Papers

We introduce a cumulant-expansion framework for quantifying how large language models (LLMs) internalize higher-order statistical structure during next-token prediction. By treating the softmax entropy of each layer's logit distribution as a perturbation around its "center" distribution, we derive closed-form cumulant observables that isolate successively higher-order correlations. Empirically, we track these cumulants in GPT-2 and Pythia models on Pile-10K prompts. (i) Structured prompts exhibit a characteristic rise-and-plateau profile across layers, whereas token-shuffled prompts remain flat, revealing the dependence of the cumulant profile on meaningful context. (ii) During training, all cumulants increase monotonically before saturating, directly visualizing the model's progression from capturing variance to learning skew, kurtosis, and higher-order statistical structures. (iii) Mathematical prompts show distinct cumulant signatures compared to general text, quantifying how models employ fundamentally different processing mechanisms for mathematical versus linguistic content. Together, these results establish cumulant analysis as a lightweight, mathematically grounded probe of feature-learning dynamics in high-dimensional neural networks.

The sample complexity of estimating or maximising an unknown function in a reproducing kernel Hilbert space is known to be linked to both the effective dimension and the information gain associated with the kernel. While the information gain has an attractive information-theoretic interpretation, the effective dimension typically results in better rates. We introduce a new quantity called the relative information gain, which measures the sensitivity of the information gain with respect to the observation noise. We show that the relative information gain smoothly interpolates between the effective dimension and the information gain, and that the relative information gain has the same growth rate as the effective dimension. In the second half of the paper, we prove a new PAC-Bayesian excess risk bound for Gaussian process regression. The relative information gain arises naturally from the complexity term in this PAC-Bayesian bound. We prove bounds on the relative information gain that depend on the spectral properties of the kernel. When these upper bounds are combined with our excess risk bound, we obtain minimax-optimal rates of convergence.

Learning graphical conditional independence structures from nonlinear, continuous or mixed data is a central challenge in machine learning and the sciences, and many existing methods struggle to scale to thousands of samples or hundreds of variables. We introduce two basis-expansion tools for scalable causal discovery. First, the Basis Function BIC (BF-BIC) score uses truncated additive expansions to approximate nonlinear dependencies. BF-BIC is theoretically consistent under additive models and extends to post-nonlinear (PNL) models via an invertible reparameterization. It remains robust under moderate interactions and supports mixed data through a degenerate-Gaussian embedding for discrete variables. In simulations with fully nonlinear neural causal models (NCMs), BF-BIC outperforms kernel- and constraint-based methods (e.g., KCI, RFCI) in both accuracy and runtime. Second, the Basis Function Likelihood Ratio Test (BF-LRT) provides an approximate conditional independence test that is substantially faster than kernel tests while retaining competitive accuracy. Extensive simulations and a real-data application to Canadian wildfire risk show that, when integrated into hybrid searches, BF-based methods enable interpretable and scalable causal discovery. Implementations are available in Python, R, and Java.

Pass$@k$ is widely used to report the reasoning performance of LLMs, but it often produces unstable and potentially misleading rankings, especially when the number of trials (samples) is limited and computational resources are constrained. We present a principled Bayesian evaluation framework that replaces Pass$@k$ and average accuracy over $N$ trials (avg$@N$) with posterior estimates of a model's underlying success probability and credible intervals, yielding stable rankings and a transparent decision rule for differences. Evaluation outcomes are modeled as categorical (not just 0/1) with a Dirichlet prior, giving closed-form expressions for the posterior mean and uncertainty of any weighted rubric and enabling the use of prior evidence when appropriate. Theoretically, under a uniform prior, the Bayesian posterior mean is order-equivalent to average accuracy (Pass$@1$), explaining its empirical robustness while adding principled uncertainty. Empirically, in simulations with known ground-truth success rates and on AIME'24/'25, HMMT'25, and BrUMO'25, the posterior-based procedure achieves faster convergence and greater rank stability than Pass$@k$ and recent variants, enabling reliable comparisons at far smaller sample counts. The framework clarifies when observed gaps are statistically meaningful (non-overlapping credible intervals) versus noise, and it naturally extends to graded, rubric-based evaluations. Together, these results recommend replacing Pass$@k$ for LLM evaluation and ranking with a posterior-based, compute-efficient protocol that unifies binary and non-binary evaluation while making uncertainty explicit. Source code is available at https://github.com/mohsenhariri/scorio

