Papers

We derive explicit non-asymptotic PAC-Bayes generalization bounds for Gibbs posteriors, that is, data-dependent distributions over model parameters obtained by exponentially tilting a prior with the empirical risk. Unlike classical worst-case complexity bounds based on uniform laws of large numbers, which require explicit control of the model space in terms of metric entropy (integrals), our analysis yields posterior-averaged risk bounds that can be applied to overparameterized models and adapt to the data structure and the intrinsic model complexity. The bound involves a marginal-type integral over the parameter space, which we analyze using tools from singular learning theory to obtain explicit and practically meaningful characterizations of the posterior risk. Applications to low-rank matrix completion and ReLU neural network regression and classification show that the resulting bounds are analytically tractable and substantially tighter than classical complexity-based bounds. Our results highlight the potential of PAC-Bayes analysis for precise finite-sample generalization guarantees in modern overparameterized and singular models.

Inductive bias refers to restrictions on the hypothesis class that enable a learning method to generalize effectively from limited data. A canonical example in control is linearity, which underpins low sample-complexity guarantees for stabilization and optimal control. For general nonlinear dynamics, by contrast, guarantees often rely on smoothness assumptions (e.g., Lipschitz continuity) which, when combined with covering arguments, can lead to data requirements that grow exponentially with the ambient dimension. In this paper we argue that data-efficient nonlinear control demands exploiting inductive bias embedded in nature itself, namely, structure imposed by physical laws. Focusing on Hamiltonian systems, we leverage symplectic geometry and intrinsic recurrence on energy level sets to solve target reachability problems. Our approach combines the recurrence property with a recently proposed class of policies, called chain policies, which composes locally certified trajectory segments extracted from demonstrations to achieve target reachability. We provide sufficient conditions for reachability under this construction and show that the resulting data requirements depend on explicit geometric and recurrence properties of the Hamiltonian rather than the state dimension.

Converting betting odds into accurate outcome probabilities is a fundamental challenge in order to use betting odds as a benchmark for sports forecasting and market efficiency analysis. In this study, we propose two methods to overcome the limitations of existing conversion methods. Firstly, we propose an odds-only method to convert betting odds to probabilities without using historical data for model fitting. While existing odds-only methods, such as Multiplicative, Shin, and Power exist, they do not adjust for biases or relationships we found in our betting odds dataset, which consists of 90014 football matches across five different bookmakers. To overcome these limitations, our proposed Odds-Only-Equal-Profitability-Confidence (OO-EPC) method aligns with the bookmakers' pricing objectives of having equal confidence in profitability for each outcome. We provide empirical evidence from our betting odds dataset that, for the majority of bookmakers, our proposed OO-EPC method outperforms the existing odds-only methods. Beyond controlled experiments, we applied the OO-EPC method under real-world uncertainty by using it for six iterations of an annual basketball outcome forecasting competition. Secondly, we propose a generalised linear model that utilises historical data for model fitting and then converts betting odds to probabilities. Existing generalised linear models attempt to capture relationships that the Efficient Market Hypothesis already captures. To overcome this shortcoming, our proposed Favourite-Longshot-Bias-Adjusted Generalised Linear Model (FL-GLM) fits just one parameter to capture the favourite-longshot bias, providing a more interpretable alternative. We provide empirical evidence from historical football matches where, for all bookmakers, our proposed FL-GLM outperforms the existing multinomial and logistic generalised linear models.

