Papers

Pretrained models are typically ranked on multi-task leaderboards to assess their effectiveness across diverse tasks. Rank confidence intervals were recently introduced as a method to quantify the uncertainty in these rankings by aggregating pairwise hypothesis tests. In this work, we analyze the sources of uncertainty in the knowledge evaluation benchmark MMLU and show how hypothesis tests can be modified to account for their effects. We demonstrate that ranking variability across MMLU subjects is substantial and should be considered when comparing LLMs or identifying the top-performing models.

Characterizing fade duration in wireless channels is fundamental for designing robust communication systems. Classical approaches – Rice's level-crossing theory and Monte Carlo simulation – lack precision for tail events and are computationally prohibitive for rare-event probability estimation. This paper introduces a rigorous framework combining Wiener Chaos Expansion (WCE), Malliavin Calculus, and importance sampling with adaptive weights to analyze fade duration $Z(T)$ distributions. Main contributions include: (i) high-accuracy moment estimation and CCDF characterization via WCE minimizing Monte Carlo variance; (ii) Markovian projection reducing infinite-dimensional dynamics to tractable systems ($\dim \leq 3$) for Rayleigh, Rician, and Nakagami models under stated assumptions; (iii) asymptotically optimal importance sampling weights derived from Malliavin sensitivities, achieving 839 to 2516x variance reductions; (iv) a theoretically grounded and provably efficient adaptive SGD algorithm with Robbins-Monro step size schedule for parameter estimation. Numerical experiments validate our approach with relative errors below 0.5%, enabling gradient-based optimization of fade duration statistics even for regimes where $P \sim 10^{-15}$, without requiring $\mathcal{O}(1/P)$ samples, by evaluating sensitivities through analytical Malliavin weights.

Transfer learning leverages knowledge from related source domains to improve learning in a target domain. Recent theoretical advances cover a broad range of regression settings within (generalized) linear models. Despite their diversity, these methods share two common constraints: they assume a known link function or linear structure and require direct access to raw source data. To move beyond these constraints, we propose a source-data-free transfer learning framework based on the single-index model (SIM). Instead of requiring raw source data, our method transfers only summary statistics derived from a generalized Stein's lemma in a one-time communication. This design preserves privacy and avoids side effects caused by dissimilarities of unknown nonlinear link functions across domains. To capture flexible, unknown nonlinearity, we employ a multilayer perceptron guided by the pre-estimated index from the transferred statistics, which significantly mitigates overfitting. Extensive experiments on synthetic data and a real-world application demonstrate consistent improvements over existing (generalized) linear model-based approaches. The proposed framework thus offers a practical, privacy-preserving, and nonlinear-adaptive solution for transfer learning.

Latent class models are central tools for multivariate categorical data from heterogeneous populations, but their standard local-independence assumption is often unrealistic in modern high-dimensional applications. We propose a high-dimensional latent class graphical model for ordinal responses with block-structured local dependence. The model retains the interpretability and parsimony of classical latent class analysis by imposing a shared block partition of variables, while allowing class-specific graphical dependence within each block. We develop a scalable three-step estimator that first recovers latent classes by spectral clustering of a flattened response matrix, then estimates class-specific latent covariance matrices and aggregates them to recover the shared block partition, and finally estimates sparse within-block precision matrices. We establish finite-sample error bounds for clustering, covariance estimation, block recovery, and precision-matrix estimation, yielding end-to-end consistency of all model components under high-dimensional scaling. Simulations demonstrate accurate recovery of latent classes, the shared block partition, and class-specific dependence graphs with scalable computation. Applications to American National Election Studies survey data and HapMap3 genotype data show that the method uncovers interpretable local dependence structures while accounting for latent heterogeneity.

