Wendao Wu

Wendao Wu, Fangqing Zhang, Haihan Zhang et al.

Teacher-Student Knowledge Transfer (KT) is ubiquitous in modern machine learning, ranging from classical model compression via Knowledge Distillation (KD) to the emergent phenomenon of Weak-to-Strong (W2S) generalization. While existing studies offer isolated insights, a unified theoretical framework explaining the efficacy of KT across these disparate regimes remains lacking. In this work, we establish a unified spectral analysis of SGD dynamics in high-dimensional linear regression, elucidating the efficiency of KT across seemingly disparate regimes. We characterize KT efficiency through two distinct mechanisms: \emph{Spectral Horizon Expansion} in KD, which enables the capture of statistically inaccessible high-frequency signals, and \emph{Spectral Denoising} in W2S, where the student acts as a filter for optimization noise. Our framework unifies these phenomena, revealing that the efficacy of transfer is governed by the interplay between implicit regularization and heterogeneous spectral learning speeds over the spectrum.

Honam Wong, Wendao Wu, Fanghui Liu et al.

Recent advances in machine learning have inspired a surge of research into reconstructing specific quantities of interest from measurements that comply with certain physical laws. These efforts focus on inverse problems that are governed by partial differential equations (PDEs). In this work, we develop an asymptotic Sobolev norm learning curve for kernel ridge(less) regression when addressing (elliptical) linear inverse problems. Our results show that the PDE operators in the inverse problem can stabilize the variance and even behave benign overfitting for fixed-dimensional problems, exhibiting different behaviors from regression problems. Besides, our investigation also demonstrates the impact of various inductive biases introduced by minimizing different Sobolev norms as a form of implicit regularization. For the regularized least squares estimator, we find that all considered inductive biases can achieve the optimal convergence rate, provided the regularization parameter is appropriately chosen. The convergence rate is actually independent to the choice of (smooth enough) inductive bias for both ridge and ridgeless regression. Surprisingly, our smoothness requirement recovered the condition found in Bayesian setting and extend the conclusion to the minimum norm interpolation estimators.