Energy-Based Learning for Cooperative Games, with Applications to Valuation Problems in Machine Learning

Valuation problems, such as feature interpretation, data valuation and model valuation for ensembles, become increasingly more important in many machine learning applications. Such problems are commonly addressed via well-known gametheoretic criteria, such as the Shapley value or Banzhaf index. In this work, we present a novel energy-based treatment for cooperative games, with a theoretical justification by the maximum entropy framework. Surprisingly, by conducting variational inference of the energy-based model, we recover classical game-theoretic valuation criteria through conducting one-step gradient ascent for maximizing the mean-field ELBO objective. This observation also verifies the rationality of existing criteria, as they are all attempting to decouple the correlations among players through the mean-field approach. By running gradient ascent for multiple steps, we achieve a trajectory of the variational valuations, among which we define the valuation with the best conceivable decoupling error as the Variational Index. We empirically demonstrate that the proposed Variational Index enjoys intriguing properties on certain synthetic and real-world valuation problems.