Papers

Matching-adjusted indirect comparison (MAIC) has been increasingly employed in health technology assessments (HTA). By reweighting subjects from a trial with individual participant data (IPD) to match the covariate summary statistics of another trial with only aggregate data (AgD), MAIC facilitates the estimation of a treatment effect defined with respect to the AgD trial population. This manuscript introduces a new class of methods, termed arbitrated indirect treatment comparisons, designed to address the “MAIC paradox” – a phenomenon highlighted by Jiang et al.~(2025). The MAIC paradox arises when different sponsors, analyzing the same data, reach conflicting conclusions regarding which treatment is more effective. The underlying issue is that each sponsor implicitly targets a different population. To resolve this inconsistency, the proposed methods focus on estimating treatment effects in a common target population, specifically chosen to be the overlap population.

Directional data consists of unit vectors in q-dimensions that can be described in polar or Cartesian coordinates. Axial data can be viewed as a pair of directions pointed in opposite directions or as a projection matrix of rank 1. Historically, their statistical analysis has largely been based on a few low-order exponential family models of distributions for random directions or axes. A lack of tractable algebraic forms for the normalizing constants has hindered the use of higher-order exponential families for less constrained modeling. Of interest are functionals of the unknown distribution of the directional/axial data, such as the directional/axial mean, dispersion, or distribution itself. This paper outlines nonparametric estimators and bootstrap confidence sets for such functionals. The procedures are based on the empirical distribution of the directional/axial sample expressed in Cartesian coordinates. Sketched as well are nonparametric comparisons among multiple mean directions or axes, estimation of trend in mean directions, and analysis of q-dimensional observations restricted to lie in a specified compact subset.

Argo is an international program that collects temperature and salinity observations in the upper two kilometers of the global ocean. Most existing approaches for modeling Argo temperature rely on localized modeling within moving windows, first estimating a prescribed mean structure and then fitting Gaussian processes (GPs) to the mean-subtracted anomalies. Such strategies introduce challenges in designing suitable mean structures and defining local moving windows, often resulting in case-specific modeling choices. In this work, we propose a one-stop Gaussian process regression framework with a flexible mean structure and a generic spatio-temporal covariance function to jointly model Argo temperature data across broad spatial domains. Our fully data-driven approach achieves predictive performance that compares favorably with the established benchmarks that require moving-window regression and separate parametric mean estimation. To ensure scalability over large spatial regions, we employ the Vecchia approximation, which reduces the computational complexity from cubic to quasi-linear in the number of observations while preserving predictive accuracy. Using Argo data from January to March over the years 2007-2016, the same dataset used in prior benchmark studies, we demonstrate that our approach provides a unified and data-driven alternative for large-scale oceanographic analysis.

We investigate the computational efficiency of agnostic learning for several fundamental geometric concept classes in the plane. While the sample complexity of agnostic learning is well understood, its time complexity has received much less attention. We study the class of triangles and, more generally, the class of convex polygons with $k$ vertices for small $k$, as well as the class of convex sets in a square. We present a proper agnostic learner for the class of triangles that has optimal sample complexity and runs in time $\tilde O({\epsilon^{-6}})$, improving on the algorithm of Dobkin and Gunopulos (COLT `95) that runs in time $\tilde O({\epsilon^{-10}})$. For 4-gons and 5-gons, we improve the running time from $O({\epsilon^{-12}})$, achieved by Fischer and Kwek (eCOLT `96), to $\tilde O({\epsilon^{-8}})$ and $\tilde O({\epsilon^{-10}})$, respectively. We also design a proper agnostic learner for convex sets under the uniform distribution over a square with running time $\tilde O({\epsilon^{-5}})$, improving on the previous $\tilde O(\epsilon^{-8})$ bound at the cost of slightly higher sample complexity. Notably, agnostic learning of convex sets in $[0,1]^2$ under general distributions is impossible because this concept class has infinite VC-dimension. Our agnostic learners use data structures and algorithms from computational geometry and their analysis relies on tools from geometry and probabilistic combinatorics. Because our learners are proper, they yield tolerant property testers with matching running times. Our results raise a fundamental question of whether a gap between the sample and time complexity is inherent for agnostic learning of these and other natural concept classes.