This work presents a new bidirectional autoregressive latent diffusion approach for predicting the evolution of multiple fields (mass density, pressure, velocity, and magnetic field components) for magnetohydrodynamics. We show that this bidirectional flow can be used as a self-supervised consistency metric for uncertainty and error estimation, which enables the model to estimate test-time uncertainty and error without access to ground truth, by comparing how closely flowing forwards and backwards in time returns to the same predicted fields. We also demonstrate this methods's potential to serve as a non-invasive plasma diagnostic, and show how adaptive feedback can be used to make the model more robust based on sparse diagnostics or limited views/measurements.

In 1937, Stefan Kaczmarz proposed a simple algorithm for solving systems of linear equations. This algorithm turned out to be the earliest known example of stochastic gradient descent, a ubiquitous computing paradigm that drives the training of modern AI models such as ChatGPT and Gemini. Now, those AI models have joined forces to discover the worst-case complexity of the Kaczmarz algorithm. This paper tells the story of how it happened.

Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(\eta^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $\tau=\eta k$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(\eta)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(\eta^2)$; a bare fixed-$\beta$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$\beta$ momentum, and 4.00 for SGD; matching the memory clock to $\tau$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

In this paper, we propose a model-based framework to robustify inference for circular data in the presence of anomalous observations, distinguishing between mild and gross anomalies. Starting from a unimodal and symmetric reference model on $[0,2\pi)$, parametrized by a mean direction and concentration, we construct a family of finite mixtures: a gross-anomaly model obtained by adding a circular uniform component; a mild-anomaly (contaminated) model obtained by mixing the reference distribution with a less concentrated version sharing the same mean direction; and a general three-component specification combining both models, the double-contaminated model. Posterior component probabilities provide an automatic classification of observations without ad hoc thresholds, while the mixing weights yield interpretable measures of anomaly prevalence and dispersion inflation. For illustration, we consider two classical circular reference distributions, the wrapped normal and von Mises. The methodology is evaluated through an extensive simulation study and three real-data applications involving animal movement directions and wind directions. The results indicate that jointly modelling mild and gross departures improves model fit and yields an informative decomposition of the directional data, demonstrating that mixture-based robustness is valuable not only for anomaly detection but also for the interpretation and the identification of latent structure in directional data.

Deep learning problems rarely involve objectives that are equal in importance. A primary objective defines the goal, whilst secondary objectives, such as sparsity, compression, or robustness constrain the solution. While existing multi-objective methods have proven effective in practice, they have a clear symmetry problem and neglect the inherent objective hierarchy built into these objective spaces. We introduce Priority-Constrained Descent (PCD), a gradient-based optimization framework designed to explicitly exploit hierarchical objective structures. PCD preserves the direction of primary descent whilst allowing for the minimal distortion necessary to guarantee progress on secondary objectives, controlled by a single $\tau \in [0, 1]$ that dictates the strength of the distortion. The resulting formulation is invariant to objective scaling and admits exact closed-form solutions for problems with two and three objectives. We evaluate PCD within structured network compression settings, unstructured sparsity and low-rankness, and across a variety of synthetic experiments, showing Pareto dominance and better per-objective performance with secondary progress guarantees over existing methods, further exhibiting the interpretable trade-off that $\tau$ provides.

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

Accurate and scalable land cover classification is essential for global conservation monitoring and policy-making. While remote sensing images provide a cost-effective alternative to ground surveys, current methods often lack principled uncertainty quantification and require substantial labelled data, limiting their usability and reliability in new regions with distribution shifts. We propose a Bayesian spatial mixture modelling approach for image segmentation, extending the classical Potts model by allowing for a generalised spatial dependence structure and incorporating informative priors estimated from pre-existing labelled data. Our framework, called POTTERS (Potts Model for Enhanced Remote Sensing), enables robust uncertainty quantification, accounts for class interactions, and can detect new clusters in the target region of interest. Crucially, our model does not require labelled data from the target region; instead, it incorporates prior information about the labels from pre-existing externally labelled images. To ensure scalability to large remote sensing images, we develop an efficient variational inference algorithm for posterior approximation. We demonstrate the benefits of our approach in simulation studies and apply it to land cover classification in a case study in Scotland, leveraging publicly available remote sensing data from England.

