Singularity-Induced Local Complexity for High-Dimensional Learning Models

Many overparameterized models that have achieved remarkable success are singular models. That is, the set of optimal parameters consists of multiple points, or the Hessian of the generalization error at such parameters is degenerate, reflecting redundancy in the model parametrization. Therefore, understanding the effective dimension and intrinsic model complexity, beyond the apparent dimension of the parameter space, is essential for studying the generalization ability of such overparameterized models. In this work, we focus on the real log canonical threshold (RLCT) and its localized counterpart, the local RLCT, which are fundamental quantities for characterizing the generalization of singular models. We clarify the theoretical relationship between the local RLCT and existing two key metrics related to generalization under some localization assumption: (I) the rank of the Hessian matrix of the generalization error and (II) the covering number of a family of loss functions. This result provides novel insights into effective dimension and intrinsic complexity of overparameterized models.