Stomata play a crucial role in regulating plant physiological processes and reflecting environmental responses. However, accurate and high-throughput stomatal phenotyping remains challenging, as conventional approaches rely on destructive sampling and manual annotation, restricting large-scale and field deployment. To overcome these limitations, a noninvasive restoration-detection integrated framework, termed StomaD2, is developed to achieve accurate and fast stomatal phenotyping under complex imaging conditions. The framework incorporates a diffusion-based restoration module to recover degraded images and a specialized rotated object detection network tailored to the small, dense, and cluttered characteristics of stomata. The proposed network enhances feature representation through three key innovations: a column-wise structure for global feature interaction, context-aware resampling and reweighting mechanism to improve multi-scale consistency, and a feature reassembly module to boost discrimination against complex backgrounds. In extensive comparisons, StomaD2 demonstrated state-of-the-art performance. On public Maize and Wheat datasets, it achieved accuracies of 0.994 and 0.992, respectively, significantly outperforming existing benchmarks. When benchmarked against ten other advanced models, including Oriented Former and YOLOv12, StomaD2 achieved a top-tier F1-score/mAP of 0.989. The framework is integrated into a user-friendly, field-operable system that supports the fast extraction of eight stomatal phenotypes, such as density and conductance. Validated on more than 130 plant species, StomaD2's results highlight its strong generalizability and potential for large-scale phenotyping, plant physiology analysis, and precision agriculture applications.

A family of random probabilities is defined and studied. This family contains the Dirichlet process as a special case, corresponding to an inner point in the appropriate parameter space. The extension makes it possible to have random means with larger or smaller skewnesses as compared to skewnesses under the Dirichlet prior, and also in other ways amounts to additional modelling flexibility. The usefulness of such random probabilities for nonparametric Bayesian statistics is discussed. The posterior distribution is complicated, but inference can nevertheless be carried out via simulation, and some exact formulae are derived for the case of random means. The class of nonparametric priors provides an instructive example where the speed with which the posterior forgets its prior with increasing data sample size depends on special aspects of the prior, which is a different situation from that of parametric inference.

Models with fewer parameters are often easier to interpret and more robust. Parsimony can be achieved through optimizing objectives like the AIC or BIC, which are functions of the the number of free parameters in the model. Optimizing this discrete objective is a challenge, often relying on discrete optimization. We construct smooth functions with optima that reach the same optima of these objectives but permit continuous rather than discrete optimization, relieving some selection burden. Proofs of convergence are provided and a novel method of clustering through explicit overparamterization shows promising results.

Computer simulations play an important role in scientific discovery and engineering innovation. Reliable computer models enable virtual experimentation that reduces the need for costly and time-consuming physical testing. However, the credibility of such models hinges on rigorous statistical validation against real-world data. This paper develops a formal frequentist framework for both global and subdomain validation of computer models. We propose the Fourier Maximum Modulus Test (FMMT), which leverages kernel ridge regression (KRR) to estimate the discrepancy between the computer model and the physical process, followed by a frequency-domain test based on weighted generalized Fourier coefficients. The theoretical analysis establishes the asymptotic normality of these coefficients, allowing for closed-form p-values. Simulation studies and a shear-layer experiment demonstrate that FMMT achieves high power, accurate Type I error control, and strong sensitivity to localized discrepancies.

We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical sufficient conditions for normality by concatenation do not apply to the Fibonacci sequence because of its exponential growth, while a criterion of Pollack and Vandehey implies that the normality of $\mathcal{F}$ in base $10$ would follow if almost all Fibonacci numbers were $(\varepsilon,k)$-normal in base $10$. The Benford bias of leading digits and the Pisano periodicity of trailing digits are shown to contribute asymptotically negligible fractions of the total digits, isolating the distribution of the deep digits of large Fibonacci numbers as the remaining obstruction. Large-scale numerical experiments on the first $500{,}000$ Fibonacci numbers in bases $10$ and $2$ indicate that global single-digit counts and $k$-block statistics for $k = 2, 3, 4$ are compatible with i.i.d.-like fluctuations at the scales tested, and that a positional decomposition concentrates the visible structured deviation at the boundaries between consecutive Fibonacci numbers, while pooled interior blocks remain close to uniform. Our computations suggest that any obstruction to normality lies in the asymptotic behavior of the deep digits of $F_{n}$.

