We prove tight lower bounds for online multicalibration, establishing an information-theoretic separation from marginal calibration. In the general setting where group functions can depend on both context and the learner's predictions, we prove an $Ω(T^{2/3})$ lower bound on expected multicalibration error using just three disjoint binary groups. This matches the upper bounds of Noarov et al. (2025) up to logarithmic factors and exceeds the $O(T^{2/3-\varepsilon})$ upper bound for marginal calibration (Dagan et al., 2025), thereby separating the two problems. We then turn to lower bounds for the more difficult case of group functions that may depend on context but not on the learner's predictions. In this case, we establish an $\widetildeΩ(T^{2/3})$ lower bound for online multicalibration via a $Θ(T)$-sized group family constructed using orthogonal function systems, again matching upper bounds up to logarithmic factors.

翻译：我们证明了在线多标定的紧下界，从而建立了其与边缘标定在信息论意义上的分离。在群函数可同时依赖于上下文和学习器预测的一般设定下，我们仅使用三个互斥二元群证明了期望多标定误差的$Ω(T^{2/3})$下界。该结果与Noarov等人（2025）的上界在对数因子内匹配，且超越了边缘标定的$O(T^{2/3-\varepsilon})$上界（Dagan等人，2025），从而分离了这两个问题。随后我们转向研究更困难情形——群函数可依赖于上下文但不能依赖于学习器预测——的下界。在此情形下，我们通过利用正交函数系统构造的$Θ(T)$规模群族，建立了在线多标定的$\widetildeΩ(T^{2/3})$下界，该结果同样在对数因子内匹配现有上界。