Online Multivalid Learning: Means, Moments, and Prediction Intervals
This work provides stronger uncertainty quantification guarantees for black-box algorithms in online adversarial settings, which is important for practitioners needing robust and fair predictions.
This paper introduces a general, efficient online learning technique that provides contextual predictions, specifically for means, moments, and prediction intervals, that are "multivalid" across arbitrary intersecting groups of adversarially chosen examples. The framework instantiates multicalibration for mean prediction, mean-conditioned moment multicalibration for variance and higher moment prediction, and a new notion of prediction interval multivalidity.
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 Hebert-Johnson et al. The second is variance and higher moment prediction, which corresponds to an online algorithm satisfying the notion of mean-conditioned 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.