), has mostly focused on undirected graphs, without fully leveraging node attributes. Here, we fill this gap by proposing Gravity-GraphSAGE (GG-SAGE), a modified version of GraphSAGE, a GDL model for node embeddings, composed of a gravity-inspired decoder. This implementation is the first example in the literature of a GraphSAGE backbone adopted for directed link prediction. Using the benchmark datasets Cora, Citeseer, PubMed and 16 real-world graphs from the online Netzschleuder repository, we show that our proposed model outperforms state-of-the-art GDL link prediction techniques. Using further experimental evidence, we relate the quality of the output of our model with various characteristics of the graph, suggesting that our framework scales well when applied to data of increasing complexity.

This paper relaxes the restrictive symmetry conditions adopted in [4], [5] and extends their universal feature selection framework to accommodate noisy observations as well as attribute structures that may exhibit directional preferences. We introduce the notion of weak spherical symmetry, quantified by second-moment distances, which allows controlled deviations from rotational invariance. Under this relaxed condition, we develop a universal feature selection framework based on the singular value decomposition of the canonical dependence matrix computed from noisy data. Our main result shows that the selected features achieve asymptotically optimal error exponents up to a residual term that depends on the symmetry deviation $\delta$ and the noise levels $\eta_1, \eta_2$. When $\delta, \eta_1, \eta_2$ are relatively small, our result recovers that of [5], thereby demonstrating that exact spherical symmetry is unnecessary. Overall, our findings highlight the robustness of the selection framework against second-moment deviations and observation noise, thereby broadening its applicability across diverse inference tasks and providing a theoretically grounded tool for universal feature selection in practical scenarios.

Interactive assessments generate sequential process data that are not well handled by conventional item response models. Existing MDP-based measurement approaches, such as the Markov decision process measurement model (MDP-MM, LaMar, 2018), link action choices to state-action values, but their reliance on person-specific tabular value functions makes them difficult to scale beyond small, fully enumerated tasks. We propose the Reinforcement Learning Measurement Model (RLMM), a measurement framework that decouples person-level choice sensitivity from task-level value representation through a shared parametric action-value function, making estimation more computationally efficient for larger process-data settings. The model combines a Boltzmann choice rule with normalized advantages, a soft Bellman consistency penalty, and a block-coordinate MAP procedure for joint estimation, while also yielding step-level influence diagnostics for identifying behaviorally critical decisions. In peg-solitaire simulations, the RLMM achieved higher estimation accuracy and substantially lower runtime than the original MDP-MM, with advantages increasing as task complexity grew. In AQUALAB gameplay logs, the estimated person parameter was positively associated with cumulative reward, task completion, and behavioral efficiency. These results show that the RLMM extends decision-process-based psychometric models to larger and more behaviorally realistic environments while preserving an interpretable latent trait tied to decision making steps.

Identifying covariates that modify treatment effects is a central problem in causal inference. Yet existing data-adaptive procedures do not provide finite-sample control over the expected number of false discoveries, risking spurious findings that fail to replicate. We introduce causal stability selection, an algorithm that combines cross-fitted estimation of conditional average treatment effects with integrated path stability selection. The method accommodates arbitrary treatment effect estimators and arbitrary base selectors, and produces a selection set with an explicit, non-asymptotic bound on the expected number of false positives. Under standard causal identifying assumptions and regularity conditions on the base selector, we prove that the estimated selection probabilities converge to their oracle counterparts at the rate of the underlying treatment effect estimator. This establishes a direct connection between treatment effect estimation and effect modifier discovery. We illustrate the method on simulated data and on real-world data from a randomized oncology trial and an observational study of maternal smoking and infant birthweight.