Neural activity forecasting is central to understanding neural systems and enabling closed-loop control. While deep learning has recently advanced the state-of-the-art in the time series forecasting literature, its application to neural activity forecasting remains limited. To bridge this gap, we systematically evaluated eight probabilistic deep learning models, including two foundation models, that have demonstrated strong performance on general forecasting benchmarks. We compared them against four classical statistical models and two baseline methods on spontaneous neural activity recorded from mouse cortex via widefield imaging. Across prediction horizons, several deep learning models consistently outperformed classical approaches, with the best model producing informative forecasts up to 1.5 seconds into the future. Our findings point toward future control applications and open new avenues for probing the intrinsic temporal structure of neural activity.

The assessment of monotone dependence between random variables is a classical problem in statistics and a gamut of application domains. Consequently, researchers have sought measures of association that are invariant under strictly increasing transformations of the margins, with the extant literature being splintered. For continuous variables, symmetric rank correlation coefficients, such as Spearman's Rho and Kendall's Tau, have been studied at great length in the statistical literature. For dichotomous outcomes, the asymmetric area under the curve (AUC) measure is used to assess monotone dependence. We unify and complete thus far disconnected strands of literature, by establishing common population level theory, common estimators, and common tests that bridge continuous and dichotomous settings and apply to all linearly ordered outcomes. Originating in the biomedical literature, the C index provides a bridge between AUC, to which it reduces for a dichotomous outcome, and Kendall's Tau, to which it relates linearly under continuity. To establish the same kind of bridge between AUC and Spearman's Rho, we introduce asymmetric grade correlation, AGC$(X,Y)$, as the covariance of the mid distribution function transforms, or grades, of $X$ and $Y$, divided by the variance of the grade of $Y$. The coefficient of monotone association then is CMA$(X,Y) = \frac{1}{2} ($AGC$(X,Y) + 1)$. When $X$ and $Y$ are continuous, AGC is symmetric and equals Spearman's Rho. When $Y$ is dichotomous, CMA equals AUC. We establish central limit theorems for the sample versions of these measures, and we develop tests of DeLong type for their equality with a shared outcome $Y$. In case studies, we assess progress in data-driven weather prediction and evaluate methods of uncertainty quantification for large language models.

Mathematical models in computational physics contain uncertain parameters that impact prediction accuracy. In turbulence modeling, this challenge is especially significant: Reynolds averaged Navier-Stokes (RANS) models, such as the Spalart-Allmaras (SA) model, are widely used for their speed and robustness but often suffer from inaccuracies and associated uncertainties due to imperfect model parameters. Reliable quantification of these uncertainties is becoming increasingly important in aircraft certification by analysis, where predictive credibility is critical. Bayesian inference provides a framework to estimate these parameters and quantify output uncertainty, but traditional methods are prohibitively expensive, especially when relying on high-fidelity simulations. We address the challenge of expensive Bayesian parameter estimation by developing a multi-fidelity framework that combines Markov chain Monte Carlo (MCMC) methods with multilevel Monte Carlo (MLMC) estimators to efficiently solve inverse problems. The MLMC approach requires correlated samples across different fidelity levels, achieved through a novel transport map-based coupling algorithm. We demonstrate a 50% reduction in inference cost compared to traditional single-fidelity methods on the challenging NACA0012 airfoil at high angles of attack near stall, while delivering realistic uncertainty bounds for model predictions in complex separated flow regimes. These results demonstrate that multi-fidelity approaches significantly improve turbulence parameter calibration, paving the way for more accurate and efficient aircraft certification by analysis.