The present study aims to investigate a cluster cleaning algorithm that is both computationally simple and capable of solving the PU classification when the SCAR condition is unsatisfied. A secondary objective of this study is to determine the robustness of the LassoJoint method to perturbations of the SCAR condition. In the first step of our algorithm, we obtain cleaning labels from 2-means clustering. Subsequently, we perform logistic regression on the cleaned data, assigning positive labels from the cleaning algorithm with additional true positive observations. The remaining observations are assigned the negative label. The proposed algorithm is evaluated by comparing 11 real data sets from machine learning repositories and a synthetic set. The findings obtained from this study demonstrate the efficacy of the clustering algorithm in scenarios where the SCAR condition is violated and further underscore the moderate robustness of the LassoJoint algorithm in this context.

Possibilities are considered to simplify the computation of several statistical functions used to test statistical hypotheses when processing observations: the inverse normal distribution, the Student's t-distribution, and the criterion for rejecting outliers. For these three cases, simple approximation expressions are proposed for the quantiles of these statistical distributions, which are accurate enough for most practical applications.

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can be arbitrarily conservative and fail to capture the geometry actually encountered by optimization trajectories. In this paper, we develop a trajectory-restricted framework for linear convergence based on localized geometric regularity. We introduce restricted variants of the Polyak–Łojasiewicz inequality, error bound, and quadratic growth conditions that are required to hold only on subsets of the domain. We show that classical convergence guarantees extend under these localized conditions, and in key cases, we develop new arguments that yield explicit relationships between the corresponding constants. The resulting rates are governed by geometric quantities associated with the regions traversed by the algorithm. For polyhedral composite problems, we prove that convergence is controlled by restricted Hoffman constants corresponding to the active polyhedral faces visited along the trajectory. Once the iterates enter a well-conditioned face, the effective condition number improves accordingly. Our work provides a geometric quantification for fast local convergence after active-set or manifold identification and more broadly suggests that linear convergence is fundamentally governed by the geometry of the subsets explored by the algorithm, rather than by worst-case global conditioning.

Recent work has shown that parameterizing and optimizing coordinate transformations using normalizing flows, i.e., invertible neural networks, can significantly accelerate the convergence of spectral approximations. We present the first error estimates for approximating functions using Hermite expansions composed with adaptive coordinate transformations. Our analysis establishes an equivalence principle: approximating a function $f$ in the span of the transformed basis is equivalent to approximating the pullback of $f$ in the span of Hermite functions. This allows us to leverage the classical approximation theory of Hermite expansions to derive error estimates in transformed coordinates in terms of the regularity of the pullback. We present an example demonstrating how a nonlinear coordinate transformation can enhance the convergence of Hermite expansions. Focusing on smooth functions decaying along the real axis, we construct a monotone transport map that aligns the decay of the target function with the Hermite basis. This guarantees spectral convergence rates for the corresponding Hermite expansion. Our analysis provides theoretical insight into the convergence behavior of adaptive Hermite approximations based on normalizing flows, as recently explored in the computational quantum physics literature.

The paper proposes new algorithms for computing the regularization path of linear-model fitting (i) with L1 costs for the independent variables and L2 costs for the dependent variables, and (ii) with L2 costs for the independent variables and L1 costs for the dependent variables. Unlike previously proposed algorithms, the proposed algorithms for (i) and (ii) are exact duals of each other. The heart of the proposed algorithms is parametric Gaussian message passing (i.e., Kalman-type forward-backward recursions) in the pertinent factor graphs. We first develop such algorithms for linear models with general (but preferably low-dimensional) state space representations. We then specialize to specific state space representations of generic linear models; in this case, the structure of the factor graph allows to refine the proposed algorithms such that the computational complexity drops to linear scale.

