Stochastic algorithms for large-scale composite optimization: the case of likelihood maximization for X-FEL imaging

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. The algorithm we use is just one of a wide range of possibilities, all of which can be formulated abstractly as random function iterations. Our framework provides a basis for evaluating and comparing different numerical 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.