Gaussian random field is a ubiquitous model for spatial phenomena in diverse scientific disciplines. Its approximation is often crucial for computational feasibility in simulation, inference, and uncertainty quantification. The Karhunen-Loève Expansion provides a theoretically optimal basis for representing a Gaussian random field as a sum of deterministic orthonormal functions weighted by uncorrelated random variables. While this is a well-established method for dimension reduction and approximation of (spatial) stochastic processes, its practical application depends on the explicit or implicit definition of the covariance structure. In this work, we propose a novel approach, referred to as regTPS-KLE, for approximating a Gaussian random field by explicitly constructing its covariance via a regularized thin plate spline (TPS) kernel. Because TPS kernels are conditionally positive definite and lack a direct spectral decomposition, we formulate the covariance as the inverse of a regularized elliptic operator. To evaluate its statistical performance, we compare its predictive accuracy and computational efficiency with a Gaussian random field approximation constructed using the stochastic partial differential equations (SPDE) method and implemented within an MCMC algorithm. In simulation studies, the predictive differences between the SPDE and regTPS-KLE models were minimal when the spatial field was generated using Matèrn and exponential covariance functions, while regTPS-KLE models consistently outperformed the SPDE approach in terms of computational efficiency. In a real data application, regTPS-KLE exhibits superior predictive accuracy compared with SPDE models based on leave-one-out cross-validation while also achieving improved computational efficiency.

In this paper, we establish the Central Limit Theorem (CLT) for linear spectral statistics (LSSs) of large-dimensional generalized spiked sample covariance matrices, where the spiked eigenvalues may be either bounded or diverge to infinity. Building upon this theorem, we derive the asymptotic distributions of linear hypothesis test statistics under the generalized spiked model, including Wilks' likelihood ratio test statistic U, the Lawley-Hotelling trace test statistic W, and the Bartlett-Nanda-Pillai trace test statistic V. Due to the complexity of the test functions, explicit solutions for the contour integrals in our calculations are generally intractable. To address this, we employ Taylor series expansions to approximate the theoretical results in the asymptotic regime. We also derive asymptotic power functions for three test criteria above, and make comparisons with Roy's largest root test under specific scenarios. Finally, numerical simulations are conducted to validate the accuracy of our asymptotic approximations.

Effective Uncertainty Quantification (UQ) represents a key aspect for reliable deployment of Large Language Models (LLMs) in automated decision-making and beyond. Yet, for LLM generation with multiple choice structure, the state-of-the-art in UQ is still dominated by the naive baseline given by the maximum softmax score. To address this shortcoming, we demonstrate that taking a principled approach via Bayesian statistics leads to improved performance despite leveraging the simplest possible model, namely linear regression. More precisely, we propose to train multiple Bayesian linear models, each predicting the output of a layer given the output of the previous one. Based on the obtained layer-level posterior distributions, we infer the global uncertainty level of the LLM by identifying a sparse combination of distributional features, leading to an efficient UQ scheme. Numerical experiments on various LLMs show consistent improvement over state-of-the-art baselines.

This article introduces the theory of offline reinforcement learning in large state spaces, where good policies are learned from historical data without online interactions with the environment. Key concepts introduced include expressivity assumptions on function approximation (e.g., Bellman completeness vs. realizability) and data coverage (e.g., all-policy vs. single-policy coverage). A rich landscape of algorithms and results is described, depending on the assumptions one is willing to make and the sample and computational complexity guarantees one wishes to achieve. We also discuss open questions and connections to adjacent areas.