Conformal prediction guarantees marginal coverage, but a pooled calibration quantile can hide systematic undercoverage across heterogeneous regions of the feature space. We introduce Self-Organized Conformal Prediction (SOCP), a calibration scheme that discovers input-space groups with an unsupervised Self-Organizing Map (SOM) trained without calibration data. At prediction time, the query's best-matching unit (BMU) retrieves a calibration buffer from one cell, a fixed grid neighborhood, or a prototype enlargement (Regimes 1 to 3). When fixed neighborhoods are too sparse, Regime 3 adds cells by prototype distance, using a global budget selected from training cell occupancies and the planned calibration size before any calibration score is observed. The predictor and nonconformity score remain unchanged. Cell-only retrieval has exact cell-conditional validity, and each fixed union of cells has exact retrieved-set validity. Interpreting a neighborhood threshold at its central cell incurs an explicit Kolmogorov-Smirnov (KS) bias term. Across ten regression and classification benchmarks, SOCP lowers the mean weighted coverage gap (WCovGap) relative to pooled split conformal prediction (SCP) on all ten datasets. The mean relative change is -14.8%, with a mean increase of 3.0% in output size. Under fixed-neighborhood retrieval, SO calibration lowers mean WCovGap across ten seeds in 43 of 50 score and dataset comparisons, while SO-SCP lowers it for every dataset at all three tested external partition granularities. These results provide local group calibration without supervised partitions or predictor retraining, while the diagnostics make the cost and limits of locality explicit. The code is available at https://github.com/LouisTier/SOCP.

Inferring the direction of a gene-regulatory relationship is harder than inferring whether a relationship exists, and most direction-inference methods are validated mainly on a single in silico benchmark. We ask which method remains reliable as the data move from a synthetic network to real organisms and as sample size decreases. We embed a copula-based measure of directional dependence (CDD) in a Bayesian framework that returns, for each candidate pair, a posterior distribution over a directional contrast, a 95% credible interval, a posterior sign-support score, and a principled no-call. We benchmark this estimator against eight direction-inference methods, including two Bayesian DAG-posterior baselines, on the three core DREAM5 networks (in silico, S. aureus, and E. coli), with S. cerevisiae used as an out-of-domain eukaryotic stress test. Across the three core networks, Bayesian CDD is the only method whose called accuracy is always above 60%, whose coverage is always above 88%, and whose direction AUROC is always above 0.6; every competing method falls to chance or below on at least one network. CDD ranks first on both real-organism networks, remains stable on the smallest-sample network where bootstrap-interval methods collapse, and is the only Bayesian method that is simultaneously above chance and high-coverage under a 95% posterior gate. We position CDD as a post-screening, uncertainty-aware direction-refinement tool for candidate regulatory pairs.

The paper "Use of roster charts in the investigation and prosecution of nurses suspected of inflicting deliberate harm on patients" by Prof. John O'Quigley explores an interesting hypothesis concerning statistical information hidden in the part of the infamous Lucy Letby roster chart pertaining to the 37 other nurses. Unfortunately, we have to point out some serious errors in his statistical analyses. The data actually contains information which strongly disproves his main modelling assumption. We do, however, strongly agree with him that from a forensic statistical point of view, the roster chart is fake evidence which should not have been shown to jurors.

Classical actuarial pricing models, such as the generalized linear model, are valued for transparency and ease of governance, but they use interactions among risk factors only when these are supplied through explicit feature engineering. We study whether embeddings from a pre-trained large language model, computed from a natural-language description of each policyholder, can replace hand-crafted features as inputs to a standard actuarial pricing model, taking Poisson claim-frequency regression as the main example. The language model is used only to construct deterministic embedding covariates; pricing is performed by a standard generalized linear model. Using French motor third-party liability data, the embedding-based model outperforms the generalized linear model, especially when data are scarce, whereas at larger sample sizes the comparison is model- and dimension-dependent. Insurance-specific fine-tuning further improves the embeddings, and a prompt-sensitivity diagnostic shows that the pipeline reacts to any appended out-of-template field, making controlled prompts a governance requirement.

