Improved Bounds for Swap Multicalibration and Swap Omniprediction

TL;DR

This paper introduces an efficient algorithm that significantly improves bounds for swap multicalibration and omniprediction errors, advancing fairness notions and loss minimization in online and distributional settings.

Contribution

It provides the first efficient algorithm achieving $O(T^{1/3})$ $ ext{ell}_2$-swap multicalibration error and improves bounds for $ ext{ell}_1$-swap multicalibration and omniprediction, with better sample complexities.

Findings

Achieves $O(T^{1/3})$ $ ext{ell}_2$-swap multicalibration error.

Improves $ ext{ell}_1$-swap multicalibration and omniprediction bounds to $O(T^{2/3})$.

Establishes new sample complexity bounds for learning swap omnipredictors.

Abstract

In this paper, we consider the related problems of multicalibration -- a multigroup fairness notion and omniprediction -- a simultaneous loss minimization paradigm, both in the distributional and online settings. The recent work of Garg et al. (2024) raised the open problem of whether it is possible to efficiently achieve $O(\sqrt{T})$ $\ell_{2}$-multicalibration error against bounded linear functions. In this paper, we answer this question in a strongly affirmative sense. We propose an efficient algorithm that achieves $O(T^{\frac{1}{3}})$ $\ell_{2}$-swap multicalibration error (both in high probability and expectation). On propagating this bound onward, we obtain significantly improved rates for $\ell_{1}$-swap multicalibration and swap omniprediction for a loss class of convex Lipschitz functions. In particular, we show that our algorithm achieves $O(T^{\frac{2}{3}})$ $\ell_{1}$-swap multicalibration and swap omniprediction errors, thereby improving upon the previous best-known bound of $O(T^{\frac{7}{8}})$. As a consequence of our improved online results, we further obtain several improved sample complexity rates in the distributional setting. In particular, we establish a $O(\varepsilon ^ {-3})$ sample complexity of efficiently learning an $\varepsilon$-swap omnipredictor for the class of convex and Lipschitz functions, $O(\varepsilon ^{-2.5})$ sample complexity of efficiently learning an $\varepsilon$-swap agnostic learner for the squared loss, and $O(\varepsilon ^ {-5}), O(\varepsilon ^ {-2.5})$ sample complexities of learning $\ell_{1}, \ell_{2}$-swap multicalibrated predictors against linear functions, all of which significantly improve on the previous best-known bounds.