Taking = p / gives the desired claim. Claim 2.7, we know that the multicalibration violation for The inequalities follow by Holder's inequality and the assumed bound on the weight of Recall that Cov[ y, z ]= E [ yz ] E [ y ] E [ z ] . Here, we give a high-level overview of the MCBoost algorithm of [ 20 ] and weak agnostic learning. Algorithm 2 MCBoost Parameters: hypothesis class C and > 0 Given: Dataset S sampled from D Initialize: p ( x) 1 / 2 . By Lemma 3.8, we know that In this Appendix, we give a full account of the definitions and results stated in Section 4 .

We introduce and study Swap Agnostic Learning.

Taking = p / gives the desired claim. Claim 2.7, we know that the multicalibration violation for The inequalities follow by Holder's inequality and the assumed bound on the weight of Recall that Cov[ y, z ]= E [ yz ] E [ y ] E [ z ] . Here, we give a high-level overview of the MCBoost algorithm of [ 20 ] and weak agnostic learning. Algorithm 2 MCBoost Parameters: hypothesis class C and > 0 Given: Dataset S sampled from D Initialize: p ( x) 1 / 2 . By Lemma 3.8, we know that In this Appendix, we give a full account of the definitions and results stated in Section 4 .

We present a new perspective on loss minimization and the recent notion of Omniprediction through the lens of Outcome Indistingusihability. For a collection of losses and hypothesis class, omniprediction requires that a predictor provide a loss-minimization guarantee simultaneously for every loss in the collection compared to the best (loss-specific) hypothesis in the class. We present a generic template to learn predictors satisfying a guarantee we call Loss Outcome Indistinguishability. For a set of statistical tests--based on a collection of losses and hypothesis class--a predictor is Loss OI if it is indistinguishable (according to the tests) from Nature's true probabilities over outcomes. By design, Loss OI implies omniprediction in a direct and intuitive manner.

A recent line of work shows that notions of multigroup fairness imply surprisingly strong notions of omniprediction: loss minimization guarantees that apply not just for a specific loss function, but for any loss belonging to a large family of losses. While prior work has derived various notions of omniprediction from multigroup fairness guarantees of varying strength, it was unknown whether the connection goes in both directions. In this work, we answer this question in the affirmative, establishing equivalences between notions of multicalibration and omniprediction. The new definitions that hold the key to this equivalence are new notions of swap omniprediction, which are inspired by swap regret in online learning. We show that these can be characterized exactly by a strengthening of multicalibration that we refer to as swap multicalibration. One can go from standard to swap multicalibration by a simple discretization; moreover all known algorithms for standard multicalibration in fact give swap multicalibration. In the context of omniprediction though, introducing the notion of swapping results in provably stronger notions, which require a predictor to minimize expected loss at least as well as an adaptive adversary who can choose both the loss function and hypothesis based on the value predicted by the predictor. Building on these characterizations, we paint a complete picture of the relationship between the various omniprediction notions in the literature by establishing implications and separations between them. Our work deepens our understanding of the connections between multigroup fairness, loss minimization and outcome indistinguishability and establishes new connections to classic notions in online learning.

We present a new perspective on loss minimization and the recent notion of Omniprediction through the lens of Outcome Indistingusihability. For a collection of losses and hypothesis class, omniprediction requires that a predictor provide a loss-minimization guarantee simultaneously for every loss in the collection compared to the best (loss-specific) hypothesis in the class. We present a generic template to learn predictors satisfying a guarantee we call Loss Outcome Indistinguishability. For a set of statistical tests--based on a collection of losses and hypothesis class--a predictor is Loss OI if it is indistinguishable (according to the tests) from Nature's true probabilities over outcomes. By design, Loss OI implies omniprediction in a direct and intuitive manner. We simplify Loss OI further, decomposing it into a calibration condition plus multiaccuracy for a class of functions derived from the loss and hypothesis classes. By careful analysis of this class, we give efficient constructions of omnipredictors for interesting classes of loss functions, including non-convex losses. This decomposition highlights the utility of a new multi-group fairness notion that we call calibrated multiaccuracy, which lies in between multiaccuracy and multicalibration. We show that calibrated multiaccuracy implies Loss OI for the important set of convex losses arising from Generalized Linear Models, without requiring full multicalibration. For such losses, we show an equivalence between our computational notion of Loss OI and a geometric notion of indistinguishability, formulated as Pythagorean theorems in the associated Bregman divergence. We give an efficient algorithm for calibrated multiaccuracy with computational complexity comparable to that of multiaccuracy. In all, calibrated multiaccuracy offers an interesting tradeoff point between efficiency and generality in the omniprediction landscape.