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

Gradient boosting in the form of decision tree ensembles has successfully been applied to a variety of problems using simple objective functions based on log-likelihoods of a single variable. The concept extends naturally to objective functions operating on vectors - for example, multinomial logistic log-likelihood for multi-class classification, where observations have a score for each class - but popular frameworks approach these functions by either updating one value of the input vectors at a time, or by using a diagonal upper bound on the second derivative. This work extends the usual gradient boosting framework to functions of vector inputs and sketches a simple algorithm that can be used efficiently with histogram-based decision trees.

In recent years, models based on the Transformer architecture have seen widespread applications and have become one of the core tools in the field of deep learning. Numerous successful techniques, such as parameter-efficient fine-tuning and efficient scaling, have been proposed surrounding their applications to further enhance performance. However, the success of these strategies has always lacked the support of rigorous mathematical theory. To study the underlying mechanisms behind Transformers and related techniques, we first propose a Transformer learning framework motivated by distribution regression, with distributions being inputs, connect a two-stage sampling process with natural language processing, and present a mathematical formulation of the attention mechanism called attention operator. We demonstrate that by the attention operator, Transformers can compress distributions into function representations without loss of information. Moreover, with the advantages of our novel attention operator, Transformers exhibit a stronger capability to learn functionals with more complex structures than convolutional neural networks and fully connected networks. Finally, we obtain a generalization bound within the distribution regression framework. Through the aforementioned theoretical results, we further discuss some successful techniques emerging with large language models (LLMs), such as prompt tuning, parameter-efficient fine-tuning, and efficient scaling. We also provide theoretical insights behind these techniques within our novel analysis framework.

Vegetable prices in Sri Lanka are highly volatile because the market is largely import-isolated, so supply disruptions quickly drive prices up. This study develops a machine learning framework to forecast such volatility by incorporating supply-chain-aware features and explicitly modelling the country's two cultivation seasons, Maha (October-April) and Yala (May-September). An integrated dataset was constructed by combining retail and farmer-gate prices with origin-aligned weather variables, diesel costs, and exchange rates across 12 vegetable varieties and 14 market centres from 2013 to 2019. A gradient-boosted ensemble model (XGBoost and LightGBM) was trained and optimised using Optuna, and unified and season-specific configurations were compared. Results show that season-specific models improve within-season fit, with the Yala-specific model achieving the highest R2 of 0.9420 (95% CI [0.690, 1.000]), while the unified model delivers the best overall predictive accuracy of 90.84% (95% CI [88.34%, 91.52%]) and an R2 of 0.9281 (95% CI [0.760, 1.000]). Notably, the unified model maintains 85.96% accuracy on a completely unseen 2024 hyperinflationary period without retraining, successfully tracking major price surges. These findings suggest that agricultural price movements in import-constrained markets are meaningfully predictable when models capture supply-chain dynamics, offering practical value for early warning and decision making by farmers, traders, and policymakers. Existing studies on Sri Lankan vegetable prices are confined to Autoregressive Integrated Moving Average (ARIMA) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) applied to single markets, with no supply-chain features, seasonal segmentation, or cross-regime validation.

