We propose new learning algorithms for building selective classifiers, which are predictors that are allowed to abstain on some fraction of the domain. We study the model where a classifier may abstain from predicting at a fixed cost. Building on the recent framework on multigroup fairness and omniprediction, given a prespecified class of loss functions, we provide an algorithm for building a single classifier that learns abstentions and predictions optimally for every loss in the entire class, where the abstentions are decided efficiently for each specific loss function by applying a fixed post-processing function. Our algorithm and theoretical guarantees generalize the previously-known algorithms for learning selective classifiers in formal learning-theoretic models. We then extend the traditional multigroup fairness algorithms to the selective classification setting and show that we can use a calibrated and multiaccurate predictor to efficiently build selective classifiers that abstain optimally not only globally but also locally within each of the groups in any pre-specified collection of possibly intersecting subgroups of the domain, and are also accurate when they do not abstain. We show how our abstention algorithms can be used as conformal prediction methods in the binary classification setting to achieve both marginal and group-conditional coverage guarantees for an intersecting collection of groups. We provide empirical evaluations for all of our theoretical results, demonstrating the practicality of our learning algorithms for abstaining optimally and fairly.

We introduce and study Swap Agnostic Learning. The problem can be phrased as a game between a predictor and an adversary: first, the predictor selects a hypothesis h; then, the adversary plays in response, and for each level set of the predictor {x X: h(x) = v} selects a loss-minimizing hypothesis cv C; the predictor wins if p competes with the adaptive adversary's loss. Despite the strength of the adversary, our main result demonstrates the feasibility Swap Agnostic Learning for any convex loss. Somewhat surprisingly, the result follows by proving an equivalence between Swap Agnostic Learning and swap variants of the recent notions Omniprediction [15] and Multicalibration [20]. Beyond this equivalence, we establish further connections to the literature on Outcome Indistinguishability [6, 14], revealing a unified notion of OI that captures all existing notions of omniprediction and multicalibration.

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.

We give the first result for agnostically learning Single-Index Models (SIMs) with arbitrary monotone and Lipschitz activations. All prior work either held only in the realizable setting or required the activation to be known. Moreover, we only require the marginal to have bounded second moments, whereas all prior work required stronger distributional assumptions (such as anticoncentration or boundedness). Our algorithm is based on recent work by Gopalan et al. [2023] on Omniprediction using predictors satisfying calibrated multiaccuracy. Our analysis is simple and relies on the relationship between Bregman divergences (or matching losses) and $\ell_p$ distances. We also provide new guarantees for standard algorithms like GLMtron and logistic regression in the agnostic setting.

Omnipredictors are simple prediction functions that encode loss-minimizing predictions with respect to a hypothesis class $\mathcal{H}$, simultaneously for every loss function within a class of losses $\mathcal{L}$. In this work, we give near-optimal learning algorithms for omniprediction, in both the online and offline settings. To begin, we give an oracle-efficient online learning algorithm that acheives $(\mathcal{L},\mathcal{H})$-omniprediction with $\tilde{O}(\sqrt{T \log |\mathcal{H}|})$ regret for any class of Lipschitz loss functions $\mathcal{L} \subseteq \mathcal{L}_\mathrm{Lip}$. Quite surprisingly, this regret bound matches the optimal regret for \emph{minimization of a single loss function} (up to a $\sqrt{\log(T)}$ factor). Given this online algorithm, we develop an online-to-offline conversion that achieves near-optimal complexity across a number of measures. In particular, for all bounded loss functions within the class of Bounded Variation losses $\mathcal{L}_\mathrm{BV}$ (which include all convex, all Lipschitz, and all proper losses) and any (possibly-infinite) $\mathcal{H}$, we obtain an offline learning algorithm that, leveraging an (offline) ERM oracle and $m$ samples from $\mathcal{D}$, returns an efficient $(\mathcal{L}_{\mathrm{BV}},\mathcal{H},\varepsilon(m))$-omnipredictor for $\varepsilon(m)$ scaling near-linearly in the Rademacher complexity of $\mathrm{Th} \circ \mathcal{H}$.

Professional networks provide invaluable entree to opportunity through referrals and introductions. A rich literature shows they also serve to entrench and even exacerbate a status quo of privilege and disadvantage. Hiring platforms, equipped with the ability to nudge link formation, provide a tantalizing opening for beneficial structural change. We anticipate that key to this prospect will be the ability to estimate the likelihood of edge formation in an evolving graph. Outcome-indistinguishable prediction algorithms ensure that the modeled world is indistinguishable from the real world by a family of statistical tests. Omnipredictors ensure that predictions can be post-processed to yield loss minimization competitive with respect to a benchmark class of predictors for many losses simultaneously, with appropriate post-processing. We begin by observing that, by combining a slightly modified form of the online K29 star algorithm of Vovk (2007) with basic facts from the theory of reproducing kernel Hilbert spaces, one can derive simple and efficient online algorithms satisfying outcome indistinguishability and omniprediction, with guarantees that improve upon, or are complementary to, those currently known. This is of independent interest. We apply these techniques to evolving graphs, obtaining online outcome-indistinguishable omnipredictors for rich -- possibly infinite -- sets of distinguishers that capture properties of pairs of nodes, and their neighborhoods. This yields, inter alia, multicalibrated predictions of edge formation with respect to pairs of demographic groups, and the ability to simultaneously optimize loss as measured by a variety of social welfare functions.

