We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of additive bidders whose values for heterogeneous items are drawn independently. For any such instance and any , we show that it is possible to learn an -Bayesian Incentive Compatible auction whose expected revenue is within of the optimal -BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive , and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well understood, our corollary for this case extends slightly the state of the art.

中文翻译：

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Up-to-ε多维收益最大化的样本复杂度

我们考虑了在不受限制的多维设置中多个投标人的收入最大化的样本复杂性。具体来说，我们研究的标准模型 附加投标人的价值为 异构项目是独立绘制的。对于任何此类情况和任何 ，我们证明有可能学习一个 -贝叶斯激励兼容拍卖，其预期收入在 最优的 -BIC 拍卖仅来自多项式多个样本。我们的完全非参数方法基于相当普遍且完全回避表征这些设置的最优（或接近最优）拍卖的困难的想法。因此，我们的结果很容易扩展到一般的多维设置，包括不一定均匀的估值次加法, 和任意分配约束。对于单个投标者和许多商品，或单个参数（好）和许多投标者的情况，我们的分析产生了精确的激励相容性（对于后者还有计算效率）。尽管单参数情况已经很好理解，但我们对这种情况的推论略微扩展了现有技术。