Optimal learning rates for distribution regression

We study a learning algorithm for distribution regression with regularized least squares. This algorithm, which contains two stages of sampling, aims at regressing from distributions to real valued outputs. The first stage sample consists of distributions and the second stage sample is obtained from these distributions. To extract information from samples, we embed distributions to a reproducing kernel Hilbert space (RKHS) and use the second stage sample to form the regressor by a tool of mean embedding. We show error bounds in the L²-norm and prove that the regressor is a good approximation to the regression function. We derive a learning rate which is optimal in the setting of standard least squares regression and improve the existing work. Our analysis is achieved by using a novel second order decomposition to bound operator norms.