Across many scientific fields, measurements often represent the number of times an event occurs. For example, a document can be represented by word occurrence counts, neural activity by spike counts per time window, or online communication by daily email counts. These measurements yield high-dimensional count data that often approximate a Poisson distribution, frequently with low rates that produce substantial sparsity and complicate downstream analysis. A useful approach is to embed the data into a low-dimensional space that preserves meaningful structure, commonly termed dimensionality reduction. Yet existing dimensionality reduction methods, including both linear (e.g., PCA) and nonlinear approaches (e.g., t-SNE), often assume continuous Euclidean geometry, thereby misaligning with the discrete, sparse nature of low-rate count data. Here, we propose p-SNE (Poisson Stochastic Neighbor Embedding), a nonlinear neighbor embedding method designed around the Poisson structure of count data, using KL divergence between Poisson distributions to measure pairwise dissimilarity and Hellinger distance to optimize the embedding. We test p-SNE on synthetic Poisson data and demonstrate its ability to recover meaningful structure in real-world count datasets, including weekday patterns in email communication, research area clusters in OpenReview papers, and temporal drift and stimulus gradients in neural spike recordings.

Computer-based assessments routinely generate detailed interaction logs – commonly referred to as process data – that record every action a respondent performs during task completion, yet systematic preprocessing guidance, integrated analytical workflows, and cross-method consistency checks remain scarce in the literature. This paper provides a unified, end-to-end analytical framework for analyzing process data from large-scale assessments – covering the full pipeline from raw log preprocessing to model-based inference – using the Programme for the International Assessment of Adult Competencies (PIAAC) Problem Solving in Technology-Rich Environments (PS-TRE) domain as an illustrative example. We first present a systematic preprocessing pipeline – including timestamp correction, duplicate removal, action block consolidation, and LLM-assisted standardization – that transforms raw event-level logs into analysis-ready action sequences. We then review and demonstrate two complementary families of analytical methods. The first consists of feature-based methods and their downstream applications, including descriptive process indicators, n-gram analysis with TF–IDF weighting, multidimensional scaling, and process data-informed differential item functioning (DIF) analysis. The second consists of model-based approaches, namely hidden Markov models and the subtask identification procedure. Empirical illustrations using the United States sample illustrate that n-gram-based behavioral clusters carry differential diagnostic information primarily among incorrect respondents, that multidimentionsl scaling-derived features comprehensively reconstruct observed behavioral variables, and that process-informed DIF analyses can identify and mitigate construct-irrelevant sources of group differences. Reproducible R code implementations are provided for all major techniques.

Factor-based Structural Equation Modeling (SEM) relies on likelihood-based estimation assuming a nonsingular sample covariance matrix, which breaks down in small-sample settings with $p>n$. To address this, we propose a novel estimation principle that reformulates the covariance structure into self-covariance and cross-covariance components. The resulting framework defines a likelihood-based feasible set combined with a relative error constraint, enabling stable estimation in small-sample settings where $p>n$ for sign and direction. Experiments on synthetic and real-world data show improved stability, particularly in recovering the sign and direction of structural parameters. These results extend covariance-based SEM to small-sample settings and provide practically useful directional information for decision-making.

In this paper, we consider the problem of extraction of most informative features from time series that are regarded as observed values of stochastic processes satisfying the Itô stochastic differential equations with unknown random drift and diffusion coefficients. We do not attract any additional information and use only the information contained in the time series as it is. Therefore, as additional features, we use the parameters of statistically adjusted mixture-type models of the observed regularities of the behavior of the time series. Several algorithms of construction of these parameters are discussed. These algorithms are based on statistical reconstruction of the coefficients which, in turn, is based on statistical separation of normal mixtures. We obtain two types of parameters by the techniques of the uniform and non-uniform statistical reconstruction of the coefficients of the underlying Itô process. The reconstructed coefficients obtained by uniform techniques do not depend on the current value of the process, while the non-uniform techniques reconstruct the coefficients with the account of their dependence on the value of the process. Actually, the non-uniform techniques used in this paper represent a stochastic analog of the Taylor expansion for the time series. The efficiency of the obtained additional features is compared by using them in the autoregressive algorithms of prediction of time series. In order to obtain pure conclusion that is not affected by unwanted factors, say, related to a special choice of the architecture of the neural network prediction methods, we used only simple autoregressive algorithms. We show that the use of additional statistical features improves the prediction.

