[论文解读] Bounding the Estimation Error of Sampling-based Shapley Value Approximation With/Without Stratifying.

Sasan Maleki, The Anh Han+3 more|arXiv (Cornell University)|Jun 18, 2013

Game Theory and Voting Systems被引用 24

一句话总结

本文在已知边际贡献的方差或取值范围的前提下，为基于抽样的Shapley值近似方法提供了非渐近误差界，涵盖无分层抽样与分层抽样两种情形。当取值范围相对于Shapley值较大时，误差界从O(r/√m)改进为O(√r/√m)，为实际应用提供了更紧致的有限样本保证。

ABSTRACT

The Shapley value is arguably the most central normative solution concept in cooperative game theory. It specifies a unique way in which the reward from cooperation can be fairly divided among players. While it has a wide range of real world applications, its use is in many cases hampered by the hardness of its computation. A number of researchers have tackled this problem by (1) focusing on classes of games where the Shapley value can be computed efficiently, or (2) proposing representation formalisms that facilitate such efficient computation, or (3) approximating the Shapley value in certain classes of games. However, given the classical extit{characteristic function} representation, the only attempt to approximate the Shapley value for the general class of games is due to Castro extit{et al.} \cite{castro}. While this algorithm provides a bound on the approximation error, this bound is extit{asymptotic}, meaning that it only holds when the number of samples increases to infinity. On the other hand, when a finite number of samples is drawn, an unquantifiable error is introduced, meaning that the bound no longer holds. With this in mind, we provide non-asymptotic bounds on the estimation error for two cases: where (1) the extit{variance}, and (2) the extit{range}, of the players' marginal contributions is known. Furthermore, for the second case, we show that when the range is significantly large relative to the Shapley value, the bound can be improved (from $O(r,\sqrt{ icefrac{1}{m}})$ to $O(\sqrt{r},\sqrt{ icefrac{1}{m}})$). Finally, we propose, and demonstrate the effectiveness of, using stratified sampling to improve the bounds.

研究动机与目标

• 解决现有基于抽样的Shapley值近似方法中缺乏有限样本误差界的问题。
• 在已知边际贡献方差或取值范围的前提下，为一般合作博弈提供非渐近估计误差界。
• 当边际贡献的取值范围显著大于Shapley值时，改进误差界。
• 研究分层抽样在收紧Shapley值近似估计误差界方面的有效性。

提出的方法

• 基于有限次样本数，推导在已知边际贡献方差条件下的Shapley值估计误差的非渐近上界。
• 提出基于边际贡献取值范围的替代界，当取值范围相对于Shapley值较大时更为紧致。
• 引入一种改进的界，使在高取值范围条件下收敛速率从O(r/√m)提升至O(√r/√m)。
• 提出并分析一种分层抽样策略，以降低方差并提高误差界的紧致性。
• 使用集中不等式形式化误差界，明确体现其对样本数m以及边际贡献方差或取值范围的依赖关系。
• 从理论上比较均匀抽样与分层抽样在误差界紧致性方面的性能差异。

实验结果

研究问题

• RQ1 当仅已知边际贡献方差时，能否为基于抽样的Shapley值近似方法推导出非渐近误差界？
• RQ2 边际贡献的取值范围如何影响估计误差界的紧致性？
• RQ3 当取值范围显著大于Shapley值时，能否改进误差界？
• RQ4 与均匀抽样相比，分层抽样在Shapley值近似中在多大程度上能减少估计误差？

主要发现

• 本文在已知边际贡献方差的前提下，建立了基于抽样的Shapley值近似方法的非渐近误差界。
• 当边际贡献的取值范围显著大于Shapley值时，误差界从O(r/√m)改进为O(√r/√m)。
• 分层抽样被证明能有效减少估计误差，相比均匀抽样可获得更紧致的误差界。
• 所推导的误差界对任意有限样本数均成立，而不同于以往依赖渐近近似的研究所采用的方法。
• 理论分析证实，误差界同时依赖于样本数和边际贡献的分布范围。
• 在高取值范围条件下误差界紧致性的提升，为聚焦于边际贡献值极端的游戏提供了理论依据。