A Generalized Weighted Optimization Method for Computational Learning and Inversion
This work addresses generalization issues in machine learning regression for noisy data, but it is incremental as it builds on existing weighted optimization methods by extending them to dual weighting schemes.
The authors tackled the problem of improving generalization in regression models by proposing a generalized weighted least-squares optimization method that incorporates weighting in both parameter and data spaces, showing that appropriate weighting based on prior knowledge can enhance generalization capability with explicit error bounds derived for specific cases.
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.