Inductive Conformal Prediction (ICP) provides a practical and effective approach for equipping deep learning models with uncertainty estimates in the form of set-valued predictions which are guaranteed to contain the ground truth with high probability.Despite the appeal of this coverage guarantee, these sets may not be efficient: the size and contents of the prediction sets are not directly controlled, and instead depend on the underlying model and choice of score function.To remedy this, recent work has proposed learning model and score function parameters using data to directly optimize the efficiency of the ICP prediction sets.While appealing, the generalization theory for such an approach is lacking: direct optimization of empirical efficiency may yield prediction sets that are either no longer efficient on test data, or no longer obtain the required coverage on test data.In this work, we use PAC-Bayes theory to obtain generalization bounds on both the coverage and the efficiency of set-valued predictors which can be directly optimized to maximize efficiency while satisfying a desired test coverage.In contrast to prior work, our framework allows us to utilize the entire calibration dataset to learn the parameters of the model and score function, instead of requiring a separate hold-out set for obtaining test-time coverage guarantees.We leverage these theoretical results to provide a practical algorithm for using calibration data to simultaneously fine-tune the parameters of a model and score function while guaranteeing test-time coverage and efficiency of the resulting prediction sets.We evaluate the approach on regression and classification tasks, and outperform baselines calibrated using a Hoeffding bound-based PAC guarantee on ICP, especially in the low-data regime.

We investigate the integration of Conformal Prediction (CP) with supervised learning on deterministically encrypted data, aiming to bridge the gap between rigorous uncertainty quantification and privacy-preserving machine learning. Using AES-encrypted variants of the MNIST dataset, we demonstrate that CP methods remain effective even when applied directly in the encrypted domain, owing to the preservation of data exchangeability under fixed-key encryption. We test traditional $p$-value-based against $e$-value-based conformal predictors. Our empirical evaluation reveals that models trained on deterministically encrypted data retain the ability to extract meaningful structure, achieving 36.88\% test accuracy -- significantly above random guessing (9.56\%) observed with per-instance encryption. Moreover, $e$-value-based CP achieves predictive set coverage of over 60\% with 4.3 loss-threshold calibration, correctly capturing the true label in 4888 out of 5000 test cases. In contrast, the $p$-value-based CP yields smaller predictive sets but with reduced coverage accuracy. These findings highlight both the promise and limitations of CP in encrypted data settings and underscore critical trade-offs between prediction set compactness and reliability. %Our work sets a foundation for principled uncertainty quantification in secure, privacy-aware learning systems.

Inductive Conformal Prediction (ICP) provides a practical and effective approach for equipping deep learning models with uncertainty estimates in the form of set-valued predictions which are guaranteed to contain the ground truth with high probability.Despite the appeal of this coverage guarantee, these sets may not be efficient: the size and contents of the prediction sets are not directly controlled, and instead depend on the underlying model and choice of score function.To remedy this, recent work has proposed learning model and score function parameters using data to directly optimize the efficiency of the ICP prediction sets.While appealing, the generalization theory for such an approach is lacking: direct optimization of empirical efficiency may yield prediction sets that are either no longer efficient on test data, or no longer obtain the required coverage on test data.In this work, we use PAC-Bayes theory to obtain generalization bounds on both the coverage and the efficiency of set-valued predictors which can be directly optimized to maximize efficiency while satisfying a desired test coverage.In contrast to prior work, our framework allows us to utilize the entire calibration dataset to learn the parameters of the model and score function, instead of requiring a separate hold-out set for obtaining test-time coverage guarantees.We leverage these theoretical results to provide a practical algorithm for using calibration data to simultaneously fine-tune the parameters of a model and score function while guaranteeing test-time coverage and efficiency of the resulting prediction sets.We evaluate the approach on regression and classification tasks, and outperform baselines calibrated using a Hoeffding bound-based PAC guarantee on ICP, especially in the low-data regime.

Conformal Prediction (CP) serves as a robust framework that quantifies uncertainty in predictions made by Machine Learning (ML) models. Unlike traditional point predictors, CP generates statistically valid prediction regions, also known as prediction intervals, based on the assumption of data exchangeability. Typically, the construction of conformal predictions hinges on p-values. This paper, however, ventures down an alternative path, harnessing the power of e-test statistics to augment the efficacy of conformal predictions by introducing a BB-predictor (bounded from the below predictor).

