AI Insight
This work establishes that fixed-bandwidth Gaussian kernel spectral algorithms achieve minimax optimal convergence rates in nonparametric regression, even when the model is misspecified, by exploiting the infinite smoothness properties of Gaussian kernels. The authors prove that with exponentially decaying regularization parameters, these algorithms attain optimal rates regardless of the algorithm's inherent qualification, and they demonstrate how this framework enables robust and adaptive transfer learning under concept shift with near-optimal convergence rates.
Why it matters
These findings provide theoretical guarantees for using Gaussian spectral algorithms in practical machine learning scenarios where model assumptions are violated or when transferring knowledge between related but different tasks. The results offer guidance on hyperparameter selection and establish conditions under which transfer learning can be expected to perform optimally despite distribution shifts between source and target domains.
Understand the Science
arXiv:2501.10870v2 Announce Type: replace-cross
Abstract: The principal objective of this work is twofold within nonparametric regression settings: (1) to establish the minimax optimal convergence rates for fixed-bandwidth Gaussian kernel spectral algorithms when the true regression function resides in a Sobolev space, and (2) to apply Gaussian spectral algorithms for achieving robust and adaptive transfer learning under concept shift. While minimax optimality of misspecified spectral algorithms has been established, existing guarantees are typically restricted to the non-saturation regime. We demonstrate that the infinite smoothness of fixed-bandwidth Gaussian kernels provides universal robustness to model misspecification by showing that this kernel choice enables any spectral algorithm to attain minimax optimal rates, provided the regularization parameter decays exponentially. This result effectively decouples optimality from the algorithm’s inherent qualification. Building on this, we then advocate Gaussian spectral algorithms as powerful components in a learning framework for robust and adaptive transfer. Specifically, we derive the adaptive convergence rate of the excess risk for this framework and show that the rates are optimal up to logarithmic factors. Our results also reveal the impact of the magnitude of the concept shift and the sample size on the generalization error.
Source: Fixed-Gaussian Spectral Algorithms: Minimax Optimal Rates for Misspecified Learning and Transfer