Conformal Calibration

Conformal prediction provides finite-sample, distribution-free coverage under exchangeability, but standard constructions may lack robustness in the presence of outliers or heavy tails. We propose a robust conformal method based on a non-conformity score defined as the half-mass radius around a point, equivalently the distance to its $(\lfloor n/2\rfloor+1)$-nearest neighbour. We show that the resulting conformal regions are marginally valid for any sample size and converge in probability to a robust population central set defined through a distance-to-a-measure functional. Under mild regularity conditions, we establish exponential concentration and tail bounds that quantify the deviation between the empirical conformal region and its population counterpart. These results provide a probabilistic justification for using robust geometric scores in conformal prediction, even for heavy-tailed or multi-modal distributions.

Large language models are increasingly deployed in settings where reliability matters, yet output-level uncertainty signals such as token probabilities, entropy, and self-consistency can become brittle under calibration--deployment mismatch. Conformal prediction provides finite-sample validity under exchangeability, but its practical usefulness depends on the quality of the nonconformity score. We propose a conformal framework for LLM question answering that uses internal representations rather than output-facing statistics: specifically, we introduce Layer-Wise Information (LI) scores, which measure how conditioning on the input reshapes predictive entropy across model depth, and use them as nonconformity scores within a standard split conformal pipeline. Across closed-ended and open-domain QA benchmarks, with the clearest gains under cross-domain shift, our method achieves a better validity--efficiency trade-off than strong text-level baselines while maintaining competitive in-domain reliability at the same nominal risk level. These results suggest that internal representations can provide more informative conformal scores when surface-level uncertainty is unstable under distribution shift.

Building on recent advances in AI, hybrid decision making (HDM) holds the promise of improving human decision quality and reducing cognitive load. We work in the context of learning to guide (LtG), a recently proposed HDM framework in which the human is always responsible for the final decision: rather than suggesting decisions, in LtG the AI supplies (textual) guidance useful for facilitating decision making. One limiting factor of existing approaches is that their guidance compounds information about all possible outcomes, and as a result it can be difficult to digest. We address this issue by introducing ConfGuide, a novel LtG approach that generates more succinct and targeted guidance. To this end, it employs conformal risk control to select a set of outcomes, ensuring a cap on the false negative rate. We demonstrate our approach on a real-world multi-label medical diagnosis task. Our empirical evaluation highlights the promise of ConfGuide.

Conformal selection (CS) uses calibration data to identify test inputs whose unobserved outcomes are likely to satisfy a pre-specified minimal quality requirement, while controlling the false discovery rate (FDR). Existing methods fix the target FDR level before observing data, which prevents the user from adapting the balance between number of selected test inputs and FDR to downstream needs and constraints based on the available data. For example, in genomics or neuroimaging, researchers often inspect the distribution of test statistics, and decide how aggressively to pursue candidates based on observed evidence strength and available follow-up resources. To address this limitation, we introduce post-hoc CS (PH-CS), which generates a path of candidate selection sets, each paired with a data-driven false discovery proportion (FDP) estimate. PH-CS lets the user select any operating point on this path by maximizing a user-specified utility, arbitrarily balancing selection size and FDR. Building on conformal e-variables and the e-Benjamini-Hochberg (e-BH) procedure, PH-CS is proved to provide a finite-sample post-hoc reliability guarantee whereby the ratio between estimated FDP level and true FDP is, on average, upper bounded by 1, so that the average estimated FDP is, to first order, a valid upper bound on the true FDR. PH-CS is extended to control quality defined in terms of a general risk. Experiments on synthetic and real-world datasets demonstrate that, unlike CS, PH-CS can consistently satisfy user-imposed utility constraints while producing reliable FDP estimates and maintaining competitive FDR control.

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.

Deep neural networks (DNNs) rely heavily on high-quality open-source datasets (e.g., ImageNet) for their success, making dataset ownership verification (DOV) crucial for protecting public dataset copyrights. In this paper, we find existing DOV methods (implicitly) assume that the verification process is faithful, where the suspicious model will directly verify ownership by using the verification samples as input and returning their results. However, this assumption may not necessarily hold in practice and their performance may degrade sharply when subjected to intentional or unintentional perturbations. To address this limitation, we propose the first certified dataset watermark (i.e., CertDW) and CertDW-based certified dataset ownership verification method that ensures reliable verification even under malicious attacks, under certain conditions (e.g., constrained pixel-level perturbation). Specifically, inspired by conformal prediction, we introduce two statistical measures, including principal probability (PP) and watermark robustness (WR), to assess model prediction stability on benign and watermarked samples under noise perturbations. We derive provable certification conditions relating WR to a PP-based calibration threshold, and a high-probability upper bound on the false positive rate, enabling ownership verification when a suspicious model's WR value significantly exceeds the PP values of multiple benign models trained on watermark-free datasets. If the number of PP values smaller than WR exceeds a threshold determined via conformal calibration, the suspicious model is regarded as having been trained on the protected dataset. Extensive experiments on benchmark datasets verify the effectiveness of our CertDW method and its resistance to potential adaptive attacks. Our codes are at \href{https://github.com/NcepuQiaoTing/CertDW}{GitHub}.

The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases. Firstly, our approach post-trains a pre-trained diffusion model to generate control sequences that better satisfy safety constraints while achieving improved control objectives via a reweighted diffusion loss, which incorporates the uncertainty quantile estimated using conformal prediction. Secondly, during inference, the diffusion model dynamically adjusts both its generation process and parameters through iterative guidance and fine-tuning, conditioned on control targets while simultaneously integrating the estimated uncertainty quantile. We evaluate SafeDiffCon on three control tasks: 1D Burgers' equation, 2D incompressible fluid, and controlled nuclear fusion problem. Results demonstrate that SafeDiffCon is the only method that satisfies all safety constraints, whereas other classical and deep learning baselines fail. Furthermore, while adhering to safety constraints, SafeDiffCon achieves the best control performance. The code can be found at https://github.com/AI4Science-WestlakeU/safediffcon.

Clinical care often relies on key laboratory indicators, yet real-world patient visits are sparse and tests are ordered irregularly, leading to pervasive missingness. While many imputation methods improve average accuracy, they provide limited guidance on which imputed values are reliable enough for high-stakes downstream use. In this work, we study reliable clinical imputation, aiming to produce accurate imputations while selectively releasing the reliable results, with statistical control over clinically unacceptable errors. To achieve this goal, we propose SafeImpute, a reliable imputation framework for irregular and sparse clinical longitudinal records. SafeImpute constructs an event graph that captures both intra-patient temporal trajectories and inter-patient clinical similarity, and learns imputations with a two-relation GNN and adaptive fusion, regularized by an auxiliary masked reconstruction objective. For reliability guarantees, SafeImpute converts a proxy risk score into conformal p-values and applies the Benjamini--Hochberg procedure to control the false discovery rate (FDR) of unacceptable errors among released imputations at a user-specified tolerance. Experiments on our Mayo Clinic data, the public MIMIC-III and MIMIC-IV datasets show that SafeImpute achieves strong imputation accuracy while providing reliable error control, outperforming diverse baselines in both standard imputation evaluation and FDR-controlled selective-release evaluation.