This paper presents conservative probabilistic bounds for the spectrum of the admittance matrix and classical linear power flow models under uncertain network parameters; for example, probabilistic line contingencies. Our proposed approach imports tools from probability theory, such as concentration inequalities for random matrices. This provides a theoretical framework for understanding error bounds of common approximations of the AC power flow equations under parameter uncertainty, including the DC and LinDistFlow approximations. Additionally, we show that the upper bounds scale as functions of nodal criticality. This network-theoretic quantity captures how uncertainty concentrates at critical nodes for use in contingency analysis. We validate these bounds on IEEE test networks, demonstrating that they correctly capture the scaling behavior of spectral perturbations up to conservative constants.

Modern probabilistic regressors often remain overconfident under distribution shift. Functional Distribution Networks (FDN) place input-conditioned distributions over network weights, producing predictive mixtures whose dispersion adapts to the input; we train them with a Monte Carlo beta-ELBO objective. We pair FDN with an evaluation protocol that separates interpolation from extrapolation and emphasizes simple OOD sanity checks. On controlled 1D tasks and small/medium UCI-style regression benchmarks, FDN remains competitive in accuracy with strong Bayesian, ensemble, dropout, and hypernetwork baselines, while providing strongly input-dependent, shift-aware uncertainty and competitive calibration under matched parameter and update budgets.

Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into $\mathbb{R}^D$, losing key manifold features when the embedding dimension $D$ approaches $d$. On the other hand, methods that directly learn the latent manifold as a differentiable atlas have been relatively underexplored. In this paper, we aim to give a proof of concept of the effectiveness and potential of atlas-based methods. To this end, we implement a generic data structure to maintain a differentiable atlas that enables Riemannian optimization over the manifold. We complement this with an unsupervised heuristic that learns a differentiable atlas from point cloud data. We experimentally demonstrate that this approach has advantages in terms of efficiency and accuracy in selected settings. Moreover, in a supervised classification task over the Klein bottle and in RNA velocity analysis of hematopoietic data, we showcase the improved interpretability and robustness of our approach.

Vision-Language Models (VLMs) extend large language models with visual reasoning, but their multimodal design also introduces new, underexplored vulnerabilities. Existing multimodal red-teaming methods largely rely on brittle templates, focus on single-attack settings, and expose only a narrow subset of vulnerabilities. To address these limitations, we introduce VERA-V, a variational inference framework that recasts multimodal jailbreak discovery as learning a joint posterior distribution over paired text-image prompts. This probabilistic view enables the generation of stealthy, coupled adversarial inputs that bypass model guardrails. We train a lightweight attacker to approximate the posterior, allowing efficient sampling of diverse jailbreaks and providing distributional insights into vulnerabilities. VERA-V further integrates three complementary strategies: (i) typography-based text prompts that embed harmful cues, (ii) diffusion-based image synthesis that introduces adversarial signals, and (iii) structured distractors to fragment VLM attention. Experiments on HarmBench and HADES benchmarks show that VERA-V consistently outperforms state-of-the-art baselines on both open-source and frontier VLMs, achieving up to 53.75% higher attack success rate (ASR) over the best baseline on GPT-4o. We include the code on the project page available here: https://github.com/kxwhiowo/VERA-V

We consider Gaussian and bootstrap approximations for the supremum of additive functionals of aperiodic Harris recurrent Markov chains. The supremum is taken over a function class that may depend on the sample size, which allows for non-Donsker settings; that is, the empirical process need not have a weak limit in the space of bounded functions. We first establish a non-asymptotic Gaussian approximation error, which holds at rates comparable to those for sums of high-dimensional independent or one-dependent vectors. Key to our derivation is the Nummelin splitting technique, which enables us to decompose the chain into either independent or one-dependent random blocks. Additionally, building upon the Nummelin splitting, we propose a Gaussian multiplier bootstrap for practical inference and establish its finite-sample guarantees in the strongly aperiodic case. Finally, we apply our bootstrap to construct a uniform confidence band for an invariant density within a certain class of diffusion processes.

