$\texttt{ShaplEIG}$: Bayesian Experimental Design for Shapley Value Estimation

Shapley values are a principled attribution measure widely used in interpretable machine learning, but their exact computation scales exponentially with the number of players, motivating a wide range of approximation methods based on value-function evaluations of sampled coalitions. This raises the question of whether approximation accuracy can be improved by *adaptively* selecting coalitions for evaluation based on previous outcomes. This is particularly relevant in settings where the value function is costly, and the number of evaluations is severely limited, such as retraining-based feature importance, data valuation, and hyperparameter importance. For this purpose, we propose $\texttt{ShaplEIG}$, a Bayesian experimental design approach that approximates the expensive value function via a Gaussian process surrogate and adaptively selects coalitions based on their expected information gain about the Shapley values. Since Shapley values are a linear function of the value function, we show that the expected information gain is available in *closed form* and *efficiently* computable. In extensive experiments across diverse costly applications, our method consistently improves estimation accuracy over state-of-the-art baselines.