Discrete flow models (DFMs) are a class of flexible generative models for generating discrete data, and diffusion large language models (dLLMs) can be viewed as a special case with a specific choice of a mixture path and a masked source distribution. While several recent works have explored reinforcement learning for dLLMs, its application to more general discrete flow models remains underexplored. In this work, we present discrete Flow-GRPO (dFlowGRPO), a unified reinforcement learning framework for discrete flow models that supports a broad family of probability paths and non-masked source distributions. We derive the full trajectory probability for DFMs and formulate the denoising process as a Markov decision process, enabling dFlowGRPO to incorporate information from both the associated conditional transition rates and the posterior model during reinforcement learning. We apply dFlowGRPO to FUDOKI, a recent multimodal discrete flow model, and evaluate it on both image generation and multimodal understanding tasks. Empirical results show that dFlowGRPO outperforms existing GRPO-type methods for dLLMs on text-to-image generation tasks and achieves performance competitive with continuous flow-based models trained using Flow-GRPO, while also demonstrating strong capabilities on understanding tasks.

We study target-population quantile treatment effects when a source study may have unmeasured treatment confounding and may not transport to a target population after conditioning on observed covariates. The observed data consist of a source sample with treatment, outcome and covariates, and a target sample with covariates only. We impose two marginal sensitivity restrictions: an odds-ratio bound \(\Gam\) for source treatment assignment and a conditional likelihood-ratio bound \(\Lam\) for source-to-target potential-outcome distribution shift. For each treatment arm and threshold \(y\), we derive a closed-form sharp target counterfactual CDF envelope. The envelope nests a source marginal-sensitivity map inside a target outcome-shift map, preserving two normalizations and generally improving on a single product likelihood-ratio relaxation. We prove process-level sharpness, so the envelopes are attainable as entire CDFs and can be inverted to obtain sharp target quantile bounds and sharp interval-hull QTE bounds. We then develop semiparametric theory for these nonsmooth bound processes. On regular index sets, we give the canonical gradient, including the source propensity contribution required in observational studies, and construct cross-fitted Neyman-orthogonal one-step estimators with uniform Gaussian approximation. On full index sets with active-set ties or mass points, we use Hadamard directional differentiability and subsampling-valid inference, with a primitive finite-support route for the required weak convergence. Finally, we invert simultaneous monotone CDF bands to obtain honest confidence sets for quantile and QTE interval-hull processes, and formulate the two-dimensional \((\Gam,\Lam)\) breakdown frontier as level-set inference for interval-hull non-refutation.

Proximal causal learning uses the treatment-inducing proxy and the outcome-inducing proxy through the outcome bridge and the treatment bridge to provide a way to address unobserved confounding in observational studies. This paper studies simultaneous inference for the marginal counterfactual distribution and quantile treatment effect under weak proxies. First, we propose a measure of weak-proxy strength and tightens the population-bias bound for the counterfactual CDF estimate. The baseline Cauchy–Schwarz bound introduces the inverse of weak-proxy strength twice, whereas the proposed bound introduces the inverse of weak-proxy strength only once and is sharp for the bridge estimators, thereby relaxing the restriction on the decay rate of weak-proxy strength. Second, we prove that, even when outcome thresholds vary continuously, the number of distinct estimated outcome bridges is at most one greater than the number of observations in the training sample, thereby establishing a uniform expansion across treatment arms and outcome thresholds. Finally, we prove that the critical value remains valid after bridge estimation, thereby obtaining simultaneous CDF bands, and constructs quantile treatment effect bands through confidence-band inversion. The proposed method provides deterministic band-width bounds and does not require counterfactual density estimation. In the numerical study, the baseline Cauchy–Schwarz bound overestimates the true population bias by a factor of approximately 50, whereas the proposed bound is very close to the true population bias. The quantile treatment effect bands constructed by the proposed method achieve coverage of 99.8%–100.0% while being only approximately 20% wider than the oracle 95% bands, compared with baseline coverage of 68.4%–89.2%.

