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 an $O(\log^3 T)$-sized group family constructed from an orthonormal basis, again matching upper bounds up to logarithmic factors.

翻译：我们证明了在线多标定（online multicalibration）问题的紧下界，确立了其与边际标定（marginal calibration）在信息论上的分离。在群函数可同时依赖于上下文和学习者预测的一般设置中，我们仅使用三个不相交二元群证明了期望多标定误差的$Ω(T^{2/3})$下界。该下界在忽略对数因子的前提下匹配Noarov等人（2025）的上界，且超出边际标定$O(T^{2/3-\varepsilon})$的上界（Dagan等人，2025），从而实现了两类问题的分离。随后，我们转向更困难情形（群函数可依赖于上下文但不可依赖于学习者预测）的下界研究。在此情形下，我们通过基于标准正交基构造的规模为$O(\log^3 T)$的群族，建立了在线多标定的$\widetildeΩ(T^{2/3})$下界，同样在忽略对数因子的意义上匹配上界。