Bayesian statistics makes inference based on Bayes' theorem, but the posterior distribution of unknown parameters is typically analytically intractable. To estimate the posterior, two widely used numerical approximation methods are Markov Chain Monte Carlo (MCMC) and variational inference (VI). MCMC methods produce asymptotically exact samples but are computationally intensive, while VI methods are faster and more scalable but may lack accuracy. This paper proposes combining MCMC and VI to construct algorithms that leverage the strengths of both. The first proposed algorithm uses Gaussian variational inference (GVI) with various covariance structures to derive a linear transformation matrix for Hamiltonian Monte Carlo (HMC). This method improves the efficiency of HMC, particularly in high-dimensional and complex target distributions. The second algorithm combines a VI-based generative model, the variational auto-encoder (VAE), with the Metropolis-Hastings (MH) sampler. The resulting VAE-MH sampler is efficient and effectively traverses the parameter space, outperforming standard MCMC methods in identifying all modes of multi-modal distributions.

We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation. Subject to an abstention budget $\alpha$, we analyze the probability of undetected error–the risk of recommending a suboptimal arm without abstaining. Our central finding is that abstention induces a phase transition: without abstention, the error probability decays polynomially in the sampling budget $T$; in contrast, introducing any small positive abstention budget shifts this to an exponential decay. For Gaussian priors and rewards, in the regime $T\to\infty$ followed by $\alpha\downarrow0$, we establish exact matching information-theoretic lower bounds and algorithmic upper bounds on the optimal error exponent, which takes the form $\exp(-\frac{\alpha^{2}T}{8\kappa_{\nu}^{2}})$. The hardness parameter $\kappa_{\nu}$ represents the prior density of the top-two gap at zero, highlighting that nearly tied instances drive the fundamental error. We introduce an adaptive algorithm, PGWS, that successfully achieves this optimal exponent by expending its abstention budget on statistically ambiguous instances. We further demonstrate that this polynomial-to-exponential improvement is exclusively a Bayesian phenomenon–in the frequentist setting, abstention only affects lower-order exponent terms. We also extend our results beyond the Gaussian model.

A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\bar\Theta = \Theta/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.

Extensible lattices are a common choice of nodes for approximating multidimensional integrals by a sample mean, typically using sample sizes of the form $n = b^m$ for a prime base $b$ and non-negative integer $m$. However, a computational time budget or early termination due to hardware failure may prevent us from choosing the preferred number of samples. In this paper, we establish an upper bound on the figure of merit $P_\alpha$ for extensible lattice sequences \emph{with arbitrary $n$}, derived as a worst-case error in a Banach space setting. We show that although the upper bound on $P_\alpha$ can decay no faster than $\mathcal O(n^{-1})$ for general $n$, restricting $n$ to the form $n =\lambda b^p$, where $\lambda$ is a fixed integer, we can recover the optimal decay of nearly $\mathcal O(n^{-\alpha})$ as $p \to \infty$. We also investigate a related figure of merit, $P_{\alpha,2}$, arising as a worst-case error for Hilbert spaces of integrands. We obtain a theoretical upper bound on $P_{\alpha,2}$ in terms of $P_{\alpha}$ for equal sample weights and show numerically that optimal sample weights can improve the cubature error bound even further.

Reflected backward stochastic differential equations (RBSDEs) provide a probabilistic formulation for obstacle constrained problems, but existing deep learning methods for their high dimensional solution remain limited. In this paper, we propose two deep learning schemes for RBSDEs, a deep forward scheme (DFS) and a deep backward scheme (DBS), by first reducing the reflected problem to a family of regularized BSDEs. Our main theoretical contribution concerns the DBS: we establish an explicit error bound showing that, for each fixed regularization parameter $\varepsilon>0$, the approximation error between the DBS solution and the solution to the regularized BSDE is controlled by the associated training loss. We prove that this training loss can be controlled by the universal approximation capability of neural networks. Together, these results yield a theoretical foundation for the deep learning-based solution and complement existing analysis for forward type methods. We illustrate the framework on high dimensional American option pricing, where the reflected formulation allows us to address the continuous time exercise feature directly rather than through a Bermudan approximation. Numerical experiments demonstrate that both DFS and DBS deliver accurate solutions in high dimensions.

