In a split conformal framework with $K$ classes, a calibration sample of $n$ labeled examples is observed for inference on the label of a new unlabeled example. We explore the setting where a ’batch’ of $m$ independent such unlabeled examples is given, and the goal is to construct a batch prediction set with 1-$\alpha$ coverage. Unlike individual prediction sets, the batch prediction set is a collection of label vectors of size $m$, while the calibration sample consists of univariate labels. A natural approach is to apply the Bonferroni correction, which concatenates individual prediction sets at level $1-\alpha/m$. We propose a uniformly more powerful solution, based on specific combinations of conformal $p$-values that exploit the Simes inequality. We provide a general recipe for valid inference with any combinations of conformal $p$-values, and compare the performance of several useful choices. Intuitively, the pooled evidence of relatively ‘easy’ examples within the batch can help provide narrower batch prediction sets. Additionally, we introduce a more computationally intensive method that aggregates batch scores and can be even more powerful. The theoretical guarantees are established when all examples are independent and identically distributed (iid), as well as more generally when iid is assumed only conditionally within each class. Notably, our results remain valid under label distribution shift, since the distribution of the labels need not be the same in the calibration sample and in the new batch. The effectiveness of the methods is highlighted through illustrative synthetic and real data examples.

Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. We consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. While classical results only concern their marginal distribution, we show that their joint distribution follows a Pólya urn model, and establish a concentration inequality for their empirical distribution function. The results hold for arbitrary exchangeable scores, including adaptive ones that can use the covariates of the test${+}$calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.

We provide new nonasymptotic false discovery proportion (FDP) confidence envelopes in several multiple testing settings relevant for modern high dimensional-data methods. We revisit the multiple testing scenarios considered in the recent work of Katsevich and Ramdas (2020): top-$k$, preordered (including knockoffs), online. Our emphasis is on obtaining FDP confidence bounds that both have non-asymptotical coverage and are asymptotically accurate in a specific sense, as the number $m$ of tested hypotheses grows. Namely, we introduce and study the property (which we call $m$-consistency) that the confidence bound converges to or below the desired level $\alpha$ when applied to a specific reference $\alpha$-level false discovery rate (FDR) controlling procedure. In this perspective, we derive new bounds that provide improvements over existing ones, both theoretically and practically, and are suitable for situations where at least a moderate number of rejections is expected. These improvements are illustrated with numerical experiments and real data examples. In particular, the improvement is significant in the knockoffs setting, which shows the impact of the method for a practical use. As side results, we introduce a new confidence envelope for the empirical cumulative distribution function of i.i.d. uniform variables, and we provide new power results in sparse cases, both being of independent interest.