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The generalization capacity of various machine learning models exhibitsdifferent phenomena in the under- and over-parameterized regimes. In thispaper, we focus on regression models such as feature regression and kernelregression and analyze a generalized weighted least-squares optimization methodfor computational learning and inversion with noisy data. The highlight of theproposed framework is that we allow weighting in both the parameter space andthe data space. The weighting scheme encodes both a priori knowledge on theobject to be learned and a strategy to weight the contribution of differentdata points in the loss function. Here, we characterize the impact of theweighting scheme on the generalization error of the learning method, where wederive explicit generalization errors for the random Fourier feature model inboth the under- and over-parameterized regimes. For more general feature maps,error bounds are provided based on the singular values of the feature matrix.We demonstrate that appropriate weighting from prior knowledge can improve thegeneralization capability of the learned model.

A Generalized Weighted Optimization Method for Computational Learning and Inversion