Modern preference alignment techniques, such as Best-of-N (BoN) sampling, rely on reward models trained with pairwise comparison data. While effective at learning relative preferences, this paradigm fails to capture a signal of response acceptability, leaving systems vulnerable to selecting the least bad of many unacceptable options. This is particularly problematic for hard prompts, where the risk of such false acceptances increases with the number of samples. In this paper, we address this critical reliability gap by introducing a new data collection and modeling framework. By augmenting preference data with an outside option, inspired by discrete choice models, we train a reward model that can distinguish not just what is better, but what is good enough. We leverage this capability to create an adaptive inference strategy, best of mini-N in-loop, which partitions the generation budget into sequential loops with a calibrated, early-exit condition. Our experiments show that when tuned as an alignment guardrail, it reduces reliability failures by 70%, and when tuned as an inference accelerator, it improves average inference speed by over 22% in IMDB-sentiment setting. We thus provide a principled and flexible framework for practitioners to explicitly manage the trade-off between reliability and computational efficiency.

Reinforcement learning (RL) has become central to enhancing reasoning in large language models (LLMs). Yet on-policy algorithms such as Group Relative Policy Optimization (GRPO) often suffer in early training: noisy gradients from low-quality rollouts lead to unstable updates and inefficient exploration. We introduce Slow-Fast Policy Optimization (SFPO), a simple yet efficient framework to address the above limitations via decomposing each step into three stages: a short fast trajectory of inner steps on the same batch, a reposition mechanism to control off-policy drift, and a final slow correction. This reposition-before-update design preserves the objective and rollout process unchanged, making SFPO plug-compatible with existing policy-gradient pipelines. Extensive experiments demonstrate that SFPO consistently improves stability, reduces number of rollouts, and accelerates convergence of reasoning RL training. Specifically, it outperforms GRPO by up to 2.80 points in average on math reasoning benchmarks. It also achieves up to 4.93\texttimes{} fewer rollouts and an up to 4.19\texttimes{} reduction in wall-clock time to match GRPO's best accuracy. Project website is available at https://slow-fast-po.github.io/.

The growing availability of large and complex datasets has increased interest in temporal stochastic processes that can capture stylized facts such as marginal skewness, non-Gaussian tails, long memory, and even non-Markovian dynamics. While such models are often easy to simulate from, parameter estimation remains challenging. Simulation-based inference (SBI) offers a promising way forward, but existing methods typically require large training datasets or complex architectures and frequently yield confidence (credible) regions that fail to attain their nominal values, raising doubts on the reliability of estimates for the very features that motivate the use of these models. To address these challenges, we propose a fast and accurate, sample-efficient SBI framework for amortized posterior inference applicable to intractable stochastic processes. The proposed approach relies on two main steps: first, we learn the posterior density by decomposing it sequentially across parameter dimensions. Then, we use Chebyshev polynomial approximations to efficiently generate independent posterior samples, enabling accurate inference even when Markov chain Monte Carlo methods mix poorly. We further develop novel diagnostic tools for SBI in this context, as well as post-hoc calibration techniques; the latter not only lead to performance improvements of the learned inferential tool, but also to the ability to reuse it directly with new time series of varying lengths, thus amortizing the training cost. We demonstrate the method's effectiveness on trawl processes, a class of flexible infinitely divisible models that generalize univariate Gaussian processes, applied to energy demand data.

This paper investigates the score-based diffusion models for density estimation when the target density admits a factorizable low-dimensional nonparametric structure. To be specific, we show that when the log density admits a $d^*$-way interaction model with $\beta$-smooth components, the vanilla diffusion model, which uses a fully connected ReLU neural network for score matching, can attain optimal $n^{-\beta/(2\beta+d^*)}$ statistical rate of convergence in total variation distance. This is, to the best of our knowledge, the first in the literature showing that diffusion models with standard configurations can adapt to the low-dimensional factorizable structures. The main challenge is that the low-dimensional factorizable structure no longer holds for most of the diffused timesteps, and it is very challenging to show that these diffused score functions can be well approximated without a significant increase in the number of network parameters. Our key insight is to demonstrate that the diffused score functions can be decomposed into a composition of either super-smooth or low-dimensional components, leading to a new approximation error analysis of ReLU neural networks with respect to the diffused score function. The rate of convergence under the 1-Wasserstein distance is also derived with a slight modification of the method.