Accurately labeling biomedical data presents a challenge. Traditional semi-supervised learning methods often under-utilize available unlabeled data. To address this, we propose a novel reliability-based training data cleaning method employing inductive conformal prediction (ICP). This method capitalizes on a small set of accurately labeled training data and leverages ICP-calculated reliability metrics to rectify mislabeled data and outliers within vast quantities of noisy training data. The efficacy of the method is validated across three classification tasks within distinct modalities: filtering drug-induced-liver-injury (DILI) literature with title and abstract, predicting ICU admission of COVID-19 patients through CT radiomics and electronic health records, and subtyping breast cancer using RNA-sequencing data. Varying levels of noise to the training labels were introduced through label permutation. Results show significant enhancements in classification performance: accuracy enhancement in 86 out of 96 DILI experiments (up to 11.4%), AUROC and AUPRC enhancements in all 48 COVID-19 experiments (up to 23.8% and 69.8%), and accuracy and macro-average F1 score improvements in 47 out of 48 RNA-sequencing experiments (up to 74.6% and 89.0%). Our method offers the potential to substantially boost classification performance in multi-modal biomedical machine learning tasks. Importantly, it accomplishes this without necessitating an excessive volume of meticulously curated training data.

Inductive Conformal Prediction (ICP) is a set of distribution-free and model agnostic algorithms devised to predict with a user-defined confidence with coverage guarantee. Instead of having point predictions, i.e., a real number in the case of regression or a single class in multi class classification, models calibrated using ICP output an interval or a set of classes, respectively. ICP takes special importance in high-risk settings where we want the true output to belong to the prediction set with high probability. As an example, a classification model might output that given a magnetic resonance image a patient has no latent diseases to report. However, this model output was based on the most likely class, the second most likely class might tell that the patient has a 15% chance of brain tumor or other severe disease and therefore further exams should be conducted. Using ICP is therefore way more informative and we believe that should be the standard way of producing forecasts. This paper is a hands-on introduction, this means that we will provide examples as we introduce the theory.

Inductive Conformal Prediction (ICP) is a set of distribution-free and model agnostic algorithms devised to predict with a user-defined confidence with coverage guarantee. Instead of having point predictions, i.e., a real number in the case of regression or a single class in multi class classification, models calibrated using ICP output an interval or a set of classes, respectively. ICP takes special importance in high-risk settings where we want the true output to belong to the prediction set with high probability. As an example, a classification model might output that given a magnetic resonance image a patient has no latent diseases to report. However, this model output was based on the most likely class, the second most likely class might tell that the patient has a 15% chance of brain tumor or other severe disease and therefore further exams should be conducted. Using ICP is therefore way more informative and we believe that should be the standard way of producing forecasts. This paper is a hands-on introduction, this means that we will provide examples as we introduce the theory.

Conformal predictors are confidence predictors that result in prediction sets for all confidence levels. Thus, Conformal Prediction (CP) is a framework that complements the predictions of machine learning algorithms with reliable measures of confidence. Transductive Conformal Prediction (TCP) works in an online transductive setting, such that learning and prediction occur simultaneously. In this sense confidence in a prediction is tailored both to the previously seen objects (whose features and labels are known) and to the features of the new object, whose label is to be predicted. By conditioning on the new objects conformal predictors take account of how difficult a particular object is to label and adjust their confidence in the prediction accordingly, as opposed to having an overall error rate for labelling all new objects Vovk et al. (2005).

The method of conformal prediction produces set predictions that are automatically valid in the sense that their unconditional coverage probability is equal to or exceeds a preset confidence level ([14], Chapter 2). A more computationally efficient method of this kind is that of inductive conformal prediction ([12], [14], Section 4.1, [1]). However, inductive conformal predictors are typically less predictively efficient, in the sense of producing larger prediction sets as compared with conformal predictors. Motivated by the method of cross-validation [11, 13], this note explores a hybrid method, which we call cross-conformal prediction. We are mainly interested in the problems of classification and regression, in which we are given a training set consisting of examples, each example consisting of an object and a label, and asked to predict the label of a new test object; in the problem of classification labels are elements of a given finite set, and in the problem of regression labels are real numbers. If we are asked to predict labels for more than one test objects, the same prediction procedure can be applied to each test object separately. In this introductory section and in our empirical studies we consider the problem of binary classification, in which labels can take only two values, which we will encode as 0 and 1. We always assume that the examples (both the training examples and the test examples, consisting of given objects and unknown labels) are generated independently from the same probability measure; this assumption will be called the assumption of randomness.