Delayed loss spikes have been reported in neural-network training, but existing theory mainly explains earlier non-monotone behavior caused by overly large fixed learning rates. We study one stylized hypothesis: normalization can postpone instability by gradually increasing the effective learning rate during otherwise stable descent. To test this hypothesis at theorem level, we analyze batch-normalized linear models. Our flagship result concerns whitened square-loss linear regression, where we derive explicit no-rising-edge and delayed-onset conditions, bound the waiting time to directional onset, and show that the rising edge self-stabilizes within finitely many iterations. Combined with a square-loss decomposition, this yields a concrete delayed-spike mechanism in the whitened regime. For logistic regression, under highly restrictive active-margin assumptions, we prove only a supporting finite-horizon directional precursor in a knife-edge regime, with an optional appendix-only loss lower bound under an extra non-degeneracy condition. The paper should therefore be read as a stylized mechanism study rather than a general explanation of neural-network loss spikes. Within that scope, the results isolate one concrete delayed-instability pathway induced by batch normalization.

Classical mixture models (MMs) are widely used tractable proposals for approximate inference settings such as variational inference (VI) and importance sampling (IS). Recently, mixture models with negative coefficients, called subtractive mixture models (SMMs), have been proposed as a potentially more expressive alternative. However, how to effectively use SMMs for VI and IS is still an open question as they do not provide latent variable semantics and therefore cannot use sampling schemes for classical MMs. In this work, we study how to circumvent this issue by designing several expectation estimators for IS and learning schemes for VI with SMMs, and we empirically evaluate them for distribution approximation. Finally, we discuss the additional challenges in estimation stability and learning efficiency that they carry and propose ways to overcome them. Code is available at: https://github.com/april-tools/delta-vi.

We study model-free reinforcement learning (RL) in non-stationary finite-horizon episodic Markov decision processes (MDPs) without prior knowledge of the non-stationarity. We focus on the piecewise stationary (PS) setting, where both rewards and transition dynamics can change at unknown times. We first revisit existing state-of-the-art approaches and identify theoretical and practical limitations that change the current landscape of performance guarantees. To characterize the difficulty of the problem, we establish the first minimax lower bounds for PS-RL in tabular and linear MDPs. We then introduce Detection Augmented Reinforcement Learning (DARLING), a modular wrapper for PS-RL that applies to both tabular and linear MDPs, without knowledge of the changes. In tabular MDPs, under change-point separability and reachability conditions, DARLING improves the best known dynamic regret bounds and matches our minimax lower bound. In linear MDPs, DARLING matches the minimax lower bound when the relevant reachability parameters are known, and our analysis clarifies the structural obstacles that distinguish this setting from the tabular case. Finally, through extensive experimentation across diverse non-stationary benchmarks, we show that DARLING consistently surpasses the state-of-the-art methods.

Online controlled experiments face growing challenges from overlapping tests on shared traffic, where interactions between concurrent experiments obscure insights into feature combinations and produce effect estimates that do not correspond to any actionable launch scenario. While traffic splitting, layering, and sequential execution (non-concurrent) mitigate some of these issues, they require coordination overhead and can reduce experimentation velocity. We propose Multi-Experiment Analysis (MEA), a methodology for consistent joint estimation in the presence of arbitrary partial or full overlaps and multiple variants. MEA produces three types of estimates: (1) corrected individual treatment effects that account for the presence of overlapping experiments, (2) combined effects of launching any desired combination of variants across experiments, and (3) conditional effects of an experiment's variant given that specific variants of other experiments are launched or deramped – all without requiring factorial pre-design or traffic restrictions. We validate the approach through comprehensive simulations confirming consistency and correct coverage. We report on production deployment at scale, illustrate the methodology through real-world use cases, and share practical lessons learned – including system design, adoption patterns, and insights from production use.