We study the optimal sample complexity of variable selection in linear regression under general design covariance, and show that subset selection is optimal while under standard complexity assumptions, efficient algorithms for this problem do not exist. Specifically, we analyze the variable selection problem and provide the optimal sample complexity with exact dependence on the problem parameters for both known and unknown sparsity settings. Moreover, we establish a sample complexity lower bound for any efficient estimator, highlighting a gap between the statistical efficiency achievable by combinatorial algorithms (such as subset selection) compared to efficient algorithms (such as those based on convex programming). The proofs rely on a finite-sample analysis of an information criterion estimator, which may be of independent interest. Our results emphasize the optimal position of subset selection, the critical role played by restricted eigenvalues, and characterize the statistical-computational trade-off in high-dimensional variable selection.

Hidden Markov models (HMMs) have been used increasingly to understand how movement patterns of animals arise from behavioural states. An animal is assumed to transition between behavioural states through time, as described by transition probabilities. Within each state, the movement typically follows a discrete-time random walk, where steps between successive observed locations are described in terms of step lengths (related to speed) and turning angles (related to tortuosity). HMMs are discrete-time models, and most of their outputs strongly depend on the temporal resolution of data. We compile known theoretical results about scale dependence in Markov chains and correlated random walks, which are the most common components of HMMs for animal movement. We also illustrate this phenomenon using simulations covering a wide range of biological scenarios. The scale dependence affects not only all model parameters, i.e., the transition probabilities and the movement parameters within each behavioural state, but also the overall classification of movement patterns into states. This highlights the importance of carefully considering the time resolution when drawing conclusions from the results of analysis. In addition, scale dependence generally precludes the analysis of tracking data collected at irregular time intervals, and the comparison (or combination) of data sets with different sampling rates. HMMs remain a valuable tool to answer questions about animal movement and behaviour, as long as these limitations are well understood.

Simulating the kinetic Langevin dynamics is a popular approach for sampling from distributions, where only their unnormalized densities are available. Various discretizations of the kinetic Langevin dynamics have been considered, where the resulting algorithm is collectively referred to as the kinetic Langevin Monte Carlo (KLMC) or underdamped Langevin Monte Carlo. Specifically, the stochastic exponential Euler discretization, or exponential integrator for short, has previously been studied under strongly log-concave and log-Lipschitz smooth potentials via the synchronous Wasserstein coupling strategy. Existing analyses, however, impose restrictions on the parameters that do not explain the behavior of KLMC under various choices of parameters. In particular, all known results fail to hold in the overdamped regime, suggesting that the exponential integrator degenerates in the overdamped limit. In this work, we revisit the synchronous Wasserstein coupling analysis of KLMC with the exponential integrator. Our refined analysis results in Wasserstein contractions and bounds on the asymptotic bias that hold under weaker restrictions on the parameters, which assert that the exponential integrator is capable of stably simulating the kinetic Langevin dynamics in the overdamped regime, as long as proper time acceleration is applied.

We present self-speculative masked diffusions, a new class of masked diffusion generative models for discrete data that require significantly fewer function evaluations to generate samples. Standard masked diffusion models predict factorized logits over currently masked positions. A number of masked positions are then sampled, however, the factorization approximation means that sampling too many positions in one go leads to poor sample quality. As a result, many simulation steps and therefore neural network function evaluations are required to generate high-quality data. We reduce the computational burden by generating non-factorized predictions over masked positions. This is achieved by modifying the final transformer attention mask from non-causal to causal, enabling draft token generation and parallel validation via a novel, model-integrated speculative sampling mechanism. This results in a non-factorized predictive distribution over masked positions in a single forward pass. We apply our method to GPT2 scale text modelling and protein sequence generation, finding that we can achieve a ~2x reduction in the required number of network forward passes relative to standard masked diffusion models.

Despite recent progress in optimal hyperparameter transfer under model and dataset scaling, no unifying explanatory principle has been established. For Adam and Scion optimizers, we discover that joint optimal scaling across model and dataset sizes is conditioned on a single invariant: the operator norm of the output layer. Across models with up to 1.3B parameters trained on up to 138B tokens, the optimal learning rate/batch size pair $(\eta^{\ast}, B^{\ast})$ consistently has the same operator norm value - a phenomenon we term norm transfer. This constant norm condition is necessary but not sufficient: while for each dataset size, multiple $(\eta, B)$ reach the optimal norm, only a unique $(\eta^{\ast}, B^{\ast})$ achieves the best loss. As a sufficient condition, we provide the first measurement of $(\eta^{\ast}, B^{\ast})$ scaling with dataset size for Scion, and find that the scaling rules are consistent with those of Adam. Tuning per-layer-group learning rates also improves model performance, with the output layer being the most sensitive and hidden layers benefiting from lower learning rates. We provide practical insights on norm-guided optimal scaling and release our Distributed Scion (Disco) implementation with logs from over two thousand runs to support research on LLM training dynamics at scale.

