An Odd Estimator for Shapley Values

Researchers have proven that Shapley values, a key framework for attribution in machine learning, depend exclusively on the odd component of set functions. This theoretical breakthrough justifies the effectiveness of paired sampling and enables OddSHAP, a new estimator that achieves state-of-the-art accuracy by performing regression solely on the odd subspace using Fourier basis decomposition.

This research addresses a fundamental challenge in explainable AI: computing Shapley values efficiently and accurately. Shapley values have become central to machine learning applications ranging from feature importance to data valuation and causal inference, yet their exact computation remains intractable for realistic problems. The paper's key contribution is proving that only the odd component of set functions matters for Shapley value calculation, providing theoretical justification for why paired sampling—the most effective practical heuristic—actually works.

The finding builds on decades of work in cooperative game theory and machine learning interpretability. Shapley values gained prominence as researchers sought principled methods to attribute model predictions to individual features or data points. While practitioners knew paired sampling improved results, the mathematical mechanism remained unclear until now. This theoretical gap has limited innovation in estimation methods.

The practical impact emerges through OddSHAP, which leverages Fourier basis decomposition to isolate the odd subspace and uses proxy models to identify high-impact interactions. This approach sidesteps the combinatorial explosion that plagues traditional higher-order polynomial approximations, enabling faster computation at larger sampling budgets. For industries relying on model interpretability—finance, healthcare, and AI safety—more efficient and accurate attribution methods directly improve risk assessment and regulatory compliance.

The research signals growing maturity in explainable AI infrastructure. As organizations face increasing pressure to justify algorithmic decisions, efficient Shapley value computation becomes competitive advantage. Future work will likely focus on implementing OddSHAP in production systems and extending these insights to dynamic or streaming contexts where feature importance evolves over time.

• →Shapley values depend exclusively on odd components of set functions, providing theoretical justification for paired sampling's empirical success.
• →OddSHAP uses Fourier basis decomposition to isolate the odd subspace, avoiding combinatorial explosion in higher-order approximations.
• →State-of-the-art estimation accuracy at larger sampling budgets enables practical deployment in computationally constrained environments.
• →The breakthrough strengthens explainable AI infrastructure, directly benefiting regulated industries requiring model interpretability.
• →Theoretical insights may accelerate innovation in attribution methods across machine learning and causal inference applications.