Predictive inference in the sparse Gaussian sequence model has received considerably less attention than its non-sparse, finite-sample counterpart. Existing work has largely been confined to discrete mixture priors. In this paper, we study predictive inference under a widely used continuous mixture prior, the Horseshoe. We provide new theoretical results establishing exact asymptotic minimax optimality of the predictive Bayes estimator when the sparsity level is known. Furthermore, through a Gaussian-mixture representation of the posterior predictive density (which we term Horseshoe spectroscopy), the phase-transition in the local shrinkage scale is inherited by the predictive mechanism, producing behavior similar to that of previous thresholding/switching estimators. When sparsity is unknown, we adopt a fully Bayesian approach using a hierarchical Horseshoe prior and show that it performs adaptive, as opposed to manual, switching. Under a theta-min condition, the resulting predictive risk admits an upper bound over a restricted parameter class that is sharper than the minimax rate over the full class. We demonstrate the practical value of predictive Horseshoe shrinkage on data such as images and time series that can be naturally modeled as sparse Gaussian sequences. We illustrate this approach on facial recognition across varying facial expressions and study region-wise atypical brain lateralization in autism spectrum disorder.

Multivariate Pearson diffusions are characterized by a linear drift and a diffusion matrix that is quadratic in the state variables. We derive closed-form expressions for the mean and covariance matrix of this class using matrix exponential integrals, and extend this framework to a broader class of nonlinear diffusions with Pearson-type multiplicative noise. The main contribution is a new parameter estimator for these nonlinear multiplicative models based on Strang splitting, which decomposes the stochastic system into a deterministic nonlinear ordinary differential equation and a multivariate Pearson diffusion. The estimator is constructed by composing their respective flows and applying a Gaussian transition approximation with exact moments from the Pearson component. We prove that the estimator is consistent and asymptotically efficient. We also introduce a new model within this class, the Student Kramers oscillator, and prove existence and uniqueness of the strong solution and of an invariant measure. We evaluate the estimator through simulation studies on this oscillator and on the multivariate Wright-Fisher diffusion from population genetics, where it outperforms the Euler-Maruyama, Gaussian approximation, and local linearization estimators. We conclude with an application to Greenland ice core data using the Student Kramers oscillator.

Single-cell CRISPR screens summarize each perturbation by how far cells move from their unperturbed state, averaging over cell-to-cell variation. Whether the cells moved together is not captured: two perturbations with identical effect magnitude can differ qualitatively, one driving cells along a shared trajectory, the other scattering them around the same mean. We define Shesha\textsubscript{P} coherence ($S_p$), the mean cosine similarity between individual cell displacement vectors and their mean, and ask what it adds across six Perturb-seq datasets (2,285 perturbations; CRISPRa, CRISPRi, Cas9 knockout). Coherence and effect magnitude are closely coupled (Spearman $\rho =0.84$–$0.98$), and the coupling persists in a foundation-model embedding, under alternative effect-size definitions, and when analysis is restricted to responding cells. Associations reported without conditioning on effect size largely recover it, shown here for two comparisons that vanish under conditioning. Effect size accounts for 89–96% of coherence variance; within the remainder, coherence is negatively associated with apoptosis and p53 signalling in the two largest screens. Two standard responder classifiers agree on only 68% of cells. Directional coherence is largely a restatement of effect magnitude, and heterogeneity measures should report their redundancy with effect size as a matter of course. $S_p$ is implemented in the open-source shesha-geometry Python package.

Most fairness-aware learning methods assume that sensitive attributes are observed, an assumption that may fail due to privacy, legal, or data-collection constraints. We develop a framework for fairness-aware generalized linear models when the sensitive attribute is latent, possibly multi-category, and the predictors may be high-dimensional. Candidate proxy variables are modeled using Gaussian mixtures for continuous predictors and product-multinomial mixtures for categorical predictors. The resulting posterior group-membership probabilities are used to residualize the predictors and reduce their association with the latent sensitive structure. For continuous outcomes, we use constrained least squares to limit the contribution of the estimated sensitive attribute to prediction variability. For binary outcomes, we use penalized logistic regression to reduce dependence between predicted probabilities and estimated group membership. In high-dimensional settings, the procedure is combined with SEMMS variable selection to obtain sparse models. We establish identifiability and latent-class recovery results for Gaussian and categorical mixtures and derive expressions that quantify the loss of predictive power after residualization.Simulations and real-data applications demonstrate favorable accuracy-fairness performance.

