  The generalization capacity of various machine learning models exhibits
different phenomena in the under- and over-parameterized regimes. In this
paper, we focus on regression models such as feature regression and kernel
regression and analyze a generalized weighted least-squares optimization method
for computational learning and inversion with noisy data. The highlight of the
proposed framework is that we allow weighting in both the parameter space and
the data space. The weighting scheme encodes both a priori knowledge on the
object to be learned and a strategy to weight the contribution of different
data points in the loss function. Here, we characterize the impact of the
weighting scheme on the generalization error of the learning method, where we
derive explicit generalization errors for the random Fourier feature model in
both 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 the
generalization capability of the learned model.
