Multi-calibration is a powerful and evolving concept originating in the field of algorithmic fairness. For a predictor $f$ that estimates the outcome $y$ given covariates $x$, and for a function class $\mathcal{C}$, multi-calibration requires that the predictor $f(x)$ and outcome $y$ are indistinguishable under the class of auditors in $\mathcal{C}$. Fairness is captured by incorporating demographic subgroups into the class of functions~$\mathcal{C}$. Recent work has shown that, by enriching the class $\mathcal{C}$ to incorporate appropriate propensity re-weighting functions, multi-calibration also yields target-independent learning, wherein a model trained on a source domain performs well on unseen, future, target domains(approximately) captured by the re-weightings. Formally, multi-calibration with respect to $\mathcal{C}$ bounds $\big|\mathbb{E}_{(x,y)\sim \mathcal{D}}[c(f(x),x)\cdot(f(x)-y)]\big|$ for all $c \in \mathcal{C}$. In this work, we view the term $(f(x)-y)$ as just one specific mapping, and explore the power of an enriched class of mappings. We propose \textit{HappyMap}, a generalization of multi-calibration, which yields a wide range of new applications, including a new fairness notion for uncertainty quantification (conformal prediction), a novel technique for conformal prediction under covariate shift, and a different approach to analyzing missing data, while also yielding a unified understanding of several existing seemingly disparate algorithmic fairness notions and target-independent learning approaches. We give a single \textit{HappyMap} meta-algorithm that captures all these results, together with a sufficiency condition for its success.

翻译：多校准是源于算法公平性领域的一个强大且不断发展的概念。对于给定协变量$x$估计结果$y$的预测器$f$，以及函数类$\mathcal{C}$，多校准要求预测器$f(x)$与结果$y$在$\mathcal{C}$中的审计函数类下不可区分。通过将人口统计子组纳入函数类$\mathcal{C}$，即可刻画公平性。近期研究表明，通过丰富函数类$\mathcal{C}$以纳入适当的倾向性重加权函数，多校准还能实现目标无关学习——即源域训练的模型能够在由重加权函数（近似）捕获的未知未来目标域上表现良好。形式上，对$\mathcal{C}$的多校准约束了对于所有$c \in \mathcal{C}$，有$\big|\mathbb{E}_{(x,y)\sim \mathcal{D}}[c(f(x),x)\cdot(f(x)-y)]\big|$的有界性。本研究将$(f(x)-y)$视为一种特定映射，并探索了更丰富映射类别的能力。我们提出HappyMap——多校准的广义化方法，由此催生了一系列新应用：包括面向不确定性量化（共形预测）的新型公平性概念、协变量偏移下共形预测的创新技术、以及缺失数据分析的新途径，同时统一解释了若干看似互异的现有算法公平性概念与目标无关学习方法。我们给出了一个统一的HappyMap元算法来囊括所有上述成果，并给出了确保其成功的充分条件。