We consider the problem of a revenue-maximizing seller with a large number of items $m$ for sale to $n$ strategic bidders, whose valuations are drawn independently from high-dimensional, unknown prior distributions. It is well-known that optimal and even approximately-optimal mechanisms for this setting are notoriously difficult to characterize or compute, and, even when they can be found, are often rife with various counter-intuitive properties. In this paper, following a model introduced recently by Cai and Daskalakis~\cite{cai2022recommender}, we consider the case that bidders' prior distributions can be well-approximated by a topic model. We design an active learning component, responsible for interacting with the bidders and outputting low-dimensional approximations of their types, and a mechanism design component, responsible for robustifying mechanisms for the low-dimensional model to work for the approximate types of the former component. On the active learning front, we cast our problem in the framework of Randomized Linear Algebra (RLA) for regression problems, allowing us to import several breakthrough results from that line of research, and adapt them to our setting. On the mechanism design front, we remove many restrictive assumptions of prior work on the type of access needed to the underlying distributions and the associated mechanisms. To the best of our knowledge, our work is the first to formulate connections between mechanism design, and RLA for active learning of regression problems, opening the door for further applications of randomized linear algebra primitives to mechanism design.

翻译：我们考虑一个收益最大化卖家的问题，该卖家拥有大量商品$m$，面向$n$个策略性买家，其估值独立地来自高维、未知的先验分布。众所周知，该情境下的最优甚至近似最优机制往往难以刻画或计算，且即便能够找到，也常充斥着各种反直觉的性质。本文沿用Cai和Daskalakis近期提出的模型~\cite{cai2022recommender}，考虑买家先验分布可被主题模型良好近似的情形。我们设计了一个主动学习组件，负责与买家交互并输出其类型的低维近似；同时设计了一个机制设计组件，负责将低维模型的机制鲁棒化，使其适用于前者提供的近似类型。在主动学习方面，我们将问题纳入回归问题的随机线性代数（RLA）框架，从而得以引入该研究方向的若干突破性成果，并使其适应我们的场景。在机制设计方面，我们消除了先前工作关于底层分布访问方式及其关联机制的诸多限制性假设。据我们所知，本文首次提出了机制设计与回归问题主动学习中RLA之间的联系，为随机线性代数原语在机制设计中的进一步应用打开了大门。