Exact Learning Coefficients for Singular Models

Researchers have achieved a significant breakthrough by developing the first deterministic algorithm capable of exactly computing local learning coefficients, also known as Real Log Canonical Thresholds (RLCTs), for two-dimensional singular models. Traditional information criteria like BIC often fail for singular models, which are common in deep learning, leading to incorrect model selection. The Widely Applicable Bayesian Information Criterion (WBIC) relies on these local learning coefficients for accurate marginal likelihood asymptotics. Previously, exact computation was limited to special cases, with general applicability relying on sampling-based estimation. This new algorithm provides ground truth for calibrating these sampling methods and uncovers algebraic structures in learning coefficients that sampling cannot reveal. The method is effective for a broad class of models, including polynomial neural networks, and demonstrates superior speed in shallow model regimes, offering a more precise understanding of model complexity.