We study phase transitions for repulsive-attractive mean-field free energies on the circle. For a $\frac{1}{n+1}$-periodic interaction whose Fourier coefficients satisfy a certain decay condition, we prove that the critical coupling strength $K_c$ coincides with the linear stability threshold $K_\#$ of the uniform distribution and that the phase transition is continuous, in the sense that the uniform distribution is the unique global minimizer at criticality. The proof is based on a sharp coercivity estimate for the free energy obtained from the constrained Lebedev–Milin inequality. We apply this result to three motivating models for which the exact value of the phase transition and its (dis)continuity in terms of the model parameters was not fully known. For the two-dimensional Doi–Onsager model $W(\theta)=-|\sin(2\pi\theta)|$, we prove that the phase transition is continuous at $K_c=K_\#=3\pi/4$. For the noisy transformer model $W_\beta(\theta)=(e^{\beta\cos(2\pi\theta)}-1)/\beta$, we identify the sharp threshold $\beta_*$ such that $K_c(\beta) = K_\#(\beta)$ and the phase transition is continuous for $\beta \leq \beta_*$, while $K_c(\beta)<K_\#(\beta)$ and the phase transition is discontinuous for $\beta > \beta_*$. We also obtain the corresponding sharp dichotomy for the noisy Hegselmann–Krause model $W_{R}(\theta) = (R-2\pi|\theta|)_{+}^2$ .

In multi-fidelity optimization, biased approximations of varying costs of the target function are available. This paper studies the problem of optimizing a locally smooth function with a limited budget, where the learner has to make a tradeoff between the cost and the bias of these approximations. We first prove lower bounds for the simple regret under different assumptions on the fidelities, based on a cost-to-bias function. We then present the Kometo algorithm which achieves, with additional logarithmic factors, the same rates without any knowledge of the function smoothness and fidelity assumptions, and improves previously proven guarantees. We finally empirically show that our algorithm outperforms previous multi-fidelity optimization methods without the knowledge of problem-dependent parameters.

Decision-makers rely on weather forecasts to plant crops, manage wildfires, allocate water and energy, and prepare for weather extremes. Today, such forecasts enjoy unprecedented accuracy out to two weeks thanks to steady advances in physics-based dynamical models and data-driven artificial intelligence (AI) models. However, model skill drops precipitously at subseasonal timescales (2 - 6 weeks ahead), due to compounding errors, systemic model biases, and the chaotic nature of the atmosphere. To counter this degradation, we introduce probabilistic bias correction (PBC), a machine learning framework that substantially reduces systematic error by learning to correct historical probabilistic forecasts. When applied to the leading dynamical and AI models from the European Centre for Medium-Range Weather Forecasts (ECMWF), PBC doubles the modest subseasonal skill of the AI Forecasting System and improves the skill of the operationally-debiased dynamical model for 91% of pressure, 92% of temperature, and 98% of precipitation targets. We designed PBC for operational deployment, and, in ECMWF's 2025 real-time forecasting competition, its global forecasts placed first for all weather variables and lead times, outperforming the dynamical models from six operational forecasting centers, an international dynamical multi-model ensemble, ECMWF's AI Forecasting System, and the forecasting systems of 34 teams worldwide. These probabilistic skill gains translate into more accurate prediction of extreme events and have the potential to improve agricultural planning, energy management, and disaster preparedness in vulnerable communities.

We show that the covariance matrix of the treatment effect estimates in a network meta-analysis can be obtained without matrix inversion using a geometric series of diffusion matrices. This property extends to the hat matrix and provides a connection between parameter estimation in regression analysis and random walks on the network graph. We also provide a number of visualization tools implemented in R.

For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices.