Daily Papers

Conformal Prediction (CP) controls the prediction uncertainty of classification systems by producing a small prediction set, ensuring a predetermined probability that the true class lies within this set. This is commonly done by defining a score, based on the model predictions, and setting a threshold on this score using a validation set. In this study, we address the problem of CP calibration when we only have access to a calibration set with noisy labels. We show how we can estimate the noise-free conformal threshold based on the noisy labeled data. We derive a finite sample coverage guarantee for uniform noise that remains effective even in tasks with a large number of classes. We dub our approach Noise-Aware Conformal Prediction (NACP). We illustrate the performance of the proposed results on several standard image classification datasets with a large number of classes.

One-shot prediction enables rapid adaptation of pretrained foundation models to new tasks using only one labeled example, but lacks principled uncertainty quantification. While conformal prediction provides finite-sample coverage guarantees, standard split conformal methods are inefficient in the one-shot setting due to data splitting and reliance on a single predictor. We propose Conformal Aggregation of One-Shot Predictors (CAOS), a conformal framework that adaptively aggregates multiple one-shot predictors and uses a leave-one-out calibration scheme to fully exploit scarce labeled data. Despite violating classical exchangeability assumptions, we prove that CAOS achieves valid marginal coverage using a monotonicity-based argument. Experiments on one-shot facial landmarking and RAFT text classification tasks show that CAOS produces substantially smaller prediction sets than split conformal baselines while maintaining reliable coverage.

Robust optimization safeguards decisions against uncertainty by optimizing against worst-case scenarios, yet their effectiveness hinges on a prespecified robustness level that is often chosen ad hoc, leading to either insufficient protection or overly conservative and costly solutions. Recent approaches using conformal prediction construct data-driven uncertainty sets with finite-sample coverage guarantees, but they still fix coverage targets a priori and offer little guidance for selecting robustness levels. We propose a new framework that provides distribution-free, finite-sample guarantees on both miscoverage and regret for any family of robust predict-then-optimize policies. Our method constructs valid estimators that trace out the miscoverage-regret Pareto frontier, enabling decision-makers to reliably evaluate and calibrate robustness levels according to their cost-risk preferences. The framework is simple to implement, broadly applicable across classical optimization formulations, and achieves sharper finite-sample performance than existing approaches. These results offer the first principled data-driven methodology for guiding robustness selection and empower practitioners to balance robustness and conservativeness in high-stakes decision-making.

Accurate probabilistic predictions are essential for optimal decision making. While neural network miscalibration has been studied primarily in classification, we investigate this in the less-explored domain of regression. We conduct the largest empirical study to date to assess the probabilistic calibration of neural networks. We also analyze the performance of recalibration, conformal, and regularization methods to enhance probabilistic calibration. Additionally, we introduce novel differentiable recalibration and regularization methods, uncovering new insights into their effectiveness. Our findings reveal that regularization methods offer a favorable tradeoff between calibration and sharpness. Post-hoc methods exhibit superior probabilistic calibration, which we attribute to the finite-sample coverage guarantee of conformal prediction. Furthermore, we demonstrate that quantile recalibration can be considered as a specific case of conformal prediction. Our study is fully reproducible and implemented in a common code base for fair comparisons.

Convolutional image classifiers can achieve high predictive accuracy, but quantifying their uncertainty remains an unresolved challenge, hindering their deployment in consequential settings. Existing uncertainty quantification techniques, such as Platt scaling, attempt to calibrate the network's probability estimates, but they do not have formal guarantees. We present an algorithm that modifies any classifier to output a predictive set containing the true label with a user-specified probability, such as 90%. The algorithm is simple and fast like Platt scaling, but provides a formal finite-sample coverage guarantee for every model and dataset. Our method modifies an existing conformal prediction algorithm to give more stable predictive sets by regularizing the small scores of unlikely classes after Platt scaling. In experiments on both Imagenet and Imagenet-V2 with ResNet-152 and other classifiers, our scheme outperforms existing approaches, achieving coverage with sets that are often factors of 5 to 10 smaller than a stand-alone Platt scaling baseline.

Conformal prediction is a framework for predictive inference with a distribution-free, finite-sample guarantee. However, it tends to provide uninformative prediction sets when calibration data are scarce. This paper introduces Synthetic-powered predictive inference (SPI), a novel framework that incorporates synthetic data -- e.g., from a generative model -- to improve sample efficiency. At the core of our method is a score transporter: an empirical quantile mapping that aligns nonconformity scores from trusted, real data with those from synthetic data. By carefully integrating the score transporter into the calibration process, SPI provably achieves finite-sample coverage guarantees without making any assumptions about the real and synthetic data distributions. When the score distributions are well aligned, SPI yields substantially tighter and more informative prediction sets than standard conformal prediction. Experiments on image classification -- augmenting data with synthetic diffusion-model generated images -- and on tabular regression demonstrate notable improvements in predictive efficiency in data-scarce settings.

Retrieval-Augmented Generation (RAG) systems remain susceptible to hallucinations despite grounding in retrieved evidence. While current detection methods leverage embedding similarity and natural language inference (NLI), their reliability in safety-critical settings remains unproven. We apply conformal prediction to RAG hallucination detection, transforming heuristic scores into decision sets with finite-sample coverage guarantees (1-alpha). Using calibration sets of n=600, we demonstrate a fundamental dichotomy: on synthetic hallucinations (Natural Questions), embedding methods achieve 95% coverage with 0% False Positive Rate (FPR). However, on real hallucinations from RLHF-aligned models (HaluEval), the same methods fail catastrophically, yielding 100% FPR at target coverage. We analyze this failure through the lens of distributional tails, showing that while NLI models achieve acceptable AUC (0.81), the "hardest" hallucinations are semantically indistinguishable from faithful responses, forcing conformal thresholds to reject nearly all valid outputs. Crucially, GPT-4 as a judge achieves 7% FPR (95% CI:[3.4%, 13.7%]) on the same data, proving the task is solvable via reasoning but opaque to surface-level semantics--a phenomenon we term the "Semantic Illusion."

Microsimulation models used by ministries of finance and central banks rely on parametric processes for lifetime earnings that capture only first and second moments of the conditional distribution and miss long-range nonlinear structure. We propose SAGA, a decoder-only transformer for irregular tabular panel sequences, paired with a split conformal calibration wrapper that delivers individual-level prediction intervals with finite-sample marginal coverage guarantees. Trained on the longitudinal Swedish LISA register over 1990 to 2022, comprising 2,143,817 individuals and 61,284,903 person-years, the model forecasts annual labor earnings at horizons of one to thirty years and aggregates them by Monte Carlo into present-discounted lifetime earnings distributions. Against the canonical Guvenen, Karahan, Ozkan, and Song parametric process and tabular and recurrent baselines, SAGA reduces continuous ranked probability score by 31.9 percent at the ten-year horizon and mean absolute error by 37.7 percent at the twenty-year horizon. Conformal intervals achieve nominal coverage to within 0.4 percentage points marginally and within 2.4 percentage points on the worst-case demographic subgroup. The reconstructed lifetime earnings Gini coefficient is 0.327 against the partially observed truth of 0.341 and the GKOS estimate of 0.378. Model weights, calibration tables, and a synthetic equivalent dataset are released for replication outside the protected SCB MONA environment.

When multi-agent systems (MAS) fail, identifying where the decisive error occurred is the first step for automated recovery to an earlier state. Error attribution remains a fundamental challenge due to the long interaction traces that large language model-based MAS generate. This paper presents a framework for error attribution based on conformal prediction (CP) which provides finite-sample, distribution-free coverage guarantees. We introduce new algorithms for filtration-based CP designed for sequential data such as agent trajectories. Unlike existing CP algorithms, our approach predicts sets that are contiguous sequences to enable efficient recovery and debugging. We verify our theoretical guarantees on a variety of agents and datasets, show that errors can be precisely isolated, then use prediction sets to rollback MAS to correct their own errors. Our overall approach is model-agnostic, and offers a principled uncertainty layer for MAS error attribution. We release code at https://github.com/layer6ai-labs/conformal-agent-error-attribution.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

Off-policy data in reinforcement learning with verifiable rewards (RLVR) can arise from asynchronous generation, rollout-training mismatch, and response reuse. Token-wise importance weighting avoids multiplying likelihood ratios across a response but generally biases policy gradients. We prove that, under coverage and local expressiveness conditions, full future KL credit makes the exact proximal optimizer the update's unique stationary policy, even for mixtures of older policies, without full-response weights or a fitted critic. We further establish convergence of continuous-time tabular population updates under persistent coverage and time-varying behavior policies, and a finite-sample reward guarantee for linear logistic models under additional assumptions. These results motivate Proximal KL, a practical off-policy RLVR method that separates the proximal reference from the sampling policy and replaces PPO clipping with bounded future KL credit. Across all seven matched off-policy comparisons on Qwen3-1.7B and Qwen3-4B, spanning two learning rates, response reuse, and rollout staleness, Proximal KL outperforms GRPO in micro-averaged accuracy over eight math benchmarks by up to 2.58%. A separate on-policy comparison yields comparable aggregate accuracy. In periodic-refresh experiments, Proximal KL avoids the collapses observed in PG and CISPO, exceeds GRPO's training accuracy, and reaches higher accuracy in fewer policy updates. Ablations show that token-wise correction outperforms sequence-level and unweighted controls. With token-wise correction retained, future KL feedback reduces sensitivity to the tested baseline perturbation relative to PPO clipping. A finite future horizon improves AIME24 mean@16 and pass@16 over the full tail, while fixed-policy measurements show that truncation reduces correction variance over the tested horizons.

We consider the problem of distribution-free predictive inference, with the goal of producing predictive coverage guarantees that hold conditionally rather than marginally. Existing methods such as conformal prediction offer marginal coverage guarantees, where predictive coverage holds on average over all possible test points, but this is not sufficient for many practical applications where we would like to know that our predictions are valid for a given individual, not merely on average over a population. On the other hand, exact conditional inference guarantees are known to be impossible without imposing assumptions on the underlying distribution. In this work we aim to explore the space in between these two, and examine what types of relaxations of the conditional coverage property would alleviate some of the practical concerns with marginal coverage guarantees while still being possible to achieve in a distribution-free setting.

Given a black-box AI system and a task, at what confidence level can a practitioner trust the system's output? We answer with a reliability level -- a single number per system-task pair, derived from self-consistency sampling and conformal calibration, that serves as a black-box deployment gate with exact, finite-sample, distribution-free guarantees. Self-consistency sampling reduces uncertainty exponentially; conformal calibration guarantees correctness within 1/(n+1) of the target level, regardless of the system's errors -- made transparently visible through larger answer sets for harder questions. Weaker models earn lower reliability levels (not accuracy -- see Definition 2.4): GPT-4.1 earns 94.6% on GSM8K and 96.8% on TruthfulQA, while GPT-4.1-nano earns 89.8% on GSM8K and 66.5% on MMLU. We validate across five benchmarks, five models from three families, and both synthetic and real data. Conditional coverage on solvable items exceeds 0.93 across all configurations; sequential stopping reduces API costs by around 50%.

Prediction sets can wrap around any ML model to cover unknown test outcomes with a guaranteed probability. Yet, it remains unclear how to use them optimally for downstream decision-making. Here, we propose a decision-theoretic framework that seeks to minimize the expected loss (risk) against a worst-case distribution consistent with the prediction set's coverage guarantee. We first characterize the minimax optimal policy for a fixed prediction set, showing that it balances the worst-case loss inside the set with a penalty for potential losses outside the set. Building on this, we derive the optimal prediction set construction that minimizes the resulting robust risk subject to a coverage constraint. Finally, we introduce Risk-Optimal Conformal Prediction (ROCP), a practical algorithm that targets these risk-minimizing sets while maintaining finite-sample distribution-free marginal coverage. Empirical evaluations on medical diagnosis and safety-critical decision-making tasks demonstrate that ROCP reduces critical mistakes compared to baselines, particularly when out-of-set errors are costly.

Scaling the amount of compute used to train language models has dramatically improved their capabilities. However, when it comes to inference, we often limit the amount of compute to only one attempt per problem. Here, we explore inference compute as another axis for scaling by increasing the number of generated samples. Across multiple tasks and models, we observe that coverage - the fraction of problems solved by any attempt - scales with the number of samples over four orders of magnitude. In domains like coding and formal proofs, where all answers can be automatically verified, these increases in coverage directly translate into improved performance. When we apply repeated sampling to SWE-bench Lite, the fraction of issues solved with DeepSeek-V2-Coder-Instruct increases from 15.9% with one sample to 56% with 250 samples, outperforming the single-attempt state-of-the-art of 43% which uses more capable frontier models. Moreover, using current API pricing, amplifying the cheaper DeepSeek model with five samples is more cost-effective and solves more issues than paying a premium for one sample from GPT-4o or Claude 3.5 Sonnet. Interestingly, the relationship between coverage and the number of samples is often log-linear and can be modelled with an exponentiated power law, suggesting the existence of inference-time scaling laws. Finally, we find that identifying correct samples out of many generations remains an important direction for future research in domains without automatic verifiers. When solving math word problems from GSM8K and MATH, coverage with Llama-3 models grows to over 95% with 10,000 samples. However, common methods to pick correct solutions from a sample collection, such as majority voting or reward models, plateau beyond several hundred samples and fail to fully scale with the sample budget.

Evaluating conditional coverage remains one of the most persistent challenges in assessing the reliability of predictive systems. Although conformal methods can give guarantees on marginal coverage, no method can guarantee to produce sets with correct conditional coverage, leaving practitioners without a clear way to interpret local deviations. To overcome sample-inefficiency and overfitting issues of existing metrics, we cast conditional coverage estimation as a classification problem. Conditional coverage is violated if and only if some classifier can achieve lower risk than the target coverage. Through the choice of a (proper) loss function, the resulting risk difference gives a conservative estimate of natural miscoverage measures such as L1 and L2 distance, and can even separate the effects of over- and under-coverage, and non-constant target coverages. We call the resulting family of metrics excess risk of the target coverage (ERT). We show experimentally that the use of modern classifiers provides much higher statistical power than simple classifiers underlying established metrics like CovGap. Additionally, we use our metric to benchmark different conformal prediction methods. Finally, we release an open-source package for ERT as well as previous conditional coverage metrics. Together, these contributions provide a new lens for understanding, diagnosing, and improving the conditional reliability of predictive systems.

Bagging is an important technique for stabilizing machine learning models. In this paper, we derive a finite-sample guarantee on the stability of bagging for any model. Our result places no assumptions on the distribution of the data, on the properties of the base algorithm, or on the dimensionality of the covariates. Our guarantee applies to many variants of bagging and is optimal up to a constant. Empirical results validate our findings, showing that bagging successfully stabilizes even highly unstable base algorithms.

You are a robot and you live in a Markov decision process (MDP) with a finite or an infinite number of transitions from state-action to next states. You got brains and so you plan before you act. Luckily, your roboparents equipped you with a generative model to do some Monte-Carlo planning. The world is waiting for you and you have no time to waste. You want your planning to be efficient. Sample-efficient. Indeed, you want to exploit the possible structure of the MDP by exploring only a subset of states reachable by following near-optimal policies. You want guarantees on sample complexity that depend on a measure of the quantity of near-optimal states. You want something, that is an extension of Monte-Carlo sampling (for estimating an expectation) to problems that alternate maximization (over actions) and expectation (over next states). But you do not want to StOP with exponential running time, you want something simple to implement and computationally efficient. You want it all and you want it now. You want TrailBlazer.

Uncertainty quantification in Knowledge Graph Embedding (KGE) methods is crucial for ensuring the reliability of downstream applications. A recent work applies conformal prediction to KGE methods, providing uncertainty estimates by generating a set of answers that is guaranteed to include the true answer with a predefined confidence level. However, existing methods provide probabilistic guarantees averaged over a reference set of queries and answers (marginal coverage guarantee). In high-stakes applications such as medical diagnosis, a stronger guarantee is often required: the predicted sets must provide consistent coverage per query (conditional coverage guarantee). We propose CondKGCP, a novel method that approximates predicate-conditional coverage guarantees while maintaining compact prediction sets. CondKGCP merges predicates with similar vector representations and augments calibration with rank information. We prove the theoretical guarantees and demonstrate empirical effectiveness of CondKGCP by comprehensive evaluations.

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the L^p variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond O(T) and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL.

The extant insurance literature demonstrates a paucity of finite-sample valid prediction intervals of future insurance claims in the regression setting. To address this challenge, this article proposes a new strategy that converts a predictive method in the unsupervised iid (independent identically distributed) setting to a predictive method in the regression setting. In particular, it enables an actuary to obtain infinitely many finite-sample valid prediction intervals in the regression setting.

Conformal prediction is a technique for constructing prediction intervals that attain valid coverage in finite samples, without making distributional assumptions. Despite this appeal, existing conformal methods can be unnecessarily conservative because they form intervals of constant or weakly varying length across the input space. In this paper we propose a new method that is fully adaptive to heteroscedasticity. It combines conformal prediction with classical quantile regression, inheriting the advantages of both. We establish a theoretical guarantee of valid coverage, supplemented by extensive experiments on popular regression datasets. We compare the efficiency of conformalized quantile regression to other conformal methods, showing that our method tends to produce shorter intervals.

Constructing prediction sets with coverage guarantees for unobserved outcomes is a core problem in modern statistics. Methods for predictive inference have been developed for a wide range of settings, but usually only consider test data points one at a time. Here we study the problem of distribution-free predictive inference for a batch of multiple test points, aiming to construct prediction sets for functions -- such as the mean or median -- of any number of unobserved test datapoints. This setting includes constructing simultaneous prediction sets with a high probability of coverage, and selecting datapoints satisfying a specified condition while controlling the number of false claims. For the general task of predictive inference on a function of a batch of test points, we introduce a methodology called batch predictive inference (batch PI), and provide a distribution-free coverage guarantee under exchangeability of the calibration and test data. Batch PI requires the quantiles of a rank ordering function defined on certain subsets of ranks. While computing these quantiles is NP-hard in general, we show that it can be done efficiently in many cases of interest, most notably for batch score functions with a compositional structure -- which includes examples of interest such as the mean -- via a dynamic programming algorithm that we develop. Batch PI has advantages over naive approaches (such as partitioning the calibration data or directly extending conformal prediction) in many settings, as it can deliver informative prediction sets even using small calibration sample sizes. We illustrate that our procedures provide informative inference across the use cases mentioned above, through experiments on both simulated data and a drug-target interaction dataset.

Mixed-precision quantization must decide which parts of a model to keep at higher precision. A common premise, shared by sensitivity-based methods such as HAWQ and CoopQ, is that the loss from quantizing a set of layers can be reconstructed from per-layer or pairwise sensitivities measured in isolation. We test this premise at the 4-bit weight-and-activation precisions now being deployed, treating the change in loss f(S) from quantizing a layer set S as a set function on the Boolean cube and analyzing it through two classical changes of basis. This analysis yields two findings. First, across configurations drawn from the deployment distribution, 85--93\% of the variance of f is explained by per-layer effects alone. Second, a monotone transform of a sum of per-layer terms reproduces f's ranking of configurations, misordering at most 2\% of pairs. We propose the coverage model f(S)=cbigl(1-prod_{iin S}(1-a_i)bigr), which reproduces the measured variance profile of f to within a few percent from its L fitted break-rates. This structure supports two predictors of a configuration's loss, each with L+1 parameters. The additive model is the optimal first-order predictor. By Parseval's identity its mean-squared error equals the variance of f left unexplained by per-layer effects, which we measure on full lattices, estimate out of sample at full-network scale, and report with every result as a certificate of how well any additive model can do. The coverage model itself is the second predictor. As allocators at matched memory, they attain the lowest KL divergence among the compared allocators on models from 30B to 355B parameters. Below four bits, the resulting allocations continue to solve code and reasoning tasks at budgets where allocations from gradient sensitivities no longer produce terminating generations.

While improving prediction accuracy has been the focus of machine learning in recent years, this alone does not suffice for reliable decision-making. Deploying learning systems in consequential settings also requires calibrating and communicating the uncertainty of predictions. To convey instance-wise uncertainty for prediction tasks, we show how to generate set-valued predictions from a black-box predictor that control the expected loss on future test points at a user-specified level. Our approach provides explicit finite-sample guarantees for any dataset by using a holdout set to calibrate the size of the prediction sets. This framework enables simple, distribution-free, rigorous error control for many tasks, and we demonstrate it in five large-scale machine learning problems: (1) classification problems where some mistakes are more costly than others; (2) multi-label classification, where each observation has multiple associated labels; (3) classification problems where the labels have a hierarchical structure; (4) image segmentation, where we wish to predict a set of pixels containing an object of interest; and (5) protein structure prediction. Lastly, we discuss extensions to uncertainty quantification for ranking, metric learning and distributionally robust learning.

We introduce a framework for calibrating machine learning models so that their predictions satisfy explicit, finite-sample statistical guarantees. Our calibration algorithms work with any underlying model and (unknown) data-generating distribution and do not require model refitting. The framework addresses, among other examples, false discovery rate control in multi-label classification, intersection-over-union control in instance segmentation, and the simultaneous control of the type-1 error of outlier detection and confidence set coverage in classification or regression. Our main insight is to reframe the risk-control problem as multiple hypothesis testing, enabling techniques and mathematical arguments different from those in the previous literature. We use the framework to provide new calibration methods for several core machine learning tasks, with detailed worked examples in computer vision and tabular medical data.

In large scale machine learning, random sampling is a popular way to approximate datasets by a small representative subset of examples. In particular, sensitivity sampling is an intensely studied technique which provides provable guarantees on the quality of approximation, while reducing the number of examples to the product of the VC dimension d and the total sensitivity mathfrak S in remarkably general settings. However, guarantees going beyond this general bound of mathfrak S d are known in perhaps only one setting, for ell_2 subspace embeddings, despite intense study of sensitivity sampling in prior work. In this work, we show the first bounds for sensitivity sampling for ell_p subspace embeddings for pneq 2 that improve over the general mathfrak S d bound, achieving a bound of roughly mathfrak S^{2/p} for 1leq p<2 and mathfrak S^{2-2/p} for 2<p<infty. For 1leq p<2, we show that this bound is tight, in the sense that there exist matrices for which mathfrak S^{2/p} samples is necessary. Furthermore, our techniques yield further new results in the study of sampling algorithms, showing that the root leverage score sampling algorithm achieves a bound of roughly d for 1leq p<2, and that a combination of leverage score and sensitivity sampling achieves an improved bound of roughly d^{2/p}mathfrak S^{2-4/p} for 2<p<infty. Our sensitivity sampling results yield the best known sample complexity for a wide class of structured matrices that have small ell_p sensitivity.

Reference-free faithfulness metrics verify each atomic claim a model makes against ground truth, and are increasingly used to evaluate grounded generation. We show they share a blind spot: they measure only precision -- are the stated claims supported? -- and therefore reward abstention, since a model can score near-perfect faithfulness by saying almost nothing. We make this measurable using Formula 1 telemetry, a domain where strategic ground truth is derived deterministically and, crucially, completely: for each decision we know the full set of facts that mattered. This completeness -- absent in open-domain faithfulness benchmarks -- lets us measure recall (coverage of the relevant facts) exactly, alongside precision. On a multilingual (EN/ES/PT) benchmark of 7,253 decision instances spanning 150 races, the most precise frontier model covers under half of the relevant facts and ranks last by F1, so requiring coverage reorders the systems; the same effect reappears in a second complete-oracle domain (NOAA weather forecasts). A prompt ablation shows the low coverage is not an under-prompting artifact: explicitly asking models to be thorough does not close the gap. We pair faithfulness with coverage into a single score, validate the metric (controlled perturbation; agreement across a model-free regex extractor and a cross-family LLM extractor, system-level Spearman 1.0), and give a verifier-guided generation method that improves precision and recall without references. We release the benchmark, structured annotations, metric, baselines, and an interactive demo.

Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source--target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

Conformal prediction has shown spurring performance in constructing statistically rigorous prediction sets for arbitrary black-box machine learning models, assuming the data is exchangeable. However, even small adversarial perturbations during the inference can violate the exchangeability assumption, challenge the coverage guarantees, and result in a subsequent decline in empirical coverage. In this work, we propose a certifiably robust learning-reasoning conformal prediction framework (COLEP) via probabilistic circuits, which comprise a data-driven learning component that trains statistical models to learn different semantic concepts, and a reasoning component that encodes knowledge and characterizes the relationships among the trained models for logic reasoning. To achieve exact and efficient reasoning, we employ probabilistic circuits (PCs) within the reasoning component. Theoretically, we provide end-to-end certification of prediction coverage for COLEP in the presence of bounded adversarial perturbations. We also provide certified coverage considering the finite size of the calibration set. Furthermore, we prove that COLEP achieves higher prediction coverage and accuracy over a single model as long as the utilities of knowledge models are non-trivial. Empirically, we show the validity and tightness of our certified coverage, demonstrating the robust conformal prediction of COLEP on various datasets, including GTSRB, CIFAR10, and AwA2. We show that COLEP achieves up to 12% improvement in certified coverage on GTSRB, 9% on CIFAR-10, and 14% on AwA2.

We study offline multi-agent reinforcement learning (RL) in Markov games, where the goal is to learn an approximate equilibrium -- such as Nash equilibrium and (Coarse) Correlated Equilibrium -- from an offline dataset pre-collected from the game. Existing works consider relatively restricted tabular or linear models and handle each equilibria separately. In this work, we provide the first framework for sample-efficient offline learning in Markov games under general function approximation, handling all 3 equilibria in a unified manner. By using Bellman-consistent pessimism, we obtain interval estimation for policies' returns, and use both the upper and the lower bounds to obtain a relaxation on the gap of a candidate policy, which becomes our optimization objective. Our results generalize prior works and provide several additional insights. Importantly, we require a data coverage condition that improves over the recently proposed "unilateral concentrability". Our condition allows selective coverage of deviation policies that optimally trade-off between their greediness (as approximate best responses) and coverage, and we show scenarios where this leads to significantly better guarantees. As a new connection, we also show how our algorithmic framework can subsume seemingly different solution concepts designed for the special case of two-player zero-sum games.

We present a new distribution-free conformal prediction algorithm for sequential data (e.g., time series), called the sequential predictive conformal inference (SPCI). We specifically account for the nature that time series data are non-exchangeable, and thus many existing conformal prediction algorithms are not applicable. The main idea is to adaptively re-estimate the conditional quantile of non-conformity scores (e.g., prediction residuals), upon exploiting the temporal dependence among them. More precisely, we cast the problem of conformal prediction interval as predicting the quantile of a future residual, given a user-specified point prediction algorithm. Theoretically, we establish asymptotic valid conditional coverage upon extending consistency analyses in quantile regression. Using simulation and real-data experiments, we demonstrate a significant reduction in interval width of SPCI compared to other existing methods under the desired empirical coverage.

Parameter inference, i.e. inferring the posterior distribution of the parameters of a statistical model given some data, is a central problem to many scientific disciplines. Generative models can be used as an alternative to Markov Chain Monte Carlo methods for conducting posterior inference, both in likelihood-based and simulation-based problems. However, assessing the accuracy of posteriors encoded in generative models is not straightforward. In this paper, we introduce `Tests of Accuracy with Random Points' (TARP) coverage testing as a method to estimate coverage probabilities of generative posterior estimators. Our method differs from previously-existing coverage-based methods, which require posterior evaluations. We prove that our approach is necessary and sufficient to show that a posterior estimator is accurate. We demonstrate the method on a variety of synthetic examples, and show that TARP can be used to test the results of posterior inference analyses in high-dimensional spaces. We also show that our method can detect inaccurate inferences in cases where existing methods fail.

Large language models perform strongly on competition mathematics, but their research-level reasoning remains difficult to evaluate systematically. Theoretical computer science (TCS) connects algorithm design to explicit guarantees and fundamental limits, providing a setting for evaluating whether models can justify computational improvements with arguments humans can inspect. We introduce TCSAlgBench, a benchmark and reusable pipeline for natural-language proof discovery, comprising 398 theorem-level challenges from 138 STOC and COLT 2026 papers. Expert-designed rules complete paper-specific context, preserve computational assumptions and quantitative guarantees, and withhold constructions when discovering an algorithm is part of the task. For each task, prover systems receive theorem statements and access to cited prior work. The pipeline supports fresh, versioned challenge batches from newly released papers. We evaluate ten model configurations from four families under direct inference and prover-verifier discussion, and compare four agent workflows under matched model-call opportunities. All evaluations use the full benchmark. In the model comparison, GPT-5.6 Sol max achieves the highest five-run verifier-accepted coverage at 23.6% after 10-round discussion. Discussion and repeated sampling improve coverage. In the separate agent comparison using GPT-5.5 xhigh, decomposition improves coverage over discussion, and agentic planning achieves the highest five-run verifier-accepted coverage at 25.4%. TCSAlgBench provides a refreshable testbed for measuring progress in model reasoning and studying how agent workflows support research-level proof discovery.

Every production model is updated, by retraining, fine-tuning, quantization, or a silent vendor swap, and each update risks being worse than what it replaced. We formalize update promotion as certified paired risk-difference auditing. Our starting point is a support identity: the risk difference between two models lives on the inputs where they disagree, observable without labels. We build DISCERN, a sequential two-tier protocol. A zero-label tier certifies benign updates whose disagreement rate is below tolerance from unlabeled traffic alone. An audited tier labels only sampled disagreements through an anytime-valid confidence sequence, valid at every stopping time and under any label-routing rule, even an adversarial judge. We prove finite-sample validity and matching label-complexity bounds of order rho^2/eps^2 at the rate level, so exploiting free disagreement provably saves a factor 1/rho over any pairing-blind auditor, and the guarantee composes across an unbounded sequence of promotions from one error budget. Across 14,000+ replayed audit streams over 785 update pairs, including LoRA fine-tunes of language models up to 1.4B parameters, miscoverage is 0.0002 (nominal 5%), power 0.986 with zero false alarms, and 56% of benign updates certify with zero labels. Each audit emits a machine-checkable evidence record for post-market monitoring.

Recent research has generated hope that inference scaling could allow weaker language models to match or exceed the accuracy of stronger models, such as by repeatedly sampling solutions to a coding problem until it passes unit tests. The central thesis of this paper is that there is no free lunch for inference scaling: indefinite accuracy improvement through resampling can only be realized if the "verifier" (in this case, a set of unit tests) is perfect. When the verifier is imperfect, as it almost always is in domains such as reasoning or coding (for example, unit tests have imperfect coverage), there is a nonzero probability of false positives: incorrect solutions that pass the verifier. Resampling cannot decrease this probability, so it imposes an upper bound to the accuracy of resampling-based inference scaling even with an infinite compute budget. We find that there is a very strong correlation between the model's single-sample accuracy (i.e. accuracy without unit tests) and its false positive rate on coding benchmarks HumanEval and MBPP, whose unit tests have limited coverage. Therefore, no amount of inference scaling of weaker models can enable them to match the single-sample accuracy of a sufficiently strong model (Fig. 1a). When we consider that false positives have a negative utility compared to abstaining from producing a solution, it bends the inference scaling curve further downward. Empirically, we find that the optimal number of samples can be less than 10 under realistic assumptions (Fig. 1b). Finally, we show that beyond accuracy, false positives may have other undesirable qualities, such as poor adherence to coding style conventions.

Achieving effective test-time scaling requires models to engage in In-Context Exploration -- the intrinsic ability to generate, verify, and refine multiple reasoning hypotheses within a single continuous context. Grounded in State Coverage theory, our analysis identifies a critical bottleneck to enabling this capability: while broader state coverage requires longer reasoning trajectories, the probability of sampling such sequences decays exponentially during autoregressive generation, a phenomenon we term the ``Shallow Exploration Trap''. To bridge this gap, we propose Length-Incentivized Exploration(\method). This simple yet effective recipe explicitly encourages models to explore more via a length-based reward coupled with a redundancy penalty, thereby maximizing state coverage in two-step manner. Comprehensive experiments across different models (Qwen3, Llama) demonstrate that \method effectively incentivize in-context exploration. As a result, our method achieves an average improvement of 4.4\% on in-domain tasks and a 2.7\% gain on out-of-domain benchmarks.

We study causal logistic bandits with counterfactual fairness constraints. The causal structure is given through known factual and counterfactual feature maps that share an unknown logistic reward parameter, but the learner observes only factual rewards. Consequently, the directions determining counterfactual feasibility need not be identifiable from the available feedback. The closest prior analyses either omit a coverage condition or impose a comparatively strong one, and do not establish matching lower bounds. We first show that some coverage condition is necessary: without a coverage-type restriction, factually indistinguishable environments with different optimal fair actions force Ω(T) expected joint loss. Under a weaker full-rank condition on the factual covariance pooled across actions, we identify a target-specific information scale V_star that measures the difficulty of estimating rewards and counterfactual effects from factual feedback. We construct worst-case families satisfying this condition on which every policy incurs expected joint loss Ωleft(left[V_starmin{log K,d}right]^{1/3}T^{2/3}right). We also give an explore--then--exploit procedure tuned using V_star and an adaptive algorithm that does not require its value. Both algorithms achieve max{R_T,V_T}=Oleft(left[V_starmin{log K,d}right]^{1/3}T^{2/3}+κd/σ_0^2right), where R_T is regret relative to the best fair action and V_T denotes the cumulative stage-wise positive violations. Thus the upper and lower bounds match in their leading dependence on T, V_star, and min{log K,d}, up to logarithmic factors.

Retrieval-augmented generation (RAG) grounds answers in retrieved passages, yet relevance does not guarantee sufficiency: a topical passage may still fail to justify the answer. We study evidence sufficiency verification for selective RAG answering, in which a verifier receives a question, a candidate answer, and retrieved evidence and decides whether the evidence supports, refutes, or is insufficient for the answer, answering only when support is established. We present SURE-RAG, an aggregation protocol that treats evidence sufficiency as a set-level property: missing hops and unresolved conflicts cannot be detected by scoring passages independently. A shared claim-evidence verifier produces a local relation distribution for each (claim, passage) pair, which SURE-RAG aggregates into four interpretable answer-level feature blocks (coverage, relation strength, uncertainty, and retrieval), producing a three-way decision and an auditable selective score. We evaluate on HotpotQA-RAG v3, a controlled multi-hop benchmark, under an artifact-aware protocol (shortcut baselines, counterfactual swaps, no-oracle checks, and GPT-4o audits). Calibrated SURE-RAG attains 0.9075 Macro-F1 (raw 0.8951 +/- 0.0069), well above DeBERTa mean-pooling (0.6516) and a GPT-4o judge (0.7284), and on par with a strong concat cross-encoder (0.8888 +/- 0.0109) while remaining fully auditable. At 30% coverage, risk falls from 0.2588 to 0.1642, a 37% relative reduction. As a boundary-mapping experiment, we contrast SURE-RAG with GPT-4o on HaluBench unsafe detection: the ranking reverses (0.3343 vs. 0.7389 unsafe-F1), indicating that controlled sufficiency verification and natural hallucination detection are distinct problems.

Researchers in explainable artificial intelligence have developed numerous methods for helping users understand the predictions of complex supervised learning models. By contrast, explaining the uncertainty of model outputs has received relatively little attention. We adapt the popular Shapley value framework to explain various types of predictive uncertainty, quantifying each feature's contribution to the conditional entropy of individual model outputs. We consider games with modified characteristic functions and find deep connections between the resulting Shapley values and fundamental quantities from information theory and conditional independence testing. We outline inference procedures for finite sample error rate control with provable guarantees, and implement efficient algorithms that perform well in a range of experiments on real and simulated data. Our method has applications to covariate shift detection, active learning, feature selection, and active feature-value acquisition.

The training of large language models (LLMs) is expensive. In this paper, we study data-efficient approaches for pre-training LLMs, i.e., techniques that aim to optimize the Pareto frontier of model quality and training resource/data consumption. We seek to understand the tradeoffs associated with data selection routines based on (i) expensive-to-compute data-quality estimates, and (ii) maximization of coverage and diversity-based measures in the feature space. Our first technique, Ask-LLM, leverages the zero-shot reasoning capabilities of instruction-tuned LLMs to directly assess the quality of a training example. To target coverage, we propose Density sampling, which models the data distribution to select a diverse sample. In our comparison of 19 samplers, involving hundreds of evaluation tasks and pre-training runs, we find that Ask-LLM and Density are the best methods in their respective categories. Coverage sampling can recover the performance of the full data, while models trained on Ask-LLM data consistently outperform full-data training -- even when we reject 90% of the original dataset, while converging up to 70% faster.

For observational studies, we study the sensitivity of causal inference when treatment assignments may depend on unobserved confounders. We develop a loss minimization approach for estimating bounds on the conditional average treatment effect (CATE) when unobserved confounders have a bounded effect on the odds ratio of treatment selection. Our approach is scalable and allows flexible use of model classes in estimation, including nonparametric and black-box machine learning methods. Based on these bounds for the CATE, we propose a sensitivity analysis for the average treatment effect (ATE). Our semi-parametric estimator extends/bounds the augmented inverse propensity weighted (AIPW) estimator for the ATE under bounded unobserved confounding. By constructing a Neyman orthogonal score, our estimator of the bound for the ATE is a regular root-n estimator so long as the nuisance parameters are estimated at the o_p(n^{-1/4}) rate. We complement our methodology with optimality results showing that our proposed bounds are tight in certain cases. We demonstrate our method on simulated and real data examples, and show accurate coverage of our confidence intervals in practical finite sample regimes with rich covariate information.

As LLM agents undertake increasingly complex, long-horizon tasks, verifying their outputs becomes increasingly challenging. We study how verification capability can be strengthened with a fixed base model, without access to reference answers or grading rubrics at test time. Repeated sampling yields multiple rollouts that can contain complementary correct claims, but we need a reliable verification mechanism to determine which claims to trust. We first find that disagreement often exposes correct alternatives, while consensus can conceal errors. These observations motivate VeriHarness, which turns the underlying LLM a generator uses into an agentic verifier by giving it a workspace, evidence tools, and reusable verification skills. A disagreement resolver checks competing claims against environmental evidence, while a consensus challenger tests shared claims and searches for omitted requirements. Their findings guide the selection and revision of the final artifact. Across five long-horizon workspace benchmarks and two frontier models, VeriHarness achieves the highest selection scores among the evaluated baselines. Evidence-backed revision further improves average performance, bringing gains over a single rollout to 6.2 points with Gemini 3.5 Flash and 6.4 points with Claude Opus 4.8. We further show that verification skills can self-improve from failure feedback, demonstrating VeriHarness as a novel and critical approach for scaling long-horizon agentic verification. We release the full pool of approximately 26,000 rollouts across all five benchmarks and both models, produced at a cost of over $100,000, to support future research on agentic verification.

Conformal prediction guarantees marginal coverage, but a single calibration threshold can still spread that coverage unevenly, over-covering easy regions and under-covering hard ones. SimplexUQ is, to our knowledge, the first benchmark and reproducible protocol for measuring this allocation problem on simplex-valued predictions; it compares existing conformal wrappers rather than proposing a new one. Its task suite, SimplexTasks-12, combines six controlled synthetic regimes with six frozen-predictor real tasks spanning class probabilities, topic mixtures, spectral abundances, cell-type fractions, age distributions, and emotion mixtures. Each comparison fixes the predictor, score, and response-free stratification map, varies only the wrapper, and reports marginal coverage, worst-stratum coverage, max disparity, and within-task radius and compute. Global calibration can look valid while failing badly: on CIFAR-10 it attains 0.900 marginal coverage but only 0.542 in the worst entropy stratum, and Mondrian calibration raises that stratum to 0.886 while reducing max disparity from 0.358 to 0.022. No wrapper dominates, however. Under smooth synthetic heterogeneity, several repairs are competitive; fixed-map analyses show that rankings depend on the evaluation groups and protocol; and in a 12-task comparison, Mondrian has lower disparity on its single target partition for all 12 tasks, whereas BatchMVP has lower disparity over overlapping groups on five. These are empirical comparisons, not new coverage guarantees. A controlled predictor-bias sweep shows that removing predictor bias only partly reduces global-threshold disparity. We release task cards, result provenance, permitted derived arrays, and rebuild instructions, and treat wrapper selection as a diagnostic comparison rather than a universal ranking.

While vision-language-action (VLA) models have advanced rapidly, running them at every control step remains expensive. Prior work accelerates VLA inference using techniques like action chunking and visual-token pruning, typically evaluating based on latency and average task success. However, acceleration may discard information and break tasks the original policy would solve, a risk hidden by average metrics. Measuring these failures is challenging because action deviations compound over closed-loop trajectories, meaning task failure is only observable across full episodes. We therefore define an acceleration-induced failure via paired rollouts from identical initial conditions, tracking when the reference succeeds but the accelerated policy fails. To manage this, we introduce CARE, an approach for certified accelerator selection. CARE uses paired rollouts on a calibration set to provide finite-sample guarantees that acceleration-induced failure risk stays below a user-specified budget. It deploys the fastest certified candidate, falling back to the reference if none qualify. By relying only on terminal outcomes and measured compute, CARE applies unchanged across diverse acceleration mechanisms, while sequential testing and failure-triggered reference rollouts keep certification affordable. On four LIBERO suites with OpenVLA-OFT, CARE certifies 9.0--10.8times speedups while guaranteeing (at 95% confidence) that at least 85.8% of reference-solved episodes are preserved. Under tight budgets, selectors without guarantees exceed the budget in up to 75% of trials, whereas CARE stays within budget and its sequential form uses 78.9% fewer rollouts than exhaustive evaluation. CARE further generalizes to flow-step reduction for π_{0.5}, and to Qwen3.5-9B and Llama-3.1-8B agents in Crafter.

Conformal prediction is often calibrated with a single pooled threshold, but this can hide cross-group heterogeneity in score distributions and distort group-wise coverage. We study this phenomenon through the population score distributions underlying split conformal calibration. First, we derive a conservation law and lower bound showing that pooled calibration incurs irreducible group-wise coverage distortion at a scale set by cross-group quantile heterogeneity. Second, we demonstrate that the two leading fairness definitions for conformal prediction, Equalized Coverage and Equalized Set Size, are fundamentally in tension. Third, we quantify the cost of moving between policies which treat groups separately or pool them. Experiments on synthetic and real data confirm the same bidirectional trade-off after finite-sample calibration. Our results show that, for the policy families studied here, calibration choice does not remove cross-group heterogeneity; it determines whether the resulting distortion appears in the coverage or size dimension, providing a principled lens for analyzing fairness-oriented calibration choices in practice.

This paper presents a new approach for assessing uncertainty in machine translation by simultaneously evaluating translation quality and providing a reliable confidence score. Our approach utilizes conformal predictive distributions to produce prediction intervals with guaranteed coverage, meaning that for any given significance level epsilon, we can expect the true quality score of a translation to fall out of the interval at a rate of 1-epsilon. In this paper, we demonstrate how our method outperforms a simple, but effective baseline on six different language pairs in terms of coverage and sharpness. Furthermore, we validate that our approach requires the data exchangeability assumption to hold for optimal performance.

We initiate the study of statistical inference and A/B testing for first-price pacing equilibria (FPPE). The FPPE model captures the dynamics resulting from large-scale first-price auction markets where buyers use pacing-based budget management. Such markets arise in the context of internet advertising, where budgets are prevalent. We propose a statistical framework for the FPPE model, in which a limit FPPE with a continuum of items models the long-run steady-state behavior of the auction platform, and an observable FPPE consisting of a finite number of items provides the data to estimate primitives of the limit FPPE, such as revenue, Nash social welfare (a fair metric of efficiency), and other parameters of interest. We develop central limit theorems and asymptotically valid confidence intervals. Furthermore, we establish the asymptotic local minimax optimality of our estimators. We then show that the theory can be used for conducting statistically valid A/B testing on auction platforms. Numerical simulations verify our central limit theorems, and empirical coverage rates for our confidence intervals agree with our theory.

We consider the problem of conformal prediction under covariate shift. Given labeled data from a source domain and unlabeled data from a covariate shifted target domain, we seek to construct prediction sets with valid marginal coverage in the target domain. Most existing methods require estimating the unknown likelihood ratio function, which can be prohibitive for high-dimensional data such as images. To address this challenge, we introduce the likelihood ratio regularized quantile regression (LR-QR) algorithm, which combines the pinball loss with a novel choice of regularization in order to construct a threshold function without directly estimating the unknown likelihood ratio. We show that the LR-QR method has coverage at the desired level in the target domain, up to a small error term that we can control. Our proofs draw on a novel analysis of coverage via stability bounds from learning theory. Our experiments demonstrate that the LR-QR algorithm outperforms existing methods on high-dimensional prediction tasks, including a regression task for the Communities and Crime dataset, an image classification task from the WILDS repository, and an LLM question-answering task on the MMLU benchmark.

We extend conformal prediction to control the expected value of any monotone loss function. The algorithm generalizes split conformal prediction together with its coverage guarantee. Like conformal prediction, the conformal risk control procedure is tight up to an O(1/n) factor. We also introduce extensions of the idea to distribution shift, quantile risk control, multiple and adversarial risk control, and expectations of U-statistics. Worked examples from computer vision and natural language processing demonstrate the usage of our algorithm to bound the false negative rate, graph distance, and token-level F1-score.

Side-channel evaluators routinely inspect leakage tests while acquisition is still running, and extend or stop the campaign based on what they see. Fixed-horizon screening such as the Welch t-test with threshold |t|>4.5 gives no error guarantee for this monitored decision rule. We study anytime-valid leakage detection based on testing by betting: SKIT-type swap-pair e-processes whose false-alarm probability is controlled uniformly over time under an explicit conditional symmetry null. In matched comparisons that share the frozen witness, rows and payoff, first-crossing detection needed 1.68-2.00times the traces of a fixed-horizon randomization test at 80% detection on synthetic streams, and 1.68-2.38times on degraded recordings from an open ML-KEM electromagnetic dataset with the primary Ridge witness at α=0.05. With the same primary witness and level, on undegraded reference and pqm4 recordings the monitored procedure stopped early: its median stopping point was 62-72 and 146-316 evaluation traces, i.e. 2-8% of a conservative 4096-trace budget. Under exact designed nulls on the recorded backgrounds, repeated-look |t|>4.5 screening over all 13000-20000 samples raised a false alarm in 2.7-12.9% of replicates, against 0.0-4.7% for terminal-only screening and no rejection by a sample-wise e-Bonferroni process, which in a prespecified follow-up detected natural-label associations in 4 of 4 backgrounds after 840-3288 traces. All recordings come from one device, and natural-label results are descriptive; we state the assumptions each claim requires.

Sequential trial emulation (STE) estimates the effect of a sustained treatment by stacking nested emulated trials with inverse-probability weighting. Additive-hazards STE estimators of the marginal risk difference recommend the nonparametric bootstrap without evaluating its coverage. Using a correctly-specifiable mechanism (up to a small, disclosed residual), we compare analytic and bootstrap standard errors for both estimands. Exploiting the closed-form linearity of the additive-hazards estimating equation, we derive the influence functions and prove the default row-level robust variance inconsistent for the marginal risk-difference curve, and show the same failure empirically for the constant hazard difference: the row-level variance omits a within-subject cross-trial covariance that is positive under a sign condition we verify across our mechanisms, and is anticonservative at every horizon except the first - only there, where the covariance is zero, is the row-level standard error unimpaired. The subject-clustered variance and multiplier bootstrap are consistent for the fixed-weight linearisation and support simultaneous confidence bands, whose measured coverage is 0.88. Across our simulations the model-based and row-level robust intervals are anticonservative and worsen with sample size, coverage falling to 0.71 at n=5000; clustering leaves the constant-hazard-difference coverage near 0.86 at n=5000, and the multiplier bootstrap leaves the risk-difference-curve coverage near 0.90 (0.86 at the longest horizon). The STE constant hazard difference is a design-weighted summary of a time-varying effect, dependent on the trial structure. We illustrate on the Stanford heart transplant data and provide them in the steCI R package.

Voting over multiple LLM responses is a common primitive in test-time scaling and ensemble inference. Collecting more responses can expand the candidate pool and increase the chance that a correct answer is discovered. Under a fixed call budget, a discovered answer still needs to accumulate enough support within the remaining calls to become the final plurality winner, creating a discovery-to-decision gap. In this work, we characterize this gap through the realized vote state and remaining call budget. We derive a sharp recoverability threshold and show that, as sampling proceeds, the observed candidate set can only expand while the set of reachable endpoint winners can only contract, inducing a candidate-level conversion window. Under a specified iid response law, the same state yields exact finite-horizon endpoint probabilities. We further show that merging wrong-answer identities preserves single-call correctness and cannot improve plurality accuracy, and that the effect of redistributing wrong-answer probability depends on the realized vote state. Singleton reachability yields a gold-free exact locking certificate. For a known answer universe, its first trigger is the earliest prefix at which all admissible continuations yield the same fixed-budget output. Empirically, most discovered-but-unselected correct answers lose reachability only after discovery. In a controlled Word16 study, input permutation improves raw-plurality accuracy by 21.1 points with essentially unchanged single-call correctness. Exact locking saves 28-30% of calls at a 16-call budget while preserving every fixed-budget output.

A service system may use the same categories to assign standard allocations and to check whether each service is fulfilled. Finer categories can match individual needs more closely, but they divide the observations available for monitoring. We study this conflict for a fixed menu from which participants select by declaring a category, with fulfilment assessed from each category's aggregate outcomes during a fixed period. We relate allocation loss to variation in preferred allocations within categories and identify conditions under which aggregate observations preserve the verification performance of individual records. For nested refinements under stated utility and observation assumptions, an allocation-loss tolerance and a per-category detection target define a feasibility band. Categories must be fine enough to provide suitable allocations but sufficiently populated to support verification. A source-dependent lower bound on declaration entropy and a minimum contributor requirement give necessary information and population constraints. For an explicit finite population with quadratic utility and binary service outcomes, we prove the exact feasible range across all categorical designs and exhibit designs attaining the information lower bound at specified tolerances. The results provide conditions for choosing categories jointly for allocation and verification.

Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates -- a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.

As large language models (LLMs) transition from research prototypes to production systems, practitioners often need reliable methods to verify that model outputs satisfy required constraints. While sampling-based estimates provide an intuition of model behavior, they offer no sound guarantees. We present BEAVER, the first practical framework for computing deterministic, sound probability bounds on LLM constraint satisfaction. Given any prefix-closed semantic constraint, BEAVER systematically explores the generation space using novel token trie and frontier data structures, maintaining provably sound bounds at every iteration. We formalize the verification problem, prove soundness of our approach, and evaluate BEAVER on correctness verification, privacy verification and secure code generation tasks across multiple state of the art LLMs. BEAVER achieves 6 to 8 times tighter probability bounds and identifies 3 to 4 times more high risk instances compared to baseline methods under identical computational budgets, enabling precise characterization and risk assessment that loose bounds or empirical evaluation cannot provide.

As machine learning (ML) enters high-stakes domains, trustworthy uncertainty quantification (UQ) is essential for safety. In this paper we introduce PCS-UQ, a framework based on the Predictability, Computability, and Stability (PCS) principles for veridical data science. Starting with a candidate set of models or algorithms, PCS-UQ integrates a rigorous prediction-check to screen out unsuitable models in the set and utilizes bootstrap samples, in order to capture both inter-sample variability and algorithmic instability for the prediction-checked algorithms. We then introduce a novel multiplicative calibration scheme to enhance local adaptivity, which basically corresponds to a new score in conformal prediction. Moreover, we produce a compilation of 17 real-world regression datasets with manually-constructed subgroups. On this benchmark, PCS-UQ maintains the target coverage while outperforming or matching conformal methods equipped with oracle-selected algorithms in interval width. PCS-UQ achieves consistent subgroup coverage, outperforming these oracle-selected conformal methods. Notably, PCS-UQ stands out in achieving both competitive interval widths and consistent subgroup coverage.Across 6 classification datasets, PCS-UQ reduces prediction set sizes by 20\%. To scale the framework for deep learning, we propose computationally efficient variants that bypass expensive retraining. On three computer vision benchmarks, these variants reduce prediction set sizes by 20\% over conformal baselines. Finally, we provide theoretical proof that a modified PCS-UQ algorithm preserves valid coverage under exchangeability as a form of split conformal inference.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

While LLMs have seen substantial improvement in reasoning capabilities, they also sometimes overthink, generating unnecessary reasoning steps, particularly under uncertainty, given ill-posed or ambiguous queries. We introduce statistically principled early stopping methods that monitor uncertainty signals during generation to mitigate this issue. Our first approach is parametric: it models inter-arrival times of uncertainty keywords as a renewal process and applies sequential testing for stopping. Our second approach is nonparametric and provides finite-sample guarantees on the probability of halting too early on well-posed queries. We conduct empirical evaluations on reasoning tasks across several domains and models. Our results indicate that uncertainty-aware early stopping can improve both efficiency and reliability in LLM reasoning, and we observe especially significant gains for math reasoning.

Generating diverse responses from large language models (LLMs) is crucial for applications such as planning/search and synthetic data generation, where diversity provides distinct answers across generations. Prior approaches rely on increasing temperature to increase diversity. However, contrary to popular belief, we show not only does this approach produce lower quality individual generations as temperature increases, but it depends on model's next-token probabilities being similar to the true distribution of answers. We propose , an alternative approach that uses the language model itself to partition the space into strata. At inference, a random stratum is selected and a sample drawn from within the strata. To measure diversity, we introduce CoverageQA, a dataset of underspecified questions with multiple equally plausible answers, and assess diversity by measuring KL Divergence between the output distribution and uniform distribution over valid ground truth answers. As computing probability per response/solution for proprietary models is infeasible, we measure recall on ground truth solutions. Our evaluation show using SimpleStrat achieves higher recall by 0.05 compared to GPT-4o and 0.36 average reduction in KL Divergence compared to Llama 3.

Policies with similar mean returns can differ sharply in rare failures, yet estimating lower-tail conditional value-at-risk (CVaR) accurately can require many costly rollouts. When different conditional components of a stochastic workflow can be queried separately, we ask how to allocate a fixed evaluation budget to estimate a fixed policy's CVaR most accurately. We derive a tail influence for each queryable conditional law that aggregates how its uncertainty affects CVaR across every Bellman reuse. Its variance yields the fixed-design efficiency bound and the oracle Neyman allocation. Tail-Influence Sampling (TIS) estimates these influence scales from a pilot model and reallocates fresh queries toward kernels that matter most for the tail; a visitation-anchored variant protects against pilot underallocation. Under fixed dimension and a positive quantile margin, TIS attains oracle asymptotic variance and first-order MSE including pilot cost, while the anchored variant is within a factor two of the oracle. We also characterize an exact-grid regime in which tail- and mean-optimal allocations coincide. On CliffWalking, TIS reduces MSE by 41% versus learned occupancy and 76% versus complete rollouts at the same charged transition budget. In frozen language-model review workflows, anchored TIS beats an equally regularized mean-influence blend in 23 of 24 MMLU-Pro settings and reaches 2.4-3.4times lower MSE than rollouts on six-call FinQA reviews.

Data-driven algorithm design automates hyperparameter tuning, but its statistical foundations remain limited because model performance can depend on hyperparameters in implicit and highly non-smooth ways. Existing guarantees focus on the simple case of a one-dimensional (scalar) hyperparameter. This leaves the practically important, multi-dimensional hyperparameter tuning setting unresolved. We address this open question by establishing the first general framework for establishing generalization guarantees for tuning multi-dimensional hyperparameters in data-driven settings. Our approach strengthens the generalization guarantee framework for semi-algebraic function classes by exploiting tools from real algebraic geometry, yielding sharper, more broadly applicable guarantees. For completeness, we also instantiate the first lower bound for this general setting. We further extend the analysis to hyperparameter tuning using the validation loss under minimal assumptions, and derive improved bounds when additional structure is available. Finally, we demonstrate the scope of the framework with new learnability results, including data-driven weighted group lasso and weighted fused lasso.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

Deep neural networks (DNN) have been widely applied in modern life, including critical domains like autonomous driving, making it essential to ensure the reliability and robustness of DNN-powered systems. As an analogy to code coverage metrics for testing conventional software, researchers have proposed neuron coverage metrics and coverage-driven methods to generate DNN test cases. However, Yan et al. doubt the usefulness of existing coverage criteria in DNN testing. They show that a coverage-driven method is less effective than a gradient-based method in terms of both uncovering defects and improving model robustness. In this paper, we conduct a replication study of the work by Yan et al. and extend the experiments for deeper analysis. A larger model and a dataset of higher resolution images are included to examine the generalizability of the results. We also extend the experiments with more test case generation techniques and adjust the process of improving model robustness to be closer to the practical life cycle of DNN development. Our experiment results confirm the conclusion from Yan et al. that coverage-driven methods are less effective than gradient-based methods. Yan et al. find that using gradient-based methods to retrain cannot repair defects uncovered by coverage-driven methods. They attribute this to the fact that the two types of methods use different perturbation strategies: gradient-based methods perform differentiable transformations while coverage-driven methods can perform additional non-differentiable transformations. We test several hypotheses and further show that even coverage-driven methods are constrained only to perform differentiable transformations, the uncovered defects still cannot be repaired by adversarial training with gradient-based methods. Thus, defensive strategies for coverage-driven methods should be further studied.

We employ random matrix theory to establish consistency of generalized cross validation (GCV) for estimating prediction risks of sketched ridge regression ensembles, enabling efficient and consistent tuning of regularization and sketching parameters. Our results hold for a broad class of asymptotically free sketches under very mild data assumptions. For squared prediction risk, we provide a decomposition into an unsketched equivalent implicit ridge bias and a sketching-based variance, and prove that the risk can be globally optimized by only tuning sketch size in infinite ensembles. For general subquadratic prediction risk functionals, we extend GCV to construct consistent risk estimators, and thereby obtain distributional convergence of the GCV-corrected predictions in Wasserstein-2 metric. This in particular allows construction of prediction intervals with asymptotically correct coverage conditional on the training data. We also propose an "ensemble trick" whereby the risk for unsketched ridge regression can be efficiently estimated via GCV using small sketched ridge ensembles. We empirically validate our theoretical results using both synthetic and real large-scale datasets with practical sketches including CountSketch and subsampled randomized discrete cosine transforms.

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

We investigate safe multi-agent reinforcement learning, where agents seek to collectively maximize an aggregate sum of local objectives while satisfying their own safety constraints. The objective and constraints are described by {\it general utilities}, i.e., nonlinear functions of the long-term state-action occupancy measure, which encompass broader decision-making goals such as risk, exploration, or imitations. The exponential growth of the state-action space size with the number of agents presents challenges for global observability, further exacerbated by the global coupling arising from agents' safety constraints. To tackle this issue, we propose a primal-dual method utilizing shadow reward and κ-hop neighbor truncation under a form of correlation decay property, where κ is the communication radius. In the exact setting, our algorithm converges to a first-order stationary point (FOSP) at the rate of Oleft(T^{-2/3}right). In the sample-based setting, we demonstrate that, with high probability, our algorithm requires mathcal{O}left(ε^{-3.5}right) samples to achieve an ε-FOSP with an approximation error of O(φ_0^{2κ}), where φ_0in (0,1). Finally, we demonstrate the effectiveness of our model through extensive numerical experiments.

This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization and improvement of the celebrated Chernoff method. At its heart, it is based on a class of composite nonnegative martingales, with strong connections to testing by betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform existing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, when sampling with and without replacement.

The inherent rigidity of fixed-size benchmarks makes them an inefficient tool for model evaluation. Diverse evaluation objectives, including model ranking, model selection and testing throughout development, demand varying levels of statistical power. The mismatch between fixed sample sizes and these diverse needs results in either excessive computational cost or compromised reliability - a critical concern for model evaluation. To overcome these limitations, we call for adoption of sequential testing in our field. We provide an adaptive evaluation framework, that provides a principled way to navigate the trade-off between efficiency and reliability in model evaluation. Our framework combines the established statistical paradigm of sequential testing with stopping criteria tailored to common evaluation needs such as diminishing returns detection, and minimum detectable effect size. We demonstrate its ability to adaptively manage the efficiency-reliability trade-off on the Open VLM Leaderboard, including, for example, a 80% reduction in computational cost compared to fixed-size evaluation (with a 2.5-point CI width allowance) while maintaining statistical significance.

Conformal prediction(CP) has become a cornerstone of distribution-free uncertainty quantification, conventionally evaluated by its coverage and interval length. This work critically examines the sufficiency of these standard metrics. We demonstrate that the interval length might be deceptively improved through a counter-intuitive approach termed Prejudicial Trick(PT), while the coverage remains valid. Specifically, for any given test sample, PT probabilistically returns an interval, which is either null or constructed using an adjusted confidence level, thereby preserving marginal coverage. While PT potentially yields a deceptively lower interval length, it introduces practical vulnerabilities: the same input can yield completely different prediction intervals across repeated runs of the algorithm. We formally derive the conditions under which PT achieves these misleading improvements and provide extensive empirical evidence across various regression and classification tasks. Furthermore, we introduce a new metric interval stability which helps detect whether a new CP method implicitly improves the length based on such PT-like techniques. Code is available at https://github.com/benben-cd/PT-Conformal-Prediction.

Large Language Models (LLMs) are increasingly deployed in discovery domains such as math and science. The usual approach is to present the problem to the model and use its answer as the proposed solution. However, beyond this best guess, discovery can be enhanced by increasing test-time compute. In a process called pass@k, the model is allowed to explore the solution space and generate diverse candidate solutions. Unfortunately, the standard approach to post-training LLMs through Reinforcement Learning (RL) may limit pass@k: the model's output distribution narrows around high-reward outputs, causing the solution coverage to collapse. The alternative is to use Evolution Strategies (ES), a population-based, gradient-free post-training method that optimizes directly in weight space through random perturbations. As this paper shows, ES achieves consistently higher pass@k than RL and produces a broader output distribution with greater solution coverage. This coverage in turn makes it possible to achieve better results in e.g. standard math benchmarks. Thus, ES provides a better foundation for post-training in discovery problems and other domains where diverse solution coverage is critical.

Despite the impressive capabilities of large language models (LLMs) across diverse applications, they still suffer from trustworthiness issues, such as hallucinations and misalignments. Retrieval-augmented language models (RAG) have been proposed to enhance the credibility of generations by grounding external knowledge, but the theoretical understandings of their generation risks remains unexplored. In this paper, we answer: 1) whether RAG can indeed lead to low generation risks, 2) how to provide provable guarantees on the generation risks of RAG and vanilla LLMs, and 3) what sufficient conditions enable RAG models to reduce generation risks. We propose C-RAG, the first framework to certify generation risks for RAG models. Specifically, we provide conformal risk analysis for RAG models and certify an upper confidence bound of generation risks, which we refer to as conformal generation risk. We also provide theoretical guarantees on conformal generation risks for general bounded risk functions under test distribution shifts. We prove that RAG achieves a lower conformal generation risk than that of a single LLM when the quality of the retrieval model and transformer is non-trivial. Our intensive empirical results demonstrate the soundness and tightness of our conformal generation risk guarantees across four widely-used NLP datasets on four state-of-the-art retrieval models.

Autonomous coding agents solve repository issues by reading code, running commands, editing files, and submitting patches. Extra inference-time compute yields gains only when it produces a useful repair and supplies reliable evidence for choosing one. Three behaviors decide both, and we argue they are teachable rather than byproducts of scale, so a policy can carry them instead of a scaffold. Location diversity remains narrow, since attempts return to the same site and extra samples add no coverage. Edit diversity is left unexploited, since methodologies that differ resolve complementary issues no single run reaches. Verification misleads, since a test the agent writes for its own patch accepts many incorrect ones. Directing search by execution feedback and scoring each patch against its own reverted tree resolves 52.8% of SWE-bench Verified using 48.1% of the agent-steps an eight-sample baseline spends. Training moves these behaviors into the policy. On the 270 issues held out from SFT and RL training, weighted supervised fine-tuning raises pass@1 from 31.9% to 35.2% and pass@8 from 46.7% to 51.1%. A reinforcement objective then trains the verifier against gold-labeled repairs and incorrect variants, crediting the assertions that detect them. It raises pass@1 to 43.0% and pass@8 to 60.7%, lifts verifier precision from 26.8% to 41.7%, and more than halves false acceptance. Resolution improves on two of three out-of-distribution suites and verifier precision on all three, and the gains hold at 7B, 14B, and 30B against published coder baselines.

The recent explosion in the capabilities of large language models has led to a wave of interest in how best to prompt a model to perform a given task. While it may be tempting to simply choose a prompt based on average performance on a validation set, this can lead to a deployment where unexpectedly poor responses are generated, especially for the worst-off users. To mitigate this prospect, we propose Prompt Risk Control, a lightweight framework for selecting a prompt based on rigorous upper bounds on families of informative risk measures. We offer methods for producing bounds on a diverse set of metrics, including quantities that measure worst-case responses and disparities in generation quality across the population of users. In addition, we extend the underlying statistical bounding techniques to accommodate the possibility of distribution shifts in deployment. Experiments on applications such as open-ended chat, medical question summarization, and code generation highlight how such a framework can foster responsible deployment by reducing the risk of the worst outcomes.

Code generated by large language models passes security checks at a rate that has barely moved in four years. In regulated backends, the defect classes that matter most are money arithmetic, time handling, retry safety, and access control. Teams answer with instruction files, yet the largest controlled study of instruction files to date found no benefit. This paper tests a narrower idea: generated code improves when the prompt carries a specification, a fixed preamble stating what must be true of the result. We pre-registered hypotheses, refuters, analysis code, and a one-shot generation rule, then ran 50 realistic backend tasks from finance, healthcare, and insurance practice through five frontier models from five vendor lineages, each task twice: bare, and preceded by a 267-word filled specification frame. Nine deterministic AST-based checkers scored the outputs. The Bandit security scanner, which knows nothing of the frame, scored them independently. The frame reduced defects in all five models (mean reduction 0.16 to 0.70 findings per task, every Holm-adjusted sign test significant, every bootstrap confidence interval excluding zero). Where the arms differed, the frame arm won 95 of 100 times. It never made any model worse in any domain. Bandit found 53 medium-or-high issues in the bare arm and 11 in the frame arm, in the same direction for every model. The effect was largest where a model's unprompted defaults were weakest: the frame supplies the discipline a model lacks. All 500 outputs, prompts, checkers, scoring code, and the pre-registration are published with a DOI, so any team can re-derive the result without trusting the author.

Candidate-scoring decision models return a probability distribution over a presented set of answers. Public documentation of one such system, Jev, and of an open counterpart, JevAny, describes three training ingredients that determine what these probabilities estimate: soft targets from larger models, a frozen pretrained backbone and temperature calibration. We analyze each ingredient in Gaussian surrogates where the finite-sample risk can be computed exactly. A student trained on teacher targets inherits the teacher's bias, which more student data does not remove and agreement with the teacher does not reveal. A fixed representation lowers the risk only below an explicit sample-size threshold. In a probit model the exact recalibration map is a temperature scaling, which removes all excess risk only when the teacher's coefficient has the correct direction and otherwise leaves a residual that calibration diagnostics do not detect. Under Luce's choice axiom, candidate scores are coherent across refinements of an event if and only if they aggregate by log-sum-exp. On 231,521 credit card complaints from the Consumer Financial Protection Bureau, where the decision is whether a complaint ends in monetary relief, a student trained on a logistic teacher's probabilities ends at the teacher's decision cost while a student trained on outcomes goes 8% below it, a 30B model read zero-shot remains worse than the teacher after recalibration, and that model departs from the coherence condition by 17.5 decibans. Pretrained representations reduce the labeled data required and calibration corrects the scale of the probabilities, but neither corrects a biased teacher, so the probabilities have to be checked against realized outcomes.

People overthink; language models over-sample, and the extra effort can talk both into a worse answer. Reasoning systems answer a hard question by sampling it many times (test-time scaling), and the more they draw, the more often a correct answer turns up somewhere, so coverage, the fraction of problems with at least one correct try, climbs and appears to be progress. But a deployed system must return one answer, and choosing it, not knowing which try is right, is selection; selection is capped, and past a point extra samples only make the model surer of a confident mistake, even as every draw adds cost. The gap between climbing coverage and stalled selection, the identifiability gap, is the answer a model can produce but not pick. So the real question is not whether to sample but how far, and the answer is: not far. For picking an answer, the vote has already settled within a few dozen draws, the modal ceiling; for scoring a benchmark, sooner still, the correlation ceiling. Beyond that, extra draws cost compute and add nothing, and can even make the answer worse. This paper turns the cutoff into a single number, the effective number of samples, that any sampling run already reveals. The bottleneck is recognizing a right answer, not generating one.

This book is about conformal prediction and related inferential techniques that build on permutation tests and exchangeability. These techniques are useful in a diverse array of tasks, including hypothesis testing and providing uncertainty quantification guarantees for machine learning systems. Much of the current interest in conformal prediction is due to its ability to integrate into complex machine learning workflows, solving the problem of forming prediction sets without any assumptions on the form of the data generating distribution. Since contemporary machine learning algorithms have generally proven difficult to analyze directly, conformal prediction's main appeal is its ability to provide formal, finite-sample guarantees when paired with such methods. The goal of this book is to teach the reader about the fundamental technical arguments that arise when researching conformal prediction and related questions in distribution-free inference. Many of these proof strategies, especially the more recent ones, are scattered among research papers, making it difficult for researchers to understand where to look, which results are important, and how exactly the proofs work. We hope to bridge this gap by curating what we believe to be some of the most important results in the literature and presenting their proofs in a unified language, with illustrations, and with an eye towards pedagogy.

A treatment may affect some individuals while leaving others unaffected. We study the reach of treatment effects by asking how large an unaffected covariate region could be. We develop efficient local randomization tests over nearest-neighbour regions to construct a finite-sample valid upper confidence bound for the size of the largest zero-effect region. A small bound provides evidence that treatment effects extend broadly across covariate values by ruling out large zero-effect regions. Under a general outcome model, we characterize how the size of the bound depends on the number of affected units, the size of their treatment effects, and the geometry of the true zero-effect set. We show that, when region size is measured as a sample fraction, the excess of the upper confidence bound over the true size decreases to zero at a minimax optimal rate up to a logarithmic factor. Numerical simulations assess the coverage, power and computational efficiency of our method. Applications to voter-turnout, HIV clinical-trial, and AI-assisted writing data illustrate how our method rules out large zero-effect regions across the covariates of interest.

Complete orthogonal predictive coordinates do not by themselves bind a latent direction to a named intervention. We present a mathematical and synthetic audit motivated by vascular device-vessel suitcordance. Capacity-matched least-squares predictors were exactly equivalent under complete fixed output transforms, whereas an anchor-only observer recovered interpretations only within the span of known perturbation signatures. Six three-dimensional configurations across 64 seeds gave a maximum paired prediction discrepancy of 6.7e-15 but a median untransported edit error of 1.513. Coordinate transport removed that error. Noisy and weak anchors constrained calibration stability, and changing the representation basis required recalibration or verified transport. Across 256 additional fits in dimensions 3-24, prediction equivalence persisted within 4.0e-15. We then evaluated nine deliberate runnable fault classes across 64 seeds. All 576 faulty executions completed, but each violated at least one reconstruction, prediction, delivered-edit or scope contract; all 320 valid control records passed. Repeating a faulty implementation gave exact self-agreement despite error against the separately computed simulator expectation. For one omitted-direction defect, probe coverage followed its analytic law, and rank-aware abstention protected unsupported interpretations. Scalar-noise experiments exposed both missed weak faults and excessive rejection under narrow relative tolerances. These controls provide an executable separation of prediction, semantic support and scientific acceptance. They are synthetic numerical audits, not clinical validation, neural JEPA-Anything replication, agent learning or patient treatment-effect estimation.

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.

We present a complete theoretical characterization of Latent Posterior Factors (LPF), a principled framework for aggregating multiple heterogeneous evidence items in probabilistic prediction tasks. Multi-evidence reasoning arises pervasively in high-stakes domains including healthcare diagnosis, financial risk assessment, legal case analysis, and regulatory compliance, yet existing approaches either lack formal guarantees or fail to handle multi-evidence scenarios architecturally. LPF encodes each evidence item into a Gaussian latent posterior via a variational autoencoder, converting posteriors to soft factors through Monte Carlo marginalization, and aggregating factors via exact Sum-Product Network inference (LPF-SPN) or a learned neural aggregator (LPF-Learned). We prove seven formal guarantees spanning the key desiderata for trustworthy AI: Calibration Preservation (ECE <= epsilon + C/sqrt(K_eff)); Monte Carlo Error decaying as O(1/sqrt(M)); a non-vacuous PAC-Bayes bound with train-test gap of 0.0085 at N=4200; operation within 1.12x of the information-theoretic lower bound; graceful degradation as O(epsilon*delta*sqrt(K)) under corruption, maintaining 88% performance with half of evidence adversarially replaced; O(1/sqrt(K)) calibration decay with R^2=0.849; and exact epistemic-aleatoric uncertainty decomposition with error below 0.002%. All theorems are empirically validated on controlled datasets spanning up to 4,200 training examples. Our theoretical framework establishes LPF as a foundation for trustworthy multi-evidence AI in safety-critical applications.

We develop new conformal inference methods for obtaining validity guarantees on the output of large language models (LLMs). Prior work in conformal language modeling identifies a subset of the text that satisfies a high-probability guarantee of correctness. These methods work by filtering claims from the LLM's original response if a scoring function evaluated on the claim fails to exceed a threshold calibrated via split conformal prediction. Existing methods in this area suffer from two deficiencies. First, the guarantee stated is not conditionally valid. The trustworthiness of the filtering step may vary based on the topic of the response. Second, because the scoring function is imperfect, the filtering step can remove many valuable and accurate claims. We address both of these challenges via two new conformal methods. First, we generalize the conditional conformal procedure of Gibbs et al. (2023) in order to adaptively issue weaker guarantees when they are required to preserve the utility of the output. Second, we show how to systematically improve the quality of the scoring function via a novel algorithm for differentiating through the conditional conformal procedure. We demonstrate the efficacy of our approach on biography and medical question-answering datasets.

Sampling-based search, a simple paradigm for utilizing test-time compute, involves generating multiple candidate responses and selecting the best one -- typically by verifying each response for correctness. In this paper, we study the scaling trends governing sampling-based search. Among our findings is that simply scaling up a minimalist implementation that uses only random sampling and direct self-verification results in sustained performance improvements that, for example, elevate the Gemini v1.5 Pro model's reasoning capabilities past that of o1-Preview on popular benchmarks. We partially attribute the scalability of sampling-based search to a phenomenon of implicit scaling, where sampling a larger pool of responses in turn improves verification accuracy. We further identify two useful principles for improving self-verification capabilities with test-time compute: (1) comparing across responses provides helpful signals about the locations of errors and hallucinations, and (2) different model output styles are useful for different contexts -- chains of thought are useful for reasoning but harder to verify. We also find that, though accurate verification can be elicited, frontier models demonstrate remarkably weak out-of-box verification capabilities and introduce a benchmark to measure progress on these deficiencies.

Randomized controlled trials (RCTs) are the benchmark for causal inference, yet field implementation can deviate. We here present a remote audit - a design-based, preregistrable diagnostic that uses only pre-treatment satellite imagery to test whether assignment is independent of local conditions. The conditional randomization test of the remote audit evaluates whether treatment assignment is more predictable from pre-treatment satellite features than expected under the experiment's registered mechanism, providing a finite-sample valid, design-based diagnostic that requires no parametric assumptions. The procedure is finite-sample valid, honors blocks and clusters, and controls multiplicity across image models and resolutions via a max-statistic. We illustrate with two RCTs: Uganda's Youth Opportunities Program, where the audit corroborates randomization and flags selection and missing-data risks; and a school-based trial in Bangladesh, where assignment is highly predictable from pre-treatment features relative to the stated design, consistent with independent concerns about irregularities. Remote audits complement balance tests, lower early-stage costs, and enable rapid design checks when baseline surveys are expensive or infeasible.

Large language models show great potential in generating and optimizing code. Widely used sampling methods such as Nucleus Sampling increase the diversity of generation but often produce repeated samples for low temperatures and incoherent samples for high temperatures. Furthermore, the temperature coefficient has to be tuned for each task, limiting its usability. We present Priority Sampling, a simple and deterministic sampling technique that produces unique samples ordered by the model's confidence. Each new sample expands the unexpanded token with the highest probability in the augmented search tree. Additionally, Priority Sampling supports generation based on regular expression that provides a controllable and structured exploration process. Priority Sampling outperforms Nucleus Sampling for any number of samples, boosting the performance of the original model from 2.87% to 5% improvement over -Oz. Moreover, it outperforms the autotuner used for the generation of labels for the training of the original model in just 30 samples.

Large language models (LLMs) are increasingly used as judges to replace costly human preference labels in pairwise evaluation. Despite their practicality, LLM judges remain prone to miscalibration and systematic biases. This paper proposes SCOPE (Selective Conformal Optimized Pairwise Evaluation), a framework for selective pairwise judging with finite-sample statistical guarantees. Under exchangeability, SCOPE calibrates an acceptance threshold such that the error rate among non-abstained judgments is at most a user-specified level α. To provide SCOPE with a bias-neutral uncertainty signal, we introduce Bidirectional Preference Entropy (BPE), which queries the judge under both response positions, aggregates the implied preference probabilities to enforce invariance to response order, and converts the aggregated probability into an entropy-based uncertainty score. Across MT-Bench, RewardBench, and Chatbot Arena, BPE improves uncertainty quality over standard confidence proxies, providing a stronger selection signal that enables SCOPE to consistently meet the target risk level while retaining good coverage across judge scales. In particular, at α= 0.10, SCOPE consistently satisfies the risk bound across all benchmarks and judge scales (empirical risk approx 0.097 to 0.099), while retaining substantial coverage, reaching 0.89 on RewardBench with Qwen-14B and 0.98 on RewardBench with Qwen-32B. Compared to naïve baselines, SCOPE accepts up to 2.4times more judgments on MT-Bench with Qwen-7B under the same target risk constraint, demonstrating that BPE enables reliable and high-coverage LLM-based evaluation.

In this paper, we introduce a conformal prediction method to construct prediction sets in a oneshot federated learning setting. More specifically, we define a quantile-of-quantiles estimator and prove that for any distribution, it is possible to output prediction sets with desired coverage in only one round of communication. To mitigate privacy issues, we also describe a locally differentially private version of our estimator. Finally, over a wide range of experiments, we show that our method returns prediction sets with coverage and length very similar to those obtained in a centralized setting. Overall, these results demonstrate that our method is particularly well-suited to perform conformal predictions in a one-shot federated learning setting.

In regression and causal inference, controlled subgroup selection aims to identify, with inferential guarantees, a subgroup (defined as a subset of the covariate space) on which the average response or treatment effect is above a given threshold. E.g., in a clinical trial, it may be of interest to find a subgroup with a positive average treatment effect. However, existing methods either lack inferential guarantees, heavily restrict the search for the subgroup, or sacrifice efficiency by naive data splitting. We propose a novel framework called chiseling that allows the analyst to interactively refine and test a candidate subgroup by iteratively shrinking it. The sole restriction is that the shrinkage direction only depends on the points outside the current subgroup, but otherwise the analyst may leverage any prior information or machine learning algorithm. Despite this flexibility, chiseling controls the probability that the discovered subgroup is null (e.g., has a non-positive average treatment effect) under minimal assumptions: for example, in randomized experiments, this inferential validity guarantee holds under only bounded moment conditions. When applied to a variety of simulated datasets and a real survey experiment, chiseling identifies substantially better subgroups than existing methods with inferential guarantees.

Large language models are often adapted through parameter efficient fine tuning, but current release practices provide weak assurances about what data were used and how updates were computed. We present Verifiable Fine Tuning, a protocol and system that produces succinct zero knowledge proofs that a released model was obtained from a public initialization under a declared training program and an auditable dataset commitment. The approach combines five elements. First, commitments that bind data sources, preprocessing, licenses, and per epoch quota counters to a manifest. Second, a verifiable sampler that supports public replayable and private index hiding batch selection. Third, update circuits restricted to parameter efficient fine tuning that enforce AdamW style optimizer semantics and proof friendly approximations with explicit error budgets. Fourth, recursive aggregation that folds per step proofs into per epoch and end to end certificates with millisecond verification. Fifth, provenance binding and optional trusted execution property cards that attest code identity and constants. On English and bilingual instruction mixtures, the method maintains utility within tight budgets while achieving practical proof performance. Policy quotas are enforced with zero violations, and private sampling windows show no measurable index leakage. Federated experiments demonstrate that the system composes with probabilistic audits and bandwidth constraints. These results indicate that end to end verifiable fine tuning is feasible today for real parameter efficient pipelines, closing a critical trust gap for regulated and decentralized deployments.

A/B tests are typically analyzed via frequentist p-values and confidence intervals; but these inferences are wholly unreliable if users endogenously choose samples sizes by *continuously monitoring* their tests. We define *always valid* p-values and confidence intervals that let users try to take advantage of data as fast as it becomes available, providing valid statistical inference whenever they make their decision. Always valid inference can be interpreted as a natural interface for a sequential hypothesis test, which empowers users to implement a modified test tailored to them. In particular, we show in an appropriate sense that the measures we develop tradeoff sample size and power efficiently, despite a lack of prior knowledge of the user's relative preference between these two goals. We also use always valid p-values to obtain multiple hypothesis testing control in the sequential context. Our methodology has been implemented in a large scale commercial A/B testing platform to analyze hundreds of thousands of experiments to date.

FSampler is a training free, sampler agnostic execution layer that accelerates diffusion sampling by reducing the number of function evaluations (NFE). FSampler maintains a short history of denoising signals (epsilon) from recent real model calls and extrapolates the next epsilon using finite difference predictors at second order, third order, or fourth order, falling back to lower order when history is insufficient. On selected steps the predicted epsilon substitutes the model call while keeping each sampler's update rule unchanged. Predicted epsilons are validated for finiteness and magnitude; a learning stabilizer rescales predictions on skipped steps to correct drift, and an optional gradient estimation stabilizer compensates local curvature. Protected windows, periodic anchors, and a cap on consecutive skips bound deviation over the trajectory. Operating at the sampler level, FSampler integrates with Euler/DDIM, DPM++ 2M/2S, LMS/AB2, and RES family exponential multistep methods and drops into standard workflows. FLUX.1 dev, Qwen Image, and Wan 2.2, FSampler reduces time by 8 to 22% and model calls by 15 to 25% at high fidelity (Structural Similarity Index (SSIM) 0.95 to 0.99), without altering sampler formulas. With an aggressive adaptive gate, reductions can reach 45 to 50% fewer model calls at lower fidelity (SSIM 0.73 to 0.74).

U-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for U-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for U-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting U-statistic, which is designed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to U-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.

Retrieval-augmented generation improves the factuality of large language models by grounding responses in retrieved evidence, yet existing evaluation frameworks struggle to provide consistent, fine-grained diagnostics across the diverse spectrum of user queries, ranging from close-ended fact-seeking to open-ended explanatory requests. We propose Q-CARE, a query-agnostic and fully reference-free framework that enables fine-grained assessment by decomposing queries into sub-queries and answers into atomic claims. Q-CARE establishes a unified evaluation principle based on query coverage and claim verifiability, yielding coverage-aware retriever metrics (C-Prec@k, C-nDCG@k) and claim-level generator metrics (Completeness, Conciseness, and Verifiableness). On a human-annotated benchmark spanning eight datasets, Q-CARE achieves higher correlation with human judgments than four existing RAG evaluation metrics, including RAGEval and RAGChecker, proving its effectiveness as a reliable, automated evaluation framework. Code and data are publicly available at https://github.com/DISL-Lab/Q-CaRE-COLM-26.

Outer-learning algorithms use infinitesimal sensitivities to propose finite changes to initialization or training parameters. For hard-ReLU training, the derivative of the finite program and the derivative of its flow limit do not by themselves specify the response at a chosen radius. We characterize the intervening regime in which the perturbation radius is proportional to the GD step. Integer event rounding then survives at leading order: smooth Euler bias shifts each discrete phase, and upstream rounding moves downstream branch boundaries. We derive the crossing indices and a uniform endpoint expansion for finitely many separated transverse events in piecewise-C^2 dynamics, away from recursive phase boundaries. In contractive affine regions, an explicit remainder and complete branch verification certify finite candidate comparisons. Scalar phase frequencies and a coupled feedback ablation test the mechanism; frozen nonlinear-network experiments show radius-dependent prediction accuracy, including incomplete branch matches and failed-word tails. Together with local AD and uniform flow consistency, the result identifies sufficient response regimes: differentiating training is a choice of perturbation resolution as well as a choice of derivative.

We study fixed-confidence best-action identification (BAI) in stochastic minimax trees. This problem is increasingly relevant in modern AI planning, where deep minimax search and Monte Carlo Tree Search (MCTS) with language model long rollouts face a fundamental tradeoff: heuristic evaluations are cheap but biased, while accurate rollouts are reliable but prohibitively expensive. We propose 2FFS, a two-fidelity tree-search algorithm that brings multi-fidelity flat bandit ideas into trees. The algorithm combines minimax-style fast expansion with MCTS-style stochastic sampling, adaptively deciding when to exploit cheap biased evaluations and when to invoke expensive accurate evaluations for local certification. We prove fixed-confidence correctness, establish finite stopping for exact identification, and give a polynomial-depth cost upper bound for general-depth trees. Across numerical stochastic-tree experiments, 2FFS uses substantially fewer samples and computational operations comparing to existing BAI-MCTS baseline.

Large Language Models (LLMs) are increasingly being used to simulate human-like decision making in agent-based financial market models (ABMs). As models become more powerful and accessible, researchers can now incorporate individual LLM decisions into ABM environments. However, integration may introduce inherent biases that need careful evaluation. In this paper we test three state-of-the-art GPT models for bias using two model sampling approaches: one-shot and few-shot API queries. We observe significant variations in distributions of outputs between specific models, and model sub versions, with GPT-4o-Mini-2024-07-18 showing notably better performance (32-43% yes responses) compared to GPT-4-0125-preview's extreme bias (98-99% yes responses). We show that sampling methods and model sub-versions significantly impact results: repeated independent API calls produce different distributions compared to batch sampling within a single call. While no current GPT model can simultaneously achieve a uniform distribution and Markovian properties in one-shot testing, few-shot sampling can approach uniform distributions under certain conditions. We explore the Temperature parameter, providing a definition and comparative results. We further compare our results to true random binary series and test specifically for the common human bias of Negative Recency - finding LLMs have a mixed ability to 'beat' humans in this one regard. These findings emphasise the critical importance of careful LLM integration into ABMs for financial markets and more broadly.

Rejection sampling is a technique for sampling from difficult distributions. However, its use is limited due to a high rejection rate. Common adaptive rejection sampling methods either work only for very specific distributions or without performance guarantees. In this paper, we present pliable rejection sampling (PRS), a new approach to rejection sampling, where we learn the sampling proposal using a kernel estimator. Since our method builds on rejection sampling, the samples obtained are with high probability i.i.d. and distributed according to f. Moreover, PRS comes with a guarantee on the number of accepted samples.

Large language models often produce plausible numerical claims from financial filings while using the wrong reporting period, unit, line item, or formula. Verifying such claims requires more than rechecking arithmetic: a verifier must ground the relevant facts, establish the authorized calculation, and determine whether the candidate value follows from both. We propose VeriFin, a neurosymbolic verification framework for numerical financial question answering that grounds operands in filed XBRL facts, derives calculations from the question, filing linkbases, or documented metric definitions, and checks claims using Z3. When a claim is inconsistent, solver-derived unsatisfiable cores identify the conflicting facts, formula, and candidate value, enabling targeted repair. We evaluate VeriFin on XBRLFiling, a new 600-question benchmark constructed from 10-K filings of 28 U.S. companies, and on FinanceBench. On fixed candidate pools shared by all verification methods, VeriFin accepts none of the incorrect claims, whereas the baselines accept 6 to 92 incorrect claims on XBRLFiling's 600 claims and 4 to 21 incorrect claims on FinanceBench's 67 claims. The zero false-accept result persists across multiple answer-generator models. Solver-derived feedback further improves repair, achieving up to 69.9% recovery among true catches. These results show that source-grounded symbolic verification can provide a reliable and auditable acceptance control for LLM-generated financial claims.

Due to the expanding capabilities and pre-training data, Large Language Models (LLMs) are facing increasingly serious evaluation challenges. On one hand, the data leakage issue cause over-estimation on existing benchmarks. On the other hand, periodically curating datasets manually is costly. In this paper, we propose to automate dataset updates for reliable and timely evaluation. The basic idea is to generate unseen and high-quality testing samples based on existing ones to mitigate leakage issues. In specific, we propose two strategies with systematically verification. First, the mimicking strategy employs LLMs to create new samples resembling existing ones, to the maximum extent preserving the stylistic of the original dataset. Our experiments demonstrate its evaluation stability across multiple instantiations and its effectiveness in dealing with data leakage issues in most cases. Second, for the cases that mimicking dataset works poorly, we design an extending strategy that adjusts the difficulty of the generated samples according to varying cognitive levels. This not only makes our evaluation more systematic, but also, with a balanced difficulty, even discern model capabilities better at fine-grained levels.

Test-time reasoning is increasingly used as a serving-time control knob, but extra reasoning is not uniformly valuable: it can repair failed attempts, waste compute on already-correct answers, or introduce harmful answer changes. We study this as a deployment allocation problem rather than a new-verifier problem. We introduce \sevra, Selective Verification for Reasoning Allocation, a serving-layer controller that decides whether to preserve a frozen solver's initial answer or invoke active verification. Using a frozen Qwen3-4B solver, we log intervention outcomes and train recoverability-aware gates from serving-visible attempt state. On \mathfive, selective verification reaches 76.3\% accuracy, compared with 75.5\% for always verifying, while reducing post-generation tokens by 26.8\% and harmful flips from 2.2\% to 1.0\%. However, an 8,192-token initial solve reaches 76.0\% accuracy with 28\% fewer total model tokens, showing that selective recovery is useful but not the best tested cost frontier. In frozen transfer to \gsm, the selective policy verifies only 3.0\% of examples, improves accuracy from 93.4\% to 94.5\%, and reduces verification tokens by 91.2\% relative to always verifying; again, a longer initial solve matches its accuracy with fewer realized tokens. On CommonsenseQA, always-on verification hurts, while Self-Consistency@5 improves accuracy at about five times the realized token cost. The resulting deployment rule is: tune the initial budget first, then use selective recovery when explicit checks, bounded retries, auditability, or regression-risk control matter.

Although randomized smoothing has demonstrated high certified robustness and superior scalability to other certified defenses, the high computational overhead of the robustness certification bottlenecks the practical applicability, as it depends heavily on the large sample approximation for estimating the confidence interval. In existing works, the sample size for the confidence interval is universally set and agnostic to the input for prediction. This Input-Agnostic Sampling (IAS) scheme may yield a poor Average Certified Radius (ACR)-runtime trade-off which calls for improvement. In this paper, we propose Input-Specific Sampling (ISS) acceleration to achieve the cost-effectiveness for robustness certification, in an adaptive way of reducing the sampling size based on the input characteristic. Furthermore, our method universally controls the certified radius decline from the ISS sample size reduction. The empirical results on CIFAR-10 and ImageNet show that ISS can speed up the certification by more than three times at a limited cost of 0.05 certified radius. Meanwhile, ISS surpasses IAS on the average certified radius across the extensive hyperparameter settings. Specifically, ISS achieves ACR=0.958 on ImageNet (sigma=1.0) in 250 minutes, compared to ACR=0.917 by IAS under the same condition. We release our code in https://github.com/roy-ch/Input-Specific-Certification.

We study the fine-grained uniform convergence behavior of halfspaces beyond worst-case VC bounds. For inhomogeneous halfspaces in R^d with dge 2, we show that standard first-order VC bounds are essentially tight: even consistent hypotheses can incur population error Θ(dln(n/d)/n), and in the agnostic setting the deviation scales as τln(1/τ) at true error τ. In contrast, homogeneous halfspaces in R^2 exhibit a markedly different behavior. In the realizable case, every hypothesis consistent with the sample has error O(1/n). In the agnostic case, we prove a bandwise, log-free deviation bound on each dyadic risk band via a critical-wedge localization argument. Unioning over bands incurs only a lnln n overhead, and we establish a matching lower bound showing this overhead is unavoidable. Together, these results give a fine-grained and nearly complete picture of uniform convergence for halfspaces, revealing sharp dimensional and structural thresholds.

Reinforcement learning with verifiable rewards improves reasoning, while the allocation of learning signal shapes which solutions remain accessible under repeated sampling. Group-relative objectives assign equal advantages to equally rewarded responses, making aggregate credit proportional to sampled mode frequency. We introduce Exploration-Preserving Policy Optimization (ExPPO), a lightweight advantage-shaping rule that redistributes credit using prompt-relative, length-normalized response surprisal and prompt pass rate. ExPPO combines bounded shaping with shared normalization to preserve verifier polarity and approximately maintain each prompt group's total absolute sequence-advantage mass. Our analysis characterizes response-level credit allocation alongside sampled mode updates, deriving local conditions for gains in entropy and correct-mode discovery. Experiments show improved in-domain and out-of-domain reasoning coverage, higher aggregate response accuracy, and strong coverage at large sampling budgets. A controlled multi-answer evaluation further demonstrates increased correct-mode yield and gains in diversity among verified-correct responses. Code is available at https://github.com/jinhangzhan/ExPPO

Repeated Sampling (RS) is a simple inference-time algorithm that has been shown to improve model performance on complex tasks. Although it is an effective way of scaling inference time, it often struggles to generate diverse solution candidates, frequently relying on the same underlying approach to solve the problem and thus producing redundant samples. To address this limitation, we propose a new inference algorithm, GuidedSampling, which decouples the exploration and generation phases during inference, increasing diversity of generated candidate solutions. The exploration phase identifies multiple concepts that can be utilized to solve the problem, while the generation phase applies a specific concept to provide final solution candidates. We first define the theoretical bounds of GuidedSampling and then empirically demonstrate that it improves the performance of base model at pass@50 by on an average ~21.6% across various benchmarks compared to RS. Furthermore, models trained on trajectories of GuidedSampling exhibit substantial performance improvements at pass@5 by on an average ~9.7%, compared to models trained on traditional RS. Additionally, models trained with GuidedSampling increases the average number of concepts per instance (1.67 -> 3.03), yielding a diverse set of candidates than traditional RS.

Decoding sits between a language model and everything we do with it, yet it is still treated as a heuristic knob-tuning exercise. We argue decoding should be understood as a principled optimisation layer: at each token, we solve a regularised problem over the probability simplex that trades off model score against structural preferences and constraints. This single template recovers greedy decoding, Softmax sampling, Top-K, Top-P, and Sparsemax-style sparsity as special cases, and explains their common structure through optimality conditions. More importantly, the framework makes it easy to invent new decoders without folklore. We demonstrate this by designing Best-of-K (BoK), a KL-anchored coverage objective aimed at multi-sample pipelines (self-consistency, reranking, verifier selection). BoK targets the probability of covering good alternatives within a fixed K-sample budget and improves empirical performance. We show that such samples can improve accuracy by, for example, +18.6% for Qwen2.5-Math-7B on MATH500 at high sampling temperatures.

Assessing the capabilities and risks of frontier AI systems is a critical area of research, and recent work has shown that repeated sampling from models can dramatically increase both. For instance, repeated sampling has been shown to increase their capabilities, such as solving difficult math and coding problems, but it has also been shown to increase their potential for harm, such as being jailbroken. Such results raise a crucial question for both capability and safety forecasting: how can one accurately predict a model's behavior when scaled to a massive number of attempts, given a vastly smaller sampling budget? This question is directly relevant to model providers, who serve hundreds of millions of users daily, and to governmental regulators, who seek to prevent harms. To answer this questions, we make three contributions. First, we find that standard methods for fitting these laws suffer from statistical shortcomings that hinder predictive accuracy, especially in data-limited scenarios. Second, we remedy these shortcomings by introducing a robust estimation framework, which uses a beta-binomial distribution to generate more accurate predictions from limited data. Third, we propose a dynamic sampling strategy that allocates a greater budget to harder problems. Combined, these innovations enable more reliable prediction of rare risks and capabilities at a fraction of the computational cost.