We introduce a use of the \(M\)-cover (or \(M\)-layer) transform for machine learning. The method replicates a model \(M\) times, but instead of coupling the copies through parameter averaging or an explicit attractive force, as in replicated SGD or Elastic SGD, it rewires the contexts in which local learning messages are computed. Each local loss is evaluated on a routed model whose parameters are drawn from different copies according to permutations sampled from a structured mixing kernel \(Q\). Training then uses the original local update rule, while the resulting learning messages are redistributed across the copies through these routed computational paths. Thus \(Q\) defines a topology for message transport and controls the long-loop structure of the lifted factor graph. We formulate this construction for perceptrons, committee machines, and multilayer perceptrons, showing that the same principle applies from discrete models to differentiable neural networks. The resulting framework provides a mechanism for improving generalization through structured message sharing rather than replica collapse or parameter-space coupling.

One-step generative modeling has emerged as a leading approach for amortizing the inference cost of diffusion and flow-matching models. Among distillation-free methods, MeanFlow training is notoriously unstable, with non-decreasing loss and unbounded gradient variance. In this work, we establish a theory that attributes this pathology to a misuse of the conditional velocity field. We show that the conditional velocity plays two distinct statistical roles in the loss: both as an unbiased regression target and as a Monte Carlo control variate in a Jacobi-vector product, with the original MeanFlow loss assigning the wrong coefficient to the latter. We derive the optimal coefficient in closed form and show that a family of fixes in concurrent works corresponds to different practical realizations of the same optimum. A controlled sweep of this coefficient on two-dimensional benchmarks and on a latent Diffusion Transformer recovers the predicted bias-variance ordering. Our DiT experiment also reveals a quantitative FID-MSE landscape mismatch. Specifically, although the gradient-MSE is minimized at an interior coefficient value near $\beta\!=\!0.94$, the coefficient that minimizes FID prefers to use conditional velocity directly at the unbiased corner. Our analysis therefore explains why MeanFlow is unstable and unifies its concurrent remedies, and shows that the variance-optimal coefficient need not coincide with the quality-optimal one.

Deep generative models provide flexible frameworks for modeling complex, structured data such as images, videos, 3D objects, and texts. However, when applied to sequences of human skeletons, standard variational autoencoders (VAEs) often allocate substantial capacity to nuisance factors-such as camera orientation, subject scale, viewpoint, and execution speed-rather than the intrinsic geometry of shapes and their motion. We propose the Elastic Shape - Variational Autoencoder (ES-VAE), a geometry-aware generative model for skeletal trajectories that leverages the transported square-root velocity field (TSRVF) representation on Kendall's shape manifold. This representation inherently removes rigid translations, rotations, and global scaling of shapes, and temporal rate variability of sequences, isolating the underlying shape dynamics. The ES-VAE encoder maps skeletal sequences to a low-dimensional latent space incorporating the Riemannian logarithm map, while the decoder reconstructs sequences using the corresponding exponential map. We demonstrate the effectiveness of ES-VAE on two datasets. First, we analyze skeletal gait cycles to predict clinical mobility scores and classify subjects into healthy and post-stroke groups. Second, we evaluate action recognition on the NTU RGB+D dataset. Across both settings, ES-VAE consistently outperforms standard VAEs and a range of sequence modeling baselines, including temporal convolutional networks, transformers, and graph convolutional networks. More broadly, ES-VAE provides a principled framework for learning generative models of longitudinal data on pose shape manifolds, offering improved latent representation and downstream performance compared to existing deep learning approaches.

We present TailedTS, a large-scale benchmark dataset derived from Wikipedia hourly page view observations throughout 2024, specifically designed to test time series forecasting models under heavy-tailed, zero-inflated, and non-Gaussian conditions. The dataset comprises approximately 24.69 billion data points spanning roughly 3 million unique Wikipedia pages per month, stored in high-efficiency Apache Parquet format. Wikipedia traffic follows a pronounced power-law distribution where roughly 5% of pages account for over 70% of total page views, creating a natural and rigorous testbed for model robustness against extreme volatility that are absent from or underrepresented in existing benchmarks such as M4, M5, and UCI electricity datasets. TailedTS enables several research tasks. First, we introduce a periodicity quantification framework based on sparse autoregression with sparsity and non-negativity constraints, revealing that frequently-viewed pages exhibit significantly weaker periodic structure than their less-viewed counterparts, showing direct implications for server allocation and traffic forecasting on large digital platforms. Second, we provide standardized prediction benchmarks evaluated under a suite of non-Gaussian loss functions, including $\ell_1$-norm, Huber, quantile, and $\ell_p$-norm losses, demonstrating that standard Gaussian-based estimators degrade substantially on high-volume page categories, while robust alternatives provide consistent gains across all traffic scales. TailedTS is publicly available at https://doi.org/10.5281/zenodo.17070469.