The recently introduced optimizer, Muon, has gained increasing attention due to its superior performance across a wide range of applications. However, its effectiveness in federated learning remains unexplored. To address this gap, this paper investigates the performance of Muon in the federated learning setting. Specifically, we propose a new algorithm, FedMuon, and establish its convergence rate for nonconvex problems. Our theoretical analysis reveals multiple favorable properties of FedMuon. In particular, due to its orthonormalized update direction, the learning rate of FedMuon is independent of problem-specific parameters, and, importantly, it can naturally accommodate heavy-tailed noise. The extensive experiments on a variety of neural network architectures validate the effectiveness of the proposed algorithm.

This paper investigates a class of games with large strategy spaces, motivated by challenges in AI alignment and language games. We introduce the hidden game problem, where for each player, an unknown subset of strategies consistently yields higher rewards compared to the rest. The central question is whether efficient regret minimization algorithms can be designed to discover and exploit such hidden structures, leading to equilibrium in these subgames while maintaining rationality in general. We answer this question affirmatively by developing a composition of regret minimization techniques that achieve optimal external and swap regret bounds. Our approach ensures rapid convergence to correlated equilibria in hidden subgames, leveraging the hidden game structure for improved computational efficiency.

Objective: Electronic health records (EHR) data are prone to missingness and errors. Previously, we devised an "enriched" chart review protocol where a "roadmap" of auxiliary diagnoses (anchors) was used to recover missing values in EHR data (e.g., a diagnosis of impaired glycemic control might imply that a missing hemoglobin A1c value would be considered unhealthy). Still, chart reviews are expensive and time-intensive, which limits the number of patients whose data can be reviewed. Now, we investigate the accuracy and scalability of a roadmap-driven algorithm, based on ICD-10 codes (International Classification of Diseases, 10th revision), to mimic expert chart reviews and recover missing values. Materials and Methods: In addition to the clinicians' original roadmap from our previous work, we consider new versions that were iteratively refined using large language models (LLM) in conjunction with clinical expertise to expand the list of auxiliary diagnoses. Using chart reviews for 100 patients from the EHR at an extensive learning health system, we examine algorithm performance with different roadmaps. Using the larger study of $1000$ patients, we applied the final algorithm, which used a roadmap with clinician-approved additions from the LLM. Results: The algorithm recovered as much, if not more, missing data as the expert chart reviewers, depending on the roadmap. Discussion: Clinically-driven algorithms (enhanced by LLM) can recover missing EHR data with similar accuracy to chart reviews and can feasibly be applied to large samples. Extending them to monitor other dimensions of data quality (e.g., plausability) is a promising future direction.

We present a theoretical framework for M-FISHER, a method for sequential distribution shift detection and stable adaptation in streaming data. For detection, we construct an exponential martingale from non-conformity scores and apply Ville's inequality to obtain time-uniform guarantees on false alarm control, ensuring statistical validity at any stopping time. Under sustained shifts, we further bound the expected detection delay as $\mathcal{O}(\log(1/\delta)/\Gamma)$, where $\Gamma$ reflects the post-shift information gain, thereby linking detection efficiency to distributional divergence. For adaptation, we show that Fisher-preconditioned updates of prompt parameters implement natural gradient descent on the distributional manifold, yielding locally optimal updates that minimize KL divergence while preserving stability and parameterization invariance. Together, these results establish M-FISHER as a principled approach for robust, anytime-valid detection and geometrically stable adaptation in sequential decision-making under covariate shift.

