  Stochastic gradient descent (SGD) provides a simple and efficient way to
solve a broad range of machine learning problems. Here, we focus on
distribution regression (DR), involving two stages of sampling: Firstly, we
regress from probability measures to real-valued responses. Secondly, we sample
bags from these distributions for utilizing them to solve the overall
regression problem. Recently, DR has been tackled by applying kernel ridge
regression and the learning properties of this approach are well understood.
However, nothing is known about the learning properties of SGD for two stage
sampling problems. We fill this gap and provide theoretical guarantees for the
performance of SGD for DR. Our bounds are optimal in a mini-max sense under
standard assumptions.
