Mathematically rigorous proofs for Shapley explanations
This work offers foundational mathematical validation for Shapley-based explanations, addressing a gap in the original literature for researchers and practitioners in interpretable AI.
The paper provides mathematically rigorous proofs for two key results on Shapley values in machine learning, showing that symmetry is essential for uniqueness and that Shapley values can be expressed as a unique solution to a weighted linear regression problem.
Machine Learning is becoming increasingly more important in today's world. It is therefore very important to provide understanding of the decision-making process of machine-learning models. A popular way to do this is by looking at the Shapley-Values of these models as introduced by Lundberg and Lee. In this thesis, we discuss the two main results by Lundberg and Lee from a mathematically rigorous standpoint and provide full proofs, which are not available from the original material. The first result of this thesis is an axiomatic characterization of the Shapley values in machine learning based on axioms by Young. We show that the Shapley values are the unique explanation to satisfy local accuracy, missingness, symmetry and consistency. Lundberg and Lee claim that the symmetry axiom is redundant for explanations. However, we provide a counterexample that shows the symmetry axiom is in fact essential. The second result shows that we can write the Shapley values as the unique solution to a weighted linear regression problem. This result is proven with the use of dimensionality reduction.