Convergence properties of ePIE and stochastic gradient methods for blind ptychography

We consider blind ptychography, an imaging technique which aims to reconstruct an object of interest from a set of its diffraction patterns each obtained by a local illumination. As the distribution of the light within the illuminated region, called the window, is not known, it has to be estimated as well. For the recovery, a minimization of amplitude-base squared loss via gradient and stochastic gradient descent methods is considered. In particular, this includes extended Ptychographic Iterative Engine as a special case of stochastic gradient descent. We show that with a proper choice of step sizes, all methods converge to a critical point at a sublinear rate and discuss possibilities for larger step sizes.