  We apply a recently developed framework for analyzing the convergence of
stochastic algorithms to the general problem of large-scale nonconvex composite
optimization more generally, and nonconvex likelihood maximization in
particular. Our theory is demonstrated on a stochastic gradient descent
algorithm for determining the electron density of a molecule from random
samples of its scattering amplitude. Numerical results on an idealized
synthetic example provide a proof of concept. This opens the door to a broad
range of algorithmic possibilities and provides a basis for evaluating and
comparing different strategies. While this case study is very specific, it
shares a structure that transfers easily to many problems of current interest,
particularly in machine learning.
