Implicit Regularization Paths of Weighted Neural Representations
This work addresses the challenge of tuning weighted neural representations for practitioners, but it is incremental as it builds on existing theories and confirms prior conjectures.
The paper tackles the problem of understanding implicit regularization effects from weighting pretrained features, showing that ridge estimators with weighted features along derived equivalence paths are asymptotically equivalent for test vectors of bounded norms, and applies this to develop an efficient cross-validation method for tuning on models like ResNet-50 and datasets like CIFAR-100.
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).