A trained transformer's weight magnitudes can be summarized by a two-parameter Weibull distribution whose shape $k \approx 1.2$ is stable across layers and models, so the scale $\lambda$ carries most training-induced movement. What corpus property sets how much $\lambda$ grows? Using the bigram conditional entropy $D = H(\text{next} \mid \text{prev})$, a training-free statistic computed before training, we find across controlled corruption families a learning-rate-conditioned law, $\lambda^2 - \lambda_0^2 = C_0(\eta) + C_1(\eta)(H_r - D)^{0.59}$, where $H_r$ is a matched-budget shuffle baseline. The convex exponent is inherited from an independently measured data-side saturation relation rather than fitted directly to the growth curve. After removing the two per-$\eta$ coefficients, 23 runs spanning an order of magnitude in learning rate collapse onto $(H_r - D)^{0.59}$ with unit slope ($R^2 = 0.941$; direct per-$\eta$ fits are weaker, $R^2 \approx 0.82$). Because $D$ is computed before training, the law is a forward predictor: an end-to-end self-validation recovers held-out within-family weight growth with 5.7% relative error. The readout holds at model and per-layer resolutions and across two tested architectures, with the functional form preserved and only the coefficients changing. It also marks its boundary: cross-corpus prediction over-predicts code, implicating redundancy as a second axis of a broader $\Phi(D,R,A,H)$ data-to-weight framework.

Streaming inference pipelines increasingly pair lightweight fast models with Large Language Models (LLMs) that provide rich semantic understanding at substantial cost. The central question of when to invoke the LLM has received limited formal treatment. We cast this as a risk-based sequential stopping problem, where a trigger policy fires when a risk functional over the observation history exceeds a threshold. Within this framework, we prove six results: a minimum inter-event time bound excluding trigger chattering; optimality of threshold policies via smooth pasting; approximate SPRT guarantees under estimated parameters; O(sqrt(T log T)) regret for stationary streams, extending to O(sqrt((C_T + 1) T log T)) under C_T changepoints; O(1/sqrt(T)) convergence of online gradient descent for adaptive thresholds; and a calibration-to-miss-rate transfer inequality. Several classical trigger families, including event-triggered, optimal stopping, SPRT, CUSUM, and Bayesian triggers, can be expressed as special cases of this framework. On turbofan degradation data (CMAPSS) with real LLM calls, we empirically verify the theoretical assumptions, ablate the risk function design, compare against six baselines including a RouteLLM-style router and contextual bandits, and analyze cost sensitivity and LLM failure modes. The results confirm sublinear regret, with alpha < 1 for all principled triggers; high diagnostic quality, with 92.9 percent of 1600 LLM diagnoses reaching grounding score >= 0.75 under our rubric; and that anomaly-score-driven risk functions dominate alternatives by roughly an order of magnitude on the Pareto AUC.

Rubin multiple imputation (MI) generates plausible data completions to account for uncertainty and statistical variability but provides little insight into their global organization. We introduce a topological reconstruction approach that complements MI by examining the ensemble of completed datasets. Individual imputations are represented as points in a reconstruction space whose coordinates summarize statistical properties. Concepts from coarse geometry and algebraic topology are then used to characterize relationships among alternative imputations across multiple scales. Coarse proximity (CP) defines large-scale neighborhoods, generating graphs in which nodes represent completed datasets and edges connect sufficiently similar imputations. Seifert van Kampen gluing provides a conceptual interpretation of how local reconstructions assemble into globally coherent structures, whereas Hurewicz-type invariants quantify persistent connectivity patterns. Synthetic multivariate biomedical datasets representing adult cardiometabolic cohorts were generated with controlled missingness levels. Multiple stochastic imputations were projected into the reconstruction space and analyzed through CP graphs, connected components, cycle descriptors and scale-dependent topological measures. MI generated structured spaces with distinct connectivity patterns rather than homogeneous clouds of solutions. Topological descriptors remained stable despite local numerical variability, whereas increasing missingness produced transitions in reconstruction-space connectivity together with progressive deterioration of reconstruction accuracy. Our approach could be applied to biological and social networks, systems medicine, ecological modeling and other domains in which large-scale structural organization contributes to reliable inference.

