We study the implicit regularization effects induced by (observation) weighting of pretrained features. For weight and feature matrices of bounded operator norms that are infinitesimally free with respect to (normalized) trace functionals, we derive equivalence paths connecting different weighting matrices and ridge regularization levels. Specifically, we show that ridge estimators trained on weighted features along the same path are asymptotically equivalent when evaluated against test vectors of bounded norms. These paths can be interpreted as matching the effective degrees of freedom of ridge estimators fitted with weighted features. For the special case of subsampling without replacement, our results apply to independently sampled random features and kernel features and confirm recent conjectures (Conjectures 7 and 8) of the authors on the existence of such paths in Patil et al. We also present an additive risk decomposition for ensembles of weighted estimators and show that the risks are equivalent along the paths when the ensemble size goes to infinity. As a practical consequence of the path equivalences, we develop an efficient cross-validation method for tuning and apply it to subsampled pretrained representations across several models (e.g., ResNet-50) and datasets (e.g., CIFAR-100).

本文研究了预训练特征加权所引起的隐式正则化效应。通过导出不同加权矩阵与岭正则化水平之间的等效路径，指出在相同路径下训练的岭估计量在评估边界范数的测试向量时渐近相等。研究结果为高效的交叉验证方法提供了理论基础，并在多个模型和数据集上进行了应用。

加权神经表示的隐式正则化路径