  In this paper, we consider the coefficient-based regularized distribution
regression which aims to regress from probability measures to real-valued
responses over a reproducing kernel Hilbert space (RKHS), where the
regularization is put on the coefficients and kernels are assumed to be
indefinite. The algorithm involves two stages of sampling, the first stage
sample consists of distributions and the second stage sample is obtained from
these distributions. Asymptotic behaviors of the algorithm in different
regularity ranges of the regression function are comprehensively studied and
learning rates are derived via integral operator techniques. We get the optimal
rates under some mild conditions, which matches the one-stage sampled minimax
optimal rate. Compared with the kernel methods for distribution regression in
the literature, the algorithm under consideration does not require the kernel
to be symmetric and positive semi-definite and hence provides a simple paradigm
for designing indefinite kernel methods, which enriches the theme of the
distribution regression. To the best of our knowledge, this is the first result
for distribution regression with indefinite kernels, and our algorithm can
improve the saturation effect.
