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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 solved by well-known game-theoretic criteria, such as Shapley value or Banzhaf value. 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 various game-theoretic valuation criteria through conducting one-step fixed point iteration 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 the players through the mean-field approach. By running fixed point iteration for multiple steps, we achieve a trajectory of the valuations, among which we define the valuation with the best conceivable decoupling error as the Variational Index. We prove that under uniform initializations, these variational valuations all satisfy a set of game-theoretic axioms. We experimentally demonstrate that the proposed Variational Index enjoys lower decoupling error and better valuation performance on certain synthetic and real-world valuation problems.