Public procurement is vulnerable to error, fraud, and corruption, particularly as high transaction volumes overwhelm oversight. While research often focuses on tender-stage anomalies, post-award payment monitoring remains underexplored. Since labelled datasets are rare and methods like Benford's Law face restrictive assumptions, there is a need for interpretable, unsupervised frameworks for high-volume procurement oversight and decision support. This paper introduces the Structural Heterogeneity Index (SHI), a composite statistic for one-dimensional samples, and its payment-specific instantiation, the Payment Heterogeneity Index (PHI), characterising payment structure and latent regimes. It incorporates Gaussian Mixture Model (GMM) parameters alongside non-parametric statistics, integrating four interpretable components: modality, asymmetry, tail behaviour, and structural dispersion. Uniquely, the tail-behaviour component captures both distributional heaviness and extreme-value concentration, while structural-dispersion combines the variability, prevalence, and separation of latent payment regimes. Applied to UK municipal procurement data, PHI identifies a financially significant cohort (0.6% of suppliers; 10.1% of high-volume vendors) with structurally distinct payment patterns. Statistical testing further supports these differences, and targeted human verification confirms the plausibility of prioritised cases. Comparative analysis shows PHI reveals regime separation obscured by the Coefficient of Variation ($\rho = 0.310$). PHI provides a transparent, decomposable, and computationally lightweight framework for procurement integrity oversight and targeted audit prioritisation.

Kullback-Leibler (KL) regularization is ubiquitous in reinforcement learning algorithms in the form of reverse or forward KL. Recent studies have demonstrated $\epsilon^{-1}$-type fast rates for decision making under reverse KL regularization, in contrast to the standard $\epsilon^{-2}$-type sample complexity. However, for forward-KL-regularized objectives, existing statistical analyses are either not applicable or result in $\tilde{O}(\epsilon^{-2})$ slow rates. We take the first step towards addressing this problem via a streamlined analysis of forward-KL-regularized offline CBs. We give the first $\tilde{O}(\epsilon^{-1})$ upper bounds in tabular and general function approximation settings, both under notions of single-policy concentrability. In particular, our convex-analytical pipeline unifies these settings by exploiting the pessimism principle in a novel way and completely bypasses the proof routines in previous works based on the mean value theorem, which might be of independent interest. Moreover, we provide rate-optimal lower bounds, manifesting the tightness of our upper bounds in terms of statistical rates. Our lower bounds also demonstrate that the forward-KL-regularized sample complexity recovers the unregularized slow rate in the low-regularization regime, similarly to the reverse-KL regularization.

Physical activity (PA) intervention studies often collect repeated intensity measurements over long observation periods. Quantifying the variation in intervention effects over the study period is critical to evaluating and improving intervention strategies, yet many analyses reduce PA data into scalar summary measures, resulting in limited insights. We propose a functional regression framework, which captures time-varying intervention effects by modeling the entire PA trajectory as a functional observation. From both methodological and practical perspectives, we demonstrate the advantages of function-on-scalar regression (FoSR) over the traditional two-step approach of applying functional principal components analysis (FPCA) followed by regressing scores on covariates. The FoSR is further extended to a function-on-function regression (FoFR) for studying the association of PA across time periods. Methods are applied to daily step counts from the Social incentives to Encourage Physical Activity and Understand Predictors (STEP UP) study, revealing distinct and highly interpretable time-varying effects of three intervention strategies on PA and differences in their sustainability. Our case study highlights the feasibility of functional data analysis techniques for uncovering novel insights in intervention studies with high-dimensional endpoints.