We study multi-item profit maximization when there is an underlying distribution over buyers' values. In practice, a full description of the distribution is typically unavailable, so we study the setting where the mechanism designer only has samples from the distribution. If the designer uses the samples to optimize over a complex mechanism class -- such as the set of all multi-item, multi-buyer mechanisms -- a mechanism may have high average profit over the samples but low expected profit. This raises the central question of this paper: how many samples are sufficient to ensure that a mechanism's average profit is close to its expected profit? To answer this question, we uncover structure shared by many pricing, auction, and lottery mechanisms: for any set of buyers' values, profit is piecewise linear in the mechanism's parameters. Using this structure, we prove new bounds for mechanism classes not yet studied in the sample-based mechanism design literature and match or improve over the best-known guarantees for many classes.

翻译：我们研究当买家估值存在潜在分布时的多商品利润最大化问题。在实践中，该分布的完整描述通常不可得，因此我们研究机制设计者仅能从该分布中获得样本的情形。若设计者利用样本对复杂机制类（例如所有多商品、多买家机制的集合）进行优化，则某个机制在样本上的平均利润可能很高，但其期望利润却可能很低。这引出了本文的核心问题：需要多少样本才能确保机制的平均利润接近其期望利润？为回答此问题，我们揭示了众多定价、拍卖和抽奖机制所共有的结构：对于任意买家的估值集，利润关于机制参数是分段线性函数。利用这一结构，我们为样本机制设计文献中尚未研究的机制类证明了新边界，并在许多机制类上匹配或改进了已知最佳保证。