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 ∈ G for any G belonging to an arbitrary intersecting collection of groups 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 Hébert-Johnson et al. [2018]. 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. [2020]. 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.

An Algorithmic Framework for Bias Bounties

“…A notable exception to the rule that fairness and accuracy must involve tradeoffs, from which we take inspiration, is the literature on multicalibration initiated by Hébert-Johnson et al [Hébert-Johnson et al, 2018, Kim et al, 2019, Gupta et al, 2021, Dwork et al, 2021 that asks that a model's predictions be calibrated not just overall, but also when restricted to a large number of protected subgroups g. Hébert-Johnson et al [Hébert-Johnson et al, 2018] and Kim, Ghorbani, and Zou [Kim et al, 2019] show that an arbitrary model f can be postprocessed to satisfy multicalibration (and the related notion of "multi-accuracy") without sacrificing (much) in terms of model accuracy. Our aim is to achieve something similar, but for predictive error, rather than model calibration.…”

“…Remark 6. We have chosen to define approximate Bayes optimality by letting the approximation term scale proportionately to the inverse probability of the group g, similar to how notions of multigroup fairness are defined in [Kearns et al, 2018, Gupta et al, 2021]. An alternative (slightly weaker) option would be to require error that is uniformly bounded by for all groups, but to only make promises for groups g that have probability µ g larger than some threshold, as is done in [Hébert-Johnson et al, 2018].…”

Scaffolding Sets

“…In particular, Hébert-Johnson et al introduced the method for constructing efficient multi-calibrated predictors in the batch setting [31]. Multi-calibration in the online setting is implicit in [52], [37] extends the notion of multi-calibration to higher moments and provides constructions, and [28] provides an efficient online solution. Multi-calibration has also been applied to solve several flavors of problems: fair ranking [13]; ommiprediction, which is roughly learning a predictor that, for a given class of loss functions, can be post-processed to minimize loss for any function in the class [23]; and providing an alternative to propensity scoring for the purposes of generalization to future populations [39].…”

Multi-group Agnostic PAC Learnability

“…because the Bayes optimal predictor always satisfies multi-group fairness (regardless of G). Notable examples include multi-calibration [Hébert-Johnson et al, 2018] and multi-accuracy , with subsequent works studying extensions in ranking [Dwork et al, 2019], regression [Jung et al, 2020] and online learning [Gupta et al, 2021]. As discussed above, Blum and Lykouris [Blum and Lykouris, 2019] study multi-group agnostic PAC learning (which, depending on the loss function, is often similarly aligned with accuracy) in the online setting.…”