When training data are fragmented across batches or federated-learned across different geographic locations, trained models manifest performance degradation. That degradation partly owes to covariate shift induced by data having been fragmented across time and space and producing dissimilar empirical training distributions. Each fragment's distribution is slightly different to a hypothetical unfragmented training distribution of covariates, and to the single validation distribution. To address this problem, we propose Fisher Information for Robust fEderated validation (FIRE). This method accumulates fragmentation-induced covariate shift divergences from the global training distribution via an approximate Fisher information. That term, which we prove to be a more computationally-tractable estimate, is then used as a per-fragment loss penalty, enabling scalable distribution alignment. FIRE outperforms importance weighting benchmarks by $5.1\%$ at maximum and federated learning (FL) benchmarks by up to $5.3\%$ on shifted validation sets.

Start-ups and product grade-changes are critical steps in continuous-process plant operation, because any misstep immediately affects product quality and drives operational losses. These transitions have long relied on manual operation by a handful of expert operators, but the progressive retirement of that workforce is leaving plant owners without the tacit know-how needed to execute them consistently. In the absence of a process model, offline reinforcement learning (RL) promises to capture and even surpass human expertise by mining historical start-up and grade-change logs, yet standard offline RL struggles with distribution shift and value-overestimation whenever a learned policy ventures outside the data envelope. We introduce HOFLON (Hybrid Offline Learning + Online Optimization) to overcome those limitations. Offline, HOFLON learns (i) a latent data manifold that represents the feasible region spanned by past transitions and (ii) a long-horizon Q-critic that predicts the cumulative reward from state-action pairs. Online, it solves a one-step optimization problem that maximizes the Q-critic while penalizing deviations from the learned manifold and excessive rates of change in the manipulated variables. We test HOFLON on two industrial case studies: a polymerization reactor start-up and a paper-machine grade-change problem, and benchmark it against Implicit Q-Learning (IQL), a leading offline-RL algorithm. In both plants HOFLON not only surpasses IQL but also delivers, on average, better cumulative rewards than the best start-up or grade-change observed in the historical data, demonstrating its potential to automate transition operations beyond current expert capability.

The task of learning a diffusion-based neural sampler for drawing samples from an unnormalized target distribution can be viewed as a stochastic optimal control problem on path measures. However, the training of neural samplers can be challenging when the target distribution is multimodal with significant barriers separating the modes, potentially leading to mode collapse. We propose a framework named Proximal Diffusion Neural Sampler (PDNS) that addresses these challenges by tackling the stochastic optimal control problem via proximal point method on the space of path measures. PDNS decomposes the learning process into a series of simpler subproblems that create a path gradually approaching the desired distribution. This staged procedure traces a progressively refined path to the desired distribution and promotes thorough exploration across modes. For a practical and efficient realization, we instantiate each proximal step with a proximal weighted denoising cross-entropy (WDCE) objective. We demonstrate the effectiveness and robustness of PDNS through extensive experiments on both continuous and discrete sampling tasks, including challenging scenarios in molecular dynamics and statistical physics. Our code is available at https://github.com/AlexandreGUO2001/PDNS.

Reinforcement Learning (RL) with PPO-like clip objectives has become the standard choice for reward-based fine-tuning of large language models (LLMs). Although recent work has explored improved estimators of advantages and normalization, the clipping mechanism itself has remained untouched. Originally introduced as a proxy for principled KL-based trust regions, clipping is a crude approximation that often causes unstable updates and suboptimal performance. We replace the clip objective with a novel discrete differentiable trust region projection, which provides principled token-level KL constraints. The projection operates on a sparse subset of the model's most important token logits to balance computational cost and projection effectiveness. Our approach, Trust Region Optimization for Large Language models (TROLL), serves as a direct replacement for PPO-like clipping during training and does not alter the model's inference behavior. Across mathematical reasoning and code generation tasks, model families, as well as advantage-estimation methods, TROLL consistently outperforms PPO-like clipping in terms of training speed, stability, and final success rates.