Recent work on supervised learning [GKR+22] defined the notion of omnipredictors, i.e., predictor functions $p$ over features that are simultaneously competitive for minimizing a family of loss functions $\mathcal{L}$ against a comparator class $\mathcal{C}$. Omniprediction requires approximating the Bayes-optimal predictor beyond the loss minimization paradigm, and has generated significant interest in the learning theory community. However, even for basic settings such as agnostically learning single-index models (SIMs), existing omnipredictor constructions require impractically-large sample complexities and runtimes, and output complex, highly-improper hypotheses. Our main contribution is a new, simple construction of omnipredictors for SIMs. We give a learner outputting an omnipredictor that is $\varepsilon$-competitive on any matching loss induced by a monotone, Lipschitz link function, when the comparator class is bounded linear predictors. Our algorithm requires $\approx \varepsilon^{-4}$ samples and runs in nearly-linear time, and its sample complexity improves to $\approx \varepsilon^{-2}$ if link functions are bi-Lipschitz. This significantly improves upon the only prior known construction, due to [HJKRR18, GHK+23], which used $\gtrsim \varepsilon^{-10}$ samples. We achieve our construction via a new, sharp analysis of the classical Isotron algorithm [KS09, KKKS11] in the challenging agnostic learning setting, of potential independent interest. Previously, Isotron was known to properly learn SIMs in the realizable setting, as well as constant-factor competitive hypotheses under the squared loss [ZWDD24]. As they are based on Isotron, our omnipredictors are multi-index models with $\approx \varepsilon^{-2}$ prediction heads, bringing us closer to the tantalizing goal of proper omniprediction for general loss families and comparators.

Consider the supervised learning setting where the goal is to learn to predict labels $\mathbf y$ given points $\mathbf x$ from a distribution. An \textit{omnipredictor} for a class $\mathcal L$ of loss functions and a class $\mathcal C$ of hypotheses is a predictor whose predictions incur less expected loss than the best hypothesis in $\mathcal C$ for every loss in $\mathcal L$. Since the work of [GKR+21] that introduced the notion, there has been a large body of work in the setting of binary labels where $\mathbf y \in \{0, 1\}$, but much less is known about the regression setting where $\mathbf y \in [0,1]$ can be continuous. Our main conceptual contribution is the notion of \textit{sufficient statistics} for loss minimization over a family of loss functions: these are a set of statistics about a distribution such that knowing them allows one to take actions that minimize the expected loss for any loss in the family. The notion of sufficient statistics relates directly to the approximate rank of the family of loss functions. Our key technical contribution is a bound of $O(1/\varepsilon^{2/3})$ on the $\epsilon$-approximate rank of convex, Lipschitz functions on the interval $[0,1]$, which we show is tight up to a factor of $\mathrm{polylog} (1/\epsilon)$. This yields improved runtimes for learning omnipredictors for the class of all convex, Lipschitz loss functions under weak learnability assumptions about the class $\mathcal C$. We also give efficient omnipredictors when the loss families have low-degree polynomial approximations, or arise from generalized linear models (GLMs). This translation from sufficient statistics to faster omnipredictors is made possible by lifting the technique of loss outcome indistinguishability introduced by [GKH+23] for Boolean labels to the regression setting.

The notion of omnipredictors (Gopalan, Kalai, Reingold, Sharan and Wieder ITCS 2021), suggested a new paradigm for loss minimization. Rather than learning a predictor based on a known loss function, omnipredictors can easily be post-processed to minimize any one of a rich family of loss functions compared with the loss of hypotheses in a class $\mathcal C$. It has been shown that such omnipredictors exist and are implied (for all convex and Lipschitz loss functions) by the notion of multicalibration from the algorithmic fairness literature. In this paper, we introduce omnipredictors for constrained optimization and study their complexity and implications. The notion that we introduce allows the learner to be unaware of the loss function that will be later assigned as well as the constraints that will be later imposed, as long as the subpopulations that are used to define these constraints are known. We show how to obtain omnipredictors for constrained optimization problems, relying on appropriate variants of multicalibration. We also investigate the implications of this notion when the constraints used are so-called group fairness notions.