仅用少量样本即可实现高效在线拍卖,突破传统分布已知假设。
Online Combinatorial Allocations and Auctions with Few Samples
- 用单一样本设计自比竞争算法,适用于子模/可加估值。
- 多项式样本下可达(2+ε)竞争力的在线公平机制。
- 适合资源受限但需高效率拍卖的应用场景。
在在线组合分配/拍卖中,n 个投标者依次到来,每个对 m 个不可分物品的子集具有组合估值(如子模或 XOS)。目标是立即分配剩余物品子集以最大化总福利(即投标者估值之和)。以往工作假设投标者估值来自已知独立分布,对子模/XOS估值已知存在 2-竞争力算法:为每件物品设定固定价格,投标者根据价格选择最优剩余子集。然而这些方法依赖分布输入,而实际中常需从有限样本中学习分布。本文研究在仅有少量样本的情况下能否实现 O(1)-竞争力算法。主要贡献:仅需每个投标者一个样本,即可获得子模/XOS估值下的 O(1)-竞争力算法;该方法基于秘书问题的扩展分析,使算法与自身竞争。其次,只需多项式数量样本,即可构造出 (2+ε)-竞争力且在线公平的机制,适用于任意常数 ε>0。该结果基于将单物品预言家不等式中的中位数算法推广至多物品组合情形。
原文摘要 · Abstract (English)
In online combinatorial allocations/auctions, n bidders sequentially arrive, each with a combinatorial valuation (such as submodular/XOS) over subsets of m indivisible items. The aim is to immediately allocate a subset of the remaining items to maximize the total welfare, defined as the sum of bidder valuations. A long line of work has studied this problem when the bidder valuations come from known independent distributions. In particular, for submodular/XOS valuations, we know 2-competitive algorithms/mechanisms that set a fixed price for each item and the arriving bidders take their favorite subset of the remaining items given these prices. However, these algorithms traditionally presume the availability of the underlying distributions as part of the input to the algorithm. Contrary to this assumption, practical scenarios often require the learning of distributions, a task complicated by limited sample availability. This paper investigates the feasibility of achieving O(1)-competitive algorithms under the realistic constraint of having access to only a limited number of samples from the underlying bidder distributions. Our first main contribution shows that a mere single sample from each bidder distribution is sufficient to yield an O(1)-competitive algorithm for submodular/XOS valuations. This result leverages a novel extension of the secretary-style analysis, employing the sample to have the algorithm compete against itself. Although online, this first approach does not provide an online truthful mechanism. Our second main contribution shows that a polynomial number of samples suffices to yield a $(2+ε)$-competitive online truthful mechanism for submodular/XOS valuations and any constant $ε>0$. This result is based on a generalization of the median-based algorithm for the single-item prophet inequality problem to combinatorial settings with multiple items.
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