On Feature Interactions Identified by Shapley Values of Binary Classification Games

For feature selection and related problems, we introduce the notion of classification game, a cooperative game, with features as players and hinge loss based characteristic function and relate a feature's contribution to Shapley value based error apportioning (SVEA) of total training error. Our major contribution is ($\star$) to show that for any dataset the threshold 0 on SVEA value identifies feature subset whose joint interactions for label prediction is significant or those features that span a subspace where the data is predominantly lying. In addition, our scheme ($\star$) identifies the features on which Bayes classifier doesn't depend but any surrogate loss function based finite sample classifier does; this contributes to the excess $0$-$1$ risk of such a classifier, ($\star$) estimates unknown true hinge risk of a feature, and ($\star$) relate the stability property of an allocation and negative valued SVEA by designing the analogue of core of classification game. Due to Shapley value's computationally expensive nature, we build on a known Monte Carlo based approximation algorithm that computes characteristic function (Linear Programs) only when needed. We address the potential sample bias problem in feature selection by providing interval estimates for SVEA values obtained from multiple sub-samples. We illustrate all the above aspects on various synthetic and real datasets and show that our scheme achieves better results than existing recursive feature elimination technique and ReliefF in most cases. Our theoretically grounded classification game in terms of well defined characteristic function offers interpretability (which we formalize in terms of final task) and explainability of our framework, including identification of important features.