Online Multivalid Learning: Means, Moments, and Prediction Intervals
Abstract
We present a general, efficient technique for providing contextual predictions that are "multivalid" in various senses, against an online sequence of adversarially chosen examples $(x,y)$. This means that the resulting estimates correctly predict various statistics of the labels $y$ not just marginally  as averaged over the sequence of examples  but also conditionally on $x \in G$ for any $G$ belonging to an arbitrary intersecting collection of groups $\mathcal{G}$. We provide three instantiations of this framework. The first is mean prediction, which corresponds to an online algorithm satisfying the notion of multicalibration from HebertJohnson et al. The second is variance and higher moment prediction, which corresponds to an online algorithm satisfying the notion of meanconditioned moment multicalibration from Jung et al. Finally, we define a new notion of prediction interval multivalidity, and give an algorithm for finding prediction intervals which satisfy it. Because our algorithms handle adversarially chosen examples, they can equally well be used to predict statistics of the residuals of arbitrary point prediction methods, giving rise to very general techniques for quantifying the uncertainty of predictions of black box algorithms, even in an online adversarial setting. When instantiated for prediction intervals, this solves a similar problem as conformal prediction, but in an adversarial environment and with multivalidity guarantees stronger than simple marginal coverage guarantees.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.01739
 Bibcode:
 2021arXiv210101739G
 Keywords:

 Computer Science  Machine Learning;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Computer Science and Game Theory;
 Economics  Econometrics