  Regularization schemes for regression have been widely studied in learning
theory and inverse problems. In this paper, we study distribution regression
(DR) which involves two stages of sampling, and aims at regressing from
probability measures to real-valued responses over a reproducing kernel Hilbert
space (RKHS). Recently, theoretical analysis on DR has been carried out via
kernel ridge regression and several learning behaviors have been observed.
However, the topic has not been explored and understood beyond the least square
based DR. By introducing a robust loss function $l_{\sigma}$ for two-stage
sampling problems, we present a novel robust distribution regression (RDR)
scheme. With a windowing function $V$ and a scaling parameter $\sigma$ which
can be appropriately chosen, $l_{\sigma}$ can include a wide range of popular
used loss functions that enrich the theme of DR. Moreover, the loss
$l_{\sigma}$ is not necessarily convex, hence largely improving the former
regression class (least square) in the literature of DR. The learning rates
under different regularity ranges of the regression function $f_{\rho}$ are
comprehensively studied and derived via integral operator techniques. The
scaling parameter $\sigma$ is shown to be crucial in providing robustness and
satisfactory learning rates of RDR.