Variable-order Markov models generate sequences over a finite alphabet by conditioning each symbol on the longest available suffix of the generated history. Regular constraints, by contrast, describe finite-horizon control requirements by an automaton: fixed positions, forced endings, metrical patterns, and forbidden copied fragments are all special cases. Existing exact methods already handle regular constraints with belief propagation for first-order Markov chains. The contribution here is the variable-order extension: identifying the state space on which the existing BP-regular machinery must be run when the generator is a variable-order/backoff model. A first-order constraint layer can enforce useful support conditions, but it computes future mass after merging histories that a variable-order generator deliberately keeps distinct. We formalize this mismatch and give the sparse construction obtained by replacing the first-order Markov state with the observed context state, then taking the standard product with the regular constraint automaton. For a fixed trained context graph and automaton, inference is linear in the sequence horizon; in general it is polynomial in the number of reachable product edges. This gives the correct variable-order distribution conditioned on regular constraints without expanding to all K-tuples. The same finite-source interface supports reversible data augmentation by inverse count lookup, matching materialized transposition augmentation without storing transformed corpora. We also separate exact BP inference from generation-time backoff policies, such as singleton avoidance, whose stochastic semantics must be made explicit if exactness is claimed.

We resolve an open problem of Hanneke on the subject of universally consistent online learning with non-i.i.d. processes and unbounded losses. The notion of an optimistically universal learning rule was defined by Hanneke in an effort to study learning theory under minimal assumptions. A given learning rule is said to be optimistically universal if it achieves a low long-run average loss whenever the data generating process makes this goal achievable by some learning rule. Hanneke posed as an open problem whether, for every unbounded loss, the family of processes admitting universal learning are precisely those having a finite number of distinct values almost surely. In this paper, we completely resolve this problem, showing that this is indeed the case. As a consequence, this also offers a dramatically simpler formulation of an optimistically universal learning rule for any unbounded loss: namely, the simple memorization rule already suffices. Our proof relies on constructing random measurable partitions of the instance space and could be of independent interest for solving other open questions. We extend the results to the non-realizable setting thereby providing an optimistically universal Bayes consistent learning rule.

Knowledge benchmarks for LLMs face three issues: scaling-driven designs that do not operationalize disciplinary representativeness; flat-payment annotation that permits lazy consensus; and unaudited ranking instability under bounded test budgets. We introduce KINA, an 899-item benchmark across 261 fine-grained disciplines, with two formal results. First, we cast representativeness as a coverage-style objective over expert-elicited anchors and operationalize disciplinary representativeness through a proxy, yielding a (1-1/e) greedy approximation (Proposition 1); the guarantee applies to the proxy, not to population representativeness. Second, we prove a bonus-on-bar tournament weakly FOSD-dominates flat payment in released-review quality, with incentive-compatibility threshold B > Delta C / Delta p_min (Theorem 1). Evaluating 42 models from 13 labs, the top model, Gemini-3.1-Pro-Preview, reaches 53.17%, followed by Claude-Opus-4.6 at 49.92% and GPT-5.4 at 48.55%, leaving substantial headroom below saturation. The full leaderboard shows a tiered structure rather than a smooth total order: a small frontier tier lies above 48%, a dense strong-model tier spans roughly 38-45%, and low-performing models remain only modestly above the 10% chance baseline. Tool augmentation adds up to 5.17 points across the five tool-use evaluations, with gains varying substantially across models. We report bootstrap ranking-stability statistics to make bounded-budget variance explicit and to discourage over-interpretation of adjacent ranks.

The use of LLMs for code generation has naturally extended to code testing and evaluation. As codebases grow in size and complexity, so does the need for automated test generation. Current approaches for LLM-based test generation rely on strategies that maximize immediate coverage gain, a greedy approach that plateaus on code where reaching deep branches requires setup steps that individually yield zero new coverage. Drawing on principles of Bayesian exploration, we treat the program's branch structure as an unknown environment, and an evolving coverage map as a proxy probabilistic posterior representing what the LLM has discovered so far. Our method, CovQValue, feeds the coverage map back to the LLM, generates diverse candidate plans in parallel, and selects the most informative plan by LLM-estimated Q-values, seeking actions that balance immediate branch discovery with future reachability. Our method outperforms greedy selection on TestGenEval Lite, achieving 51-77% higher branch coverage across three popular LLMs and winning on 77-84% of targets. In addition, we build a benchmark for iterative test generation, RepoExploreBench, where they achieve 40-74%. These results show the potential of curiosity-driven planning methods for LLM-based exploration, enabling more effective discovery of program behavior through sequential interaction

BoN improves model outputs by sampling several candidates and selecting one with a proxy score, but it assumes that complete candidates can be evaluated reliably. Many vision-language tasks instead provide only partial verification: a finding, span, value, region, or relation may be checkable even when no dependable whole-response verifier exists. Moreover, the same claim may recur across candidates with opposing stances, allowing one observation to support part of the pool and contradict another. We introduce Best-of-Evidence (BoE), an inference-time selection framework that keeps the BoN candidate pool fixed, represents reusable claims with a signed candidate--factor graph, and allocates a limited budget to evidence actions that can change the final choice. BoE formalizes selection under partial verification and provides a practical score-based controller, with the zero-budget case recovering the underlying BoN decision. Theoretically, we show that residual evidence capacity limits any evidence-driven improvement and that shared factor queries can achieve an O(log K) versus Θ(K) query separation in a factor-code model. Common-ledger experiments on four medical VQA settings show that BoE can improve fixed-pool selection and rescue some BoN failures when evidence is reliable, contrastive, and decision-relevant, while also revealing the channel-quality and candidate-generation limits that prevent universal gains.

We study subsampling-based ridge ensembles in the proportional asymptotics regime, where the feature size grows proportionally with the sample size such that their ratio converges to a constant. By analyzing the squared prediction risk of ridge ensembles as a function of the explicit penalty lambda and the limiting subsample aspect ratio phi_s (the ratio of the feature size to the subsample size), we characterize contours in the (lambda, phi_s)-plane at any achievable risk. As a consequence, we prove that the risk of the optimal full ridgeless ensemble (fitted on all possible subsamples) matches that of the optimal ridge predictor. In addition, we prove strong uniform consistency of generalized cross-validation (GCV) over the subsample sizes for estimating the prediction risk of ridge ensembles. This allows for GCV-based tuning of full ridgeless ensembles without sample splitting and yields a predictor whose risk matches optimal ridge risk.

Testing is an integral part of the software development process. Yet, writing tests is time-consuming and therefore often neglected. Classical test generation tools such as EvoSuite generate behavioral test suites by optimizing for coverage, but tend to produce tests that are hard to understand. Language models trained on code can generate code that is highly similar to that written by humans, but current models are trained to generate each file separately, as is standard practice in natural language processing, and thus fail to consider the code-under-test context when producing a test file. In this work, we propose the Aligned Code And Tests Language Model (CAT-LM), a GPT-style language model with 2.7 Billion parameters, trained on a corpus of Python and Java projects. We utilize a novel pretraining signal that explicitly considers the mapping between code and test files when available. We also drastically increase the maximum sequence length of inputs to 8,192 tokens, 4x more than typical code generation models, to ensure that the code context is available to the model when generating test code. We analyze its usefulness for realistic applications, showing that sampling with filtering (e.g., by compilability, coverage) allows it to efficiently produce tests that achieve coverage similar to ones written by developers while resembling their writing style. By utilizing the code context, CAT-LM generates more valid tests than even much larger language models trained with more data (CodeGen 16B and StarCoder) and substantially outperforms a recent test-specific model (TeCo) at test completion. Overall, our work highlights the importance of incorporating software-specific insights when training language models for code and paves the way to more powerful automated test generation.

The theories of offline and online reinforcement learning, despite having evolved in parallel, have begun to show signs of the possibility for a unification, with algorithms and analysis techniques for one setting often having natural counterparts in the other. However, the notion of density ratio modeling, an emerging paradigm in offline RL, has been largely absent from online RL, perhaps for good reason: the very existence and boundedness of density ratios relies on access to an exploratory dataset with good coverage, but the core challenge in online RL is to collect such a dataset without having one to start. In this work we show -- perhaps surprisingly -- that density ratio-based algorithms have online counterparts. Assuming only the existence of an exploratory distribution with good coverage, a structural condition known as coverability (Xie et al., 2023), we give a new algorithm (GLOW) that uses density ratio realizability and value function realizability to perform sample-efficient online exploration. GLOW addresses unbounded density ratios via careful use of truncation, and combines this with optimism to guide exploration. GLOW is computationally inefficient; we complement it with a more efficient counterpart, HyGLOW, for the Hybrid RL setting (Song et al., 2022) wherein online RL is augmented with additional offline data. HyGLOW is derived as a special case of a more general meta-algorithm that provides a provable black-box reduction from hybrid RL to offline RL, which may be of independent interest.

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population means under the constraint of local differential privacy (LDP). Given bounded observations (X_1, dots, X_n) with mean mu^star that are privatized into (Z_1, dots, Z_n), we present confidence intervals (CI) and time-uniform confidence sequences (CS) for mu^star when only given access to the privatized data. To achieve this, we introduce a nonparametric and sequentially interactive generalization of Warner's famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. For example, our results yield private analogues of Hoeffding's inequality in both fixed-time and time-uniform regimes. We extend these Hoeffding-type CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.

Conformal prediction is a powerful tool to generate uncertainty sets with guaranteed coverage using any predictive model, under the assumption that the training and test data are i.i.d.. Recently, it has been shown that adversarial examples are able to manipulate conformal methods to construct prediction sets with invalid coverage rates, as the i.i.d. assumption is violated. To address this issue, a recent work, Randomized Smoothed Conformal Prediction (RSCP), was first proposed to certify the robustness of conformal prediction methods to adversarial noise. However, RSCP has two major limitations: (i) its robustness guarantee is flawed when used in practice and (ii) it tends to produce large uncertainty sets. To address these limitations, we first propose a novel framework called RSCP+ to provide provable robustness guarantee in evaluation, which fixes the issues in the original RSCP method. Next, we propose two novel methods, Post-Training Transformation (PTT) and Robust Conformal Training (RCT), to effectively reduce prediction set size with little computation overhead. Experimental results in CIFAR10, CIFAR100, and ImageNet suggest the baseline method only yields trivial predictions including full label set, while our methods could boost the efficiency by up to 4.36times, 5.46times, and 16.9times respectively and provide practical robustness guarantee. Our codes are available at https://github.com/Trustworthy-ML-Lab/Provably-Robust-Conformal-Prediction.

Despite remarkable progress, image generation is far from solved. The dominant metric, FID, conflates sample fidelity with mode coverage and is close to being saturated. Yet a model can still exhibit mode collapse while achieving a low FID, since a handful of sharp, near-duplicate images can outscore a model that faithfully covers the full data distribution. We argue that precision and recall are essential complements to FID, and that because FID is already saturated, the more meaningful goal is to improve diversity and coverage. Achieving high recall requires a model that explicitly prioritizes mode coverage, unlike most generative models, which optimize sample fidelity. We introduce RTM, which replaces the single-pass latent mapping in style-based generators with an iterative refinement process, and show that this consistently improves both quality and diversity. Integrated with Implicit Maximum Likelihood Estimation (IMLE), which optimizes mode coverage by design, RTM achieves the highest precision and recall among current state-of-the-art approaches while maintaining competitive FID, with improvements across CIFAR-10, CelebA-HQ at 256x256, and nine few-shot benchmarks. RTM also improves StyleGAN2 and StyleGAN2-ADA on CIFAR-10 and AFHQ-v1 at 512x512, demonstrating that the benefit is not specific to IMLE. Unlike flow-matching baselines that achieve competitive FID at the expense of coverage, recursive refinement improves both quality and diversity simultaneously.

In modern large-scale deep learning, a prevalent and effective workflow for solving low-data problems is adapting powerful pre-trained foundation models (FMs) to new tasks via parameter-efficient fine-tuning (PEFT). However, while empirically effective, the resulting solutions lack generalisation guarantees to certify their accuracy - which may be required for ethical or legal reasons prior to deployment in high-importance applications. In this paper we develop a novel transfer learning approach that is designed to facilitate non-vacuous learning theoretic generalisation guarantees for downstream tasks, even in the low-shot regime. Specifically, we first use upstream tasks to train a distribution over PEFT parameters. We then learn the downstream task by a sample-and-evaluate procedure -- sampling plausible PEFTs from the trained diffusion model and selecting the one with the highest likelihood on the downstream data. Crucially, this confines our model hypothesis to a finite set of PEFT samples. In contrast to learning in the typical continuous hypothesis spaces of neural network weights, this facilitates tighter risk certificates. We instantiate our bound and show non-trivial generalization guarantees compared to existing learning approaches which lead to vacuous bounds in the low-shot regime.

Bayesian inference provides principled uncertainty quantification but is often limited by challenges of prior elicitation, likelihood misspecification, and computational burden. The martingale posterior (MGP, Fong et al., 2023) offers an alternative, replacing prior-likelihood elicitation with a predictive rule - namely, a sequence of one-step-ahead predictive distributions - for forward data generation. The utility of MGPs depends on the choice of predictive rule, yet the literature has offered few compelling examples. Foundation transformers are well-suited here, as their autoregressive generation mirrors this forward simulation and their general-purpose design enables rich predictive modeling. We introduce TabMGP, an MGP built on TabPFN, a transformer foundation model that is currently state-of-the-art for tabular data. TabMGP produces credible sets with near-nominal coverage and often outperforms both existing MGP constructions and standard Bayes.