Penalty shootouts play a crucial role in the knockout stage of major football tournaments. Their importance has substantially increased from the 2021/22 season, when the Union of European Football Associations (UEFA) scrapped the away goals rule. Our paper examines whether the outcome of a penalty shootout can be predicted in UEFA club competitions. Based on all shootouts between 2000 and 2025, no evidence is found for the effect of the kicking order, the field of the match, or psychological momentum. In contrast to previous results, we do not detect any relationship between shootout success and relative team strength, quantified by differences in Elo ratings and the implied winning probability. Thus, the hypothesis that penalty shootouts are close to a coin toss in international competitions for European football clubs cannot be rejected.

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution – such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

This paper introduces a robust and computationally efficient estimation framework for high-dimensional volatility models in the BEKK-ARCH class. The proposed approach employs data truncation to ensure robustness against heavy-tailed distributions and utilizes a regularized least squares method for efficient optimization in high-dimensional settings. Non-asymptotic error bounds are established for the resulting estimators under heavy-tailed regimes, and the minimax optimal convergence rate is derived. Moreover, a robust BIC and a Ridge-type estimator are introduced for selecting the model order and the number of BEKK components, respectively, with their selection consistency established under heavy-tailed settings. Simulation studies demonstrate finite-sample performance of the proposed method, and two empirical applications illustrate its practical utility. The results show that the new framework outperforms existing alternatives in both computational speed and forecasting accuracy.

The modified odd Burr XII-G family is developed, capable of incorporating bimodal and bathtub shapes in its baseline distributions, with properties derived from the exponentiated-G class. A regression model is developed within this family. The parameters are estimated by maximum likelihood, and simulations are performed to verify their consistency. The usefulness of the proposals is demonstrated by means of three real data sets.

We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik-Ben Arous-Peche (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.

Many Mendelian randomization (MR) papers have been conducted only in people of European ancestry, limiting transportability of results to the global population. Expanding MR to diverse ancestry groups is essential to ensure equitable biomedical insights, yet presents analytical and conceptual challenges. This review examines the practical challenges of MR analyses beyond the European only context, including use of data from multi-ancestry, mismatched ancestry, and admixed populations. We explain how apparent heterogeneity in MR estimates between populations can arise from differences in genetic variant frequencies and correlation patterns, as well as from differences in the distribution of phenotypic variables, complicating the detection of true differences in the causal pathway. We summarize published strategies for selecting genetic instruments and performing analyses when working with limited ancestry-specific data, discussing the assumptions needed in each case for incorporating external data from different ancestry populations. We conclude that differences in MR estimates by ancestry group should be interpreted cautiously, with consideration of how the identified differences may arise due to social and cultural factors. Corroborating evidence of a biological mechanism altering the causal pathway is needed to support a conclusion of differing causal pathways between ancestry groups.

Data analysis of files that are a result of linking records from multiple sources are often affected by linkage errors. Records may be linked incorrectly, or their links may be missed. In consequence, it is essential that such errors are taken into account to ensure valid post-linkage inference. Here, we propose an extension to a general framework for regression with linked covariates and responses based on a two-component mixture model, which was developed in prior work. This framework addresses the challenging case of secondary analysis in which only the linked data is available and information about the record linkage process is limited. The extension considered herein relaxes the assumption of strongly non-informative linkage in the framework according to which linkage does not depend on the covariates used in the analysis, which may be limiting in practice. The effectiveness of the proposed extension is investigated by simulations and a case study.

The main objective of dose finding trials is to find an optimal dose amongst a candidate set for further research. The trial design in oncology proceeds in stages with a decision as to how to treat the next group of patients made at every stage until a final sample size is reached or the trial stopped early. This work applies a Bayesian decision-theoretic approach to the problem, proposing a new utility function based on both efficacy and toxicity and grounded in von Neumann-Morgenstern (VNM) utility theory. Our proposed framework seeks to better capture real clinical judgements by allowing attitudes to risk to vary when the judgements are of gains or losses, which are defined with respect to an intermediate outcome known as a reference point. We call this method Reference Dependent Decision Theoretic dose finding (R2DT). A simulation study demonstrates that the framework can perform well and produce good operating characteristics. The simulation results demonstrate that R2DT is better at detecting the optimal dose in scenarios where candidate doses are around minimum acceptable efficacy and maximum acceptable toxicity thresholds. The proposed framework shows that a flexible utility function, which better captures clinician beliefs, can lead to trials with good operating characteristics, including a high probability of finding the optimal dose. Our work demonstrates proof-of-concept for this framework, which should be evaluated in a broader range of settings.

