发表机构
Sorbonne Université; Inria（索邦大学; 法国国家信息与自动化研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对二维奇异模型，提出首个可精确计算局部实对数典范阈值（RLCT）的确定性算法，为校准采样估计器提供真值，在浅层区域比采样方法更快。
AI 中文摘要
贝叶斯信息准则（BIC）等经典信息准则依赖于正则性假设，该假设在奇异模型中不成立，会导致深度学习等场景下的模型选择错误。广泛适用的贝叶斯信息准则（WBIC）依赖于局部学习系数λ，在解析情况下，该系数与模型的Kullback-Leibler散度的局部实对数典范阈值（RLCT）一致，用于捕捉正确的边际似然渐近行为。学习系数的精确计算目前仅局限于特殊情况，通常仅适用基于采样的估计方法。本文提出首个确定性算法，可针对任何Kullback-Leibler距离与多项式接触等价的二维模型精确计算局部RLCT，推导了其复杂度边界，并在广泛类别的模型上验证了其有效性，应用包括多项式神经网络。除了为校准基于采样的估计器提供真值外，精确计算还揭示了采样无法发现的学习系数的代数结构，且在浅层区域比采样方法速度更快。
英文摘要
Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $λ$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.
