Error analysis for [Formula: see text] -coefficient regularized moving least-square regression

We consider the moving least-square (MLS) method by the coefficient-based regression framework with [Formula: see text] -regularizer [Formula: see text] and the sample dependent hypothesis spaces. The data dependent characteristic of the new algorithm provides flexibility and adaptivity for MLS. We carry out a rigorous error analysis by using the stepping stone technique in the error decomposition. The concentration technique with the [Formula: see text] -empirical covering number is also employed in our study to improve the sample error. We derive the satisfactory learning rate that can be arbitrarily close to the best rate [Formula: see text] under more natural and much simpler conditions.