Modern multivariate regression problems involve several related outcomes whose regression effects are not only nonlinear, heterogeneous, and outcome-specific, but also where the residual dependence among outcomes is scientifically meaningful. Existing multivariate Bayesian tree-based methods typically address only part of this problem: some impose substantial sharing of tree architecture across outcomes, which is overly restrictive when responses depend on distinct predictors or effect modifiers, while others accommodate residual dependence but retain simpler mean structures. This paper develops multiVCBART, a multivariate varying-coefficient Bayesian additive regression tree framework that jointly models flexible outcome-specific coefficient surfaces and a sparse residual precision matrix. Each entry of the coefficient matrix $B(x)$ is represented by an independent BART ensemble, allowing predictor effects to vary nonlinearly with modifiers $x$ across outcomes, while a Graphical Horseshoe prior on the precision matrix $\Omega$ captures parsimonious residual conditional dependence. To permit efficient computation, we introduce a sampler that reduces the multivariate Gaussian likelihood to a sequence of scalar pseudo-response updates, decoupling the tree backfitting from the Graphical Horseshoe step. Theoretically, we establish the first posterior contraction rates for a multivariate BART model with jointly estimated residual dependence, proving near-minimax adaptation to underlying smoothness and structural sparsity. Empirically, multiVCBART outperforms existing multivariate tree models and Bayesian SUR competitors on sparse, high-dimensional datasets. Finally, in a re-analysis of the Genomics of Drug Sensitivity in Cancer dataset, our method identifies distinct biomarker signals and recovers a coherent residual pharmacologic network.

Recent progress in flow-based generative modeling has led to models that output high-quality samples while using only a small number of function evaluations. However, at present, there is a lack of similar advances in estimating the model likelihood. In particular, most existing methods either rely on restrictive architectures that enable exact calculations, or use stochastic approximations such as Hutchinson's trace estimator that introduce substantial variance. In this work, we introduce SCAlable LikeLihood distillation of flOw maPs (SCALLOP). SCALLOP builds on the recently proposed F2D2, a likelihood flow map model that can generate samples and their densities in a small number of function evaluations. While F2D2 uses Hutchinson's estimator during training, we introduce an alternative and more scalable likelihood distillation objective that is Hutchinson-free and admits a vectorized formulation. Empirically, we demonstrate the effectiveness of SCALLOP as a Boltzmann generator in molecular science, and further validate its benefit on image datasets. SCALLOP significantly reduces both training variance and training time while consistently improving performance compared to F2D2, and is competitive with the state-of-the-art while achieving up to 10x inference speedup over the fastest baseline.

We study the asymptotic properties of the posterior on the latent space for infinite mixtures driven by a Dirichlet process, both in terms of mixing measure and clustering behaviour. In the well-specified regime, where the data are generated by a finite mixture of location densities, we show that the posterior is adaptive to the true number of components $K$: indeed the cumulative mass assigned to weights of the stick-breaking representation beyond the $K$-th one vanishes as $n^{-1/2}$, up to terms growing slower than any polynomial. This also implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. A remarkable phase transition underlies this result: approximating the mixing measure to any precision finer than $n^{-1/2}$ requires a number of components growing logarithmically with the sample size. We show that this has a profound impact on the clustering behaviour: the number of clusters grows logarithmically, as in the prior case, but the proportion of observations outside the $K$ largest clusters vanishes polynomially fast. Finally, we turn these results into posterior guarantees for truncation-based approximations: while any truncation with at least $K$ elements recovers the optimal contraction rates for both density and mixing measure, $\mathcal{O}(\log n)$ components are both necessary and sufficient to reproduce the clustering of the exact posterior.