Multicalibration Sample Complexity Analyzed for Multilevel Properties.

Calibration in predictive models ensures unbiased predictions when conditioned on the predictions themselves. Multicalibration extends this by requiring this guarantee across multiple predefined groups, which is crucial for fairness and accuracy across diverse populations. This research delves into a more complex scenario: multicalibration for a sequence of $k$ properties, where each property's definition relies on the preceding ones, such as variance depending on the mean. The study establishes precise theoretical bounds for the sample complexity required to achieve multicalibration. It demonstrates that even with a relatively small number of binary groups, achieving a multicalibration error of $\varepsilon$ necessitates a sample size proportional to $\varepsilon^{-(k+2)}$. Conversely, the paper presents a randomized learning algorithm that can achieve this with a sample complexity of $O(\varepsilon^{-(k+2)}+\varepsilon^{-2}\log|\mathcal G|)$ for any finite group family. This means for polynomial-size group families, the sample complexity is tightly bounded by $\widetilde{\Theta}(\varepsilon^{-(k+2)})$, providing a clear understanding of the data requirements for robust and fair multi-property predictions.