基于采样的Shapley值逼近估计误差的界
Bounding the Estimation Error of Sampling-based Shapley Value Approximation
摘要：
Shapley值可以说是合作博弈论中最核心的规范解概念。它规定了一种独特的方式，在这种方式下，合作的奖励可以在玩家之间“公平”分配。虽然它在现实世界中有广泛的应用，但在许多情况下，它的使用受到其计算难度的阻碍。许多研究人员通过以下方式解决了这个问题：（i）专注于可以有效计算Shapley值的游戏类别，或（ii）提出促进这种有效计算的表示形式，或（iii）在某些游戏类别中近似Shapley值。对于经典的\textit{characteristic function}表示，唯一试图近似Shapley值的一般类的游戏是由于卡斯特罗\textit{et al.} \cite{castro}。虽然该算法提供了近似误差的界限，但该界限是\textit{asymptotic}，这意味着它仅在样本数增加到无穷大时才成立。另一方面，当抽取有限数量的样本时，引入了不可量化的误差，这意味着界限不再成立。考虑到这一点，我们提供了两种情况下的估计误差的非渐近界：其中（i）的\textit{variance}，和（ii）的\textit{range}，玩家的边际贡献是已知的。此外，对于第二种情况下，我们表明，当范围是显着较大的Shapley值，可以提高边界（从$O(\frac{r}{m})$到$O(\sqrt{\frac{r}{m}})$）。最后，我们提出，并证明了使用分层抽样的有效性，进一步提高界限。
英文摘要：
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 (i) focusing on classes of games where the Shapley value can be computed efficiently, or (ii) proposing representation formalisms that facilitate such efficient computation, or (iii) approximating the Shapley value in certain classes of games. For the classical \textit{characteristic function} representation, the only attempt to approximate the Shapley value for the general class of games is due to Castro \textit{et al.} \cite{castro}. While this algorithm provides a bound on the approximation error, this bound is \textit{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 (i) the \textit{variance}, and (ii) the \textit{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 $O(\frac{r}{m})$ to $O(\sqrt{\frac{r}{m}})$). Finally, we propose, and demonstrate the effectiveness of using stratified sampling for improving the bounds further.
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