In high-dimensional learning, models remain stable until they collapse abruptly once the sample size falls below a critical level. This instability is not algorithm-specific but a geometric mechanism: when the weakest Fisher eigendirection falls beneath sample-level fluctuations, identifiability fails. Our Fisher Threshold Theorem formalizes this by proving that stability requires the minimal Fisher eigenvalue to exceed an explicit $O(\sqrt{d/n})$ bound. Unlike prior asymptotic or model-specific criteria, this threshold is finite-sample and necessary, marking a sharp phase transition between reliable concentration and inevitable failure. To make the principle constructive, we introduce the Fisher floor, a verifiable spectral regularization robust to smoothing and preconditioning. Synthetic experiments on Gaussian mixtures and logistic models confirm the predicted transition, consistent with $d/n$ scaling. Statistically, the threshold sharpens classical eigenvalue conditions into a non-asymptotic law; learning-theoretically, it defines a spectral sample-complexity frontier, bridging theory with diagnostics for robust high-dimensional inference.

Driven by the interest on how uniformity of marginal distributions propa\-gates to properties of regression functions, in this contribution we tackle the following questions: Given a $(d-1)$-dimensional random vector $\textbf{X}$ and a random variable $Y$ such that all univariate marginals of $(\textbf{X},Y)$ are uniformly distributed on $[0,1]$, how large can the average absolute deviation of the mean and the quantile regression function of $Y$ given $\textbf{X}$ from the value $\frac{1}{2}$ be, and how much mass may sets with large deviation have? We answer these questions by deriving sharp inequalities, both in the mean as well as in the quantile setting, and sketch some cautionary consequences to nowadays quite popular pair copula constructions involving the so-called simplifying assumption. Rounding off our results, working with the so-called empirical checkerboard estimator in the bivariate setting, we show strong consistency for both regression types and illustrate the speed of convergence in terms of a simulation study.

The batched multi-armed bandit (MAB) problem, where rewards are collected in batches, is pivotal in applications like clinical trials. While prior work assumes light-tailed reward distributions, real-world scenarios often exhibit heavy-tailed outcomes. This paper addresses this gap by introducing robust batched bandit algorithms for heavy-tailed rewards in both multi-arm and Lipschitz settings. We uncover somewhat surprising phenomena for such problems – heavier tails require fewer batches to achieve near-optimal regret in the instance-independent setting, as well as the Lipschitz setting. In sharp contrast, in the instance-dependent setting, the number of batches required to achieve near-optimal regret does not depend on the tail heaviness.

Transformers have achieved remarkable successes across a wide range of applications, yet the theoretical foundation of their model efficiency remains underexplored. In this work, we investigate how the model parameters – mainly attention heads and head dimensions – should be allocated across layers to balance expressivity and efficiency. We first provide mathematical analysis on the role of early layers in information extraction from an approximation perspective, with a theoretical characterization on the trade-off between the number of heads and head dimension under a fixed parameter budget. In addition, we uncover and prove the saturation behavior of softmax activations: Continuously increasing head dimensions can lead to diminishing returns in learning errors, particularly for long sequences. Supported by both theory and experiments, this saturation pattern suggests that later layers can operate more efficiently with reduced parameters. Combining these insights, we propose principled strategies for allocating attention heads and dimensions across Transformers' layers, shedding light on theoretically-grounded model efficiency of Transformer-based architectures.

We study the problem of auditing the fairness of a given classifier under partial feedback, where true labels are available only for positively classified individuals, (e.g., loan repayment outcomes are observed only for approved applicants). We introduce a novel cost model for acquiring additional labeled data, designed to more accurately reflect real-world costs such as credit assessment, loan processing, and potential defaults. Our goal is to find optimal fairness audit algorithms that are more cost-effective than random exploration and natural baselines. In our work, we consider two audit settings: a black-box model with no assumptions on the data distribution, and a mixture model, where features and true labels follow a mixture of exponential family distributions. In the black-box setting, we propose a near-optimal auditing algorithm under mild assumptions and show that a natural baseline can be strictly suboptimal. In the mixture model setting, we design a novel algorithm that achieves significantly lower audit cost than the black-box case. Our approach leverages prior work on learning from truncated samples and maximum-a-posteriori oracles, and extends known results on spherical Gaussian mixtures to handle exponential family mixtures, which may be of independent interest. Moreover, our algorithms apply to popular fairness metrics including demographic parity, equal opportunity, and equalized odds. Empirically, we demonstrate strong performance of our algorithms on real-world fair classification datasets like Adult Income and Law School, consistently outperforming natural baselines by around 50% in terms of audit cost.