Real-world pricing mechanisms are typically optimized using training data, a setting corresponding to the \textit{pricing query complexity} problem in Mechanism Design. The previous work [LSTW23] studies the \textit{single-distribution} case, with tight bounds of $\widetildeΘ(\varepsilon^{-3})$ for a \textit{general} distribution and $\widetildeΘ(\varepsilon^{-2})$ for either a \textit{regular} or \textit{monotone-hazard-rate (MHR)} distribution, where $\varepsilon \in (0, 1)$ denotes the (additive) revenue loss of a learned uniform price relative to the Bayesian-optimal uniform price. This can be directly interpreted as ``the query complexity of the {\em \textsf{Uniform Pricing}} mechanism, in the \textit{single-distribution} case''. Yet in the \textit{multi-distribution} case, can the regularity and MHR conditions still lead to improvements over the tight bound $\widetildeΘ(\varepsilon^{-3})$ for general distributions? We answer this question in the negative, by establishing a (near-)matching lower bound $Ω(\varepsilon^{-3})$ for either \textit{two regular distributions} or \textit{three MHR distributions}. We also address the \textit{regret minimization} problem and, in comparison with the folklore upper bound $\widetilde{O}(T^{2 / 3})$ for general distributions (see, e.g., [SW24]), establish a (near-)matching lower bound $Ω(T^{2 / 3})$ for either \textit{two regular distributions} or \textit{three MHR distributions}, via a black-box reduction. Again, this is in stark contrast to the tight bound $\widetildeΘ(T^{1 / 2})$ for a single regular or MHR distribution.

翻译：现实世界中的定价机制通常使用训练数据进行优化，这一设定对应于机制设计中的\textit{定价查询复杂度}问题。先前的研究[LSTW23]探讨了\textit{单一分布}情形，对于\textit{一般}分布获得了$\widetildeΘ(\varepsilon^{-3})$的紧界，对于\textit{正则}分布或\textit{单调风险率（MHR）}分布则获得了$\widetildeΘ(\varepsilon^{-2})$的紧界，其中$\varepsilon \in (0, 1)$表示学习到的统一价格相对于贝叶斯最优统一价格的（加性）收益损失。这可以直接解释为“{\em \textsf{统一定价}}机制在\textit{单一分布}情形下的查询复杂度”。然而，在\textit{多分布}情形下，正则性和MHR条件是否仍能带来相对于一般分布紧界$\widetildeΘ(\varepsilon^{-3})$的改进？我们通过为\textit{两个正则分布}或\textit{三个MHR分布}建立（近似）匹配的下界$Ω(\varepsilon^{-3})$，对此问题给出了否定回答。我们还探讨了\textit{遗憾最小化}问题，与一般分布的经典上界$\widetilde{O}(T^{2 / 3})$（参见例如[SW24]）相比，通过黑盒归约，为\textit{两个正则分布}或\textit{三个MHR分布}建立了（近似）匹配的下界$Ω(T^{2 / 3})$。这再次与单一正则或MHR分布的紧界$\widetildeΘ(T^{1 / 2})$形成鲜明对比。