Edge intelligence enables low-latency inference via compact on-device models, but assuring reliability remains challenging. We study edge-cloud cascades that must preserve conditional coverage: whenever the edge returns a prediction set, it should contain the true label with a user-specified probability, as if produced by the cloud model. We formalize conditional coverage with respect to the cloud predictive distribution, and introduce a conformal alignment-based (CAb) cascading mechanism that certifies this property with user control over the risk level. Our method casts escalation from edge to cloud models as a multiple-hypothesis testing (MHT) problem, tailoring conformal alignment (CA) to select which inputs can be safely handled at the edge. The proposed CAb model cascading method yields statistical guarantees on the average fraction of edge decisions that satisfy cloud-level conditional coverage. The procedure applies to arbitrary edge prediction sets, including variants of conformal prediction (CP), and exposes a tunable trade-off among coverage, deferral rate, and set size. Experiments on CIFAR-100 image classification and the TeleQnA question-answering (QA) benchmark show that the proposed CAb cascade maintains the target conditional coverage for edge predictions while substantially reducing offloading to the cloud and incurring modest increases in prediction-set size.

Testing the equivalence of multiple quantiles between two populations is important in many scientific applications, such as clinical trials, where conventional mean-based methods may be inadequate. This is particularly relevant in bridging studies that compare drug responses across different experimental conditions or patient populations. These studies often aim to assess whether a proposed dose for a target population achieves pharmacokinetic levels comparable to those of a reference population where efficacy and safety have been established. The focus is on extreme quantiles which directly inform both efficacy and safety assessments. When analyzing heterogeneous Gaussian samples, where a single quantile of interest is estimated, the existing Two One-Sided Tests method for quantile equivalence testing (qTOST) tends to be overly conservative. To mitigate this behavior, we introduce $\alpha$-qTOST, a finite-sample adjustment that achieves uniformly higher power compared to qTOST while maintaining the test size at the nominal level. Moreover, we extend the quantile equivalence framework to simultaneously assess equivalence across multiple quantiles. Through theoretical guarantees and an extensive simulation study, we demonstrate that $\alpha$-qTOST offers substantial improvements, especially when testing extreme quantiles under heteroskedasticity and with small, unbalanced sample sizes. We illustrate these advantages through two case studies, one in HIV drug development, where a bridging clinical trial examines exposure distributions between male and female populations with unbalanced sample sizes, and another in assessing the reproducibility of an identical experimental protocol performed by different operators for generating biodistribution profiles of topically administered and locally acting products.

We study the online stochastic difference-of-convex (DC) program $\min_{x \in R^n} F(x) = G(x) - H(x) + r_1(x) - r_2(x)$, which covers sparse learning with nonconvex penalties, robust regression with smooth robust losses, and non-negative positive–unlabeled learning. We propose momentum-based DCA (MDCA), which combines a damped DCA step with a momentum estimator for $\nabla H$ and uses a constant per-iteration batch size. With exact solutions of the convex subproblems, MDCA with Polyak momentum uses $\mathcal{O}( \epsilon^{-4})$ stochastic gradients of $H$ to reach $\epsilon$-stationarity, and momentum-based variance reduction (MVR) uses $\mathcal{O}(\epsilon^{-3})$ under averaged smoothness, with a single $\mathcal O(\epsilon^{-1})$ warm-up batch in place of the periodic large batches of DCA-PAGE, whose $\epsilon$-dependence it matches. We also introduce a single-loop variant (S-MDCA) that replaces the subproblem solve by one stochastic prox-gradient step; for every fixed smoothing parameter $\gamma$ for which the separately smoothed surrogate $F_\gamma$ is bounded below, it reaches $\epsilon$-stationarity of $F_\gamma$ in $\mathcal{O}(\epsilon^{-4})$ iterations, which does not by itself imply stationarity of $F$. A variance–drift trade-off lemma describes how the momentum parameter trades estimator variance against iterate-drift bias in steady-state upper bounds. Guarantees are given for two criteria: the critical-distance measure, and a DCA fixed-point residual that requires no smoothness of $r_2$ and vanishes exactly at DC-critical points; both give the same $\epsilon$-dependence. Experiments on two LIBSVM classification datasets illustrate the methods' behaviour.

Recent advances such as self-consistency and test-time reinforcement learning (TTRL) improve the reliability of large language models (LLMs) without additional supervision, yet their underlying mechanisms and statistical guarantees remain poorly understood. We present a unified framework for certifiable inference in LLMs, showing that majority voting provides a statistical certificate of self-consistency: under mild assumptions, the aggregated answer coincides with the mode of the model's terminal distribution with high probability. We derive finite-sample and anytime-valid concentration bounds that quantify this confidence, and introduce the Martingale Majority Certificate (MMC), a sequential stopping rule that adaptively determines when sufficient samples have been drawn. We further prove that label-free post-training methods such as TTRL implicitly sharpen the answer distribution by exponentially tilting it toward its mode, thereby reducing the number of samples required for certification. Building on this insight, we propose new post-training objectives that explicitly optimise this trade-off between sharpness and bias. Together, these results explain and connect two central test-time scaling strategies, self-consistency and TTRL, within a single statistical framework for label-free, certifiable reliability in reasoning LLMs.

Time series data are often affected by various forms of corruption, such as missing values, noise, and outliers, which pose significant challenges for tasks such as forecasting and anomaly detection. To address these issues, inverse problems focus on reconstructing the original signal from corrupted data by leveraging prior knowledge about its underlying structure. While deep learning methods have demonstrated potential in this domain, they often require extensive pretraining and struggle to generalize under distribution shifts. In this work, we propose RINS-T (Robust Implicit Neural Solvers for Time Series Linear Inverse Problems), a novel deep prior framework that achieves high recovery performance without requiring pretraining data. RINS-T leverages neural networks as implicit priors and integrates robust optimization techniques, making it resilient to outliers while relaxing the reliance on Gaussian noise assumptions. To further improve optimization stability and robustness, we introduce three key innovations: guided input initialization, input perturbation, and convex output combination techniques. Each of these contributions strengthens the framework's optimization stability and robustness. These advancements make RINS-T a flexible and effective solution for addressing complex real-world time series challenges. Our code is available at https://github.com/EPFL-IMOS/RINS-T.

We propose feature perturbation, a simple yet effective exploration strategy for contextual bandits that injects randomness directly into feature inputs, instead of randomizing unknown parameters or adding noise to rewards. Remarkably, this algorithm achieves $\tilde{\mathcal{O}}(d\sqrt{T})$ worst-case regret bound for generalized linear contextual bandits, while avoiding the $\tilde{\mathcal{O}}(d^{3/2}\sqrt{T})$ regret typical of existing randomized bandit algorithms. Because our algorithm eschews parameter sampling, it is both computationally efficient and naturally extends to non-parametric or neural network models. We verify these advantages through empirical evaluations, demonstrating that feature perturbation not only surpasses existing methods but also unifies strong practical performance with the near-optimal regret guarantees.

Best Arm Identification (BAI) algorithms are deployed in data-sensitive applications, such as adaptive clinical trials or user studies. Driven by the privacy concerns of these applications, we study the problem of fixed-confidence BAI under global Differential Privacy (DP) for Bernoulli distributions. While numerous asymptotically optimal BAI algorithms exist in the non-private setting, a significant gap remains between the best lower and upper bounds in the global DP setting. This work reduces this gap to a small multiplicative constant, for any privacy budget $\epsilon$. First, we provide a tighter lower bound on the expected sample complexity of any $\delta$-correct and $\epsilon$-global DP strategy. Our lower bound replaces the Kullback-Leibler (KL) divergence in the transportation cost used by the non-private characteristic time with a new information-theoretic quantity that optimally trades off between the KL divergence and the Total Variation distance scaled by $\epsilon$. Second, we introduce a stopping rule based on these transportation costs and a private estimator of the means computed using an arm-dependent geometric batching. En route to proving the correctness of our stopping rule, we derive concentration results of independent interest for the Laplace distribution and for the sum of Bernoulli and Laplace distributions. Third, we propose a Top Two sampling rule based on these transportation costs. For any budget $\epsilon$, we show an asymptotic upper bound on its expected sample complexity that matches our lower bound to a multiplicative constant smaller than $8$. Our algorithm outperforms existing $\delta$-correct and $\epsilon$-global DP BAI algorithms for different values of $\epsilon$.

In this paper, we consider high-dimensional Lp-quantile regression which only requires a low order moment of the error and is also a natural generalization of the above methods and Lp-regression as well. The loss function of Lp-quantile regression circumvents the non-differentiability of the absolute loss function and the difficulty of the squares loss function requiring the finiteness of error's variance and thus promises excellent properties of Lp-quantile regression. Specifically, we first develop a new method called composite Lp-quantile regression(CLpQR). We study the oracle model selection theory based on CLpQR (call the estimator CLpQR-oracle) and show in some cases of p CLpQR-oracle behaves better than CQR-oracle (based on composite quantile regression) when error's variance is infinite. Moreover, CLpQR has high efficiency and can be sometimes arbitrarily more efficient than both CQR and the least squares regression. Second, we propose another new regression method,i.e. near quantile regression and prove the asymptotic normality of the estimator when p converges to 1 and the sample size infinity simultaneously. As its applications, a new thought of smoothing quantile objective functions and a new estimation are provided for the asymptotic covariance matrix of quantile regression. Third, we develop a unified efficient algorithm for fitting high-dimensional Lp-quantile regression by combining the cyclic coordinate descent and an augmented proximal gradient algorithm. Remarkably, the algorithm turns out to be a favourable alternative of the commonly used liner programming and interior point algorithm when fitting quantile regression.

Symmetries are known to improve the empirical performance of machine learning models, yet theoretical guarantees explaining these gains remain limited. Prior work has focused mainly on compact group symmetries and often assumes that the data distribution itself is invariant, an assumption rarely satisfied in real-world applications. In this work, we extend generalization guarantees to the broader setting of non-compact symmetries, such as translations and to non-invariant data distributions. Building on the PAC-Bayes framework, we adapt and tighten existing bounds, demonstrating the approach on McAllester's PAC-Bayes bound while showing that it applies to a wide range of PAC-Bayes bounds. We validate our theory with experiments on several datasets with non-uniform and non-compact transformations, where the derived guarantees not only hold but also improve upon prior results. These findings provide theoretical evidence that, for symmetric data, symmetric models are preferable beyond the narrow setting of compact groups and invariant distributions, opening the way to a more general understanding of symmetries in machine learning.

Data assimilation, consisting in the combination of a dynamical model with a set of noisy and incomplete observations in order to infer the state of a system over time, involves uncertainty in most settings. Building upon an existing deterministic machine learning approach, we propose a variational inference-based extension in which the predicted state follows a multivariate Gaussian distribution. Using the chaotic Lorenz-96 dynamics as a testing ground, we show that our new model enables to obtain nearly perfectly calibrated predictions, and can be integrated in a wider variational data assimilation pipeline in order to achieve greater benefit from increasing lengths of data assimilation windows. Our code is available at https://github.com/anthony-frion/Stochastic_CODA.