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 unknown, it also has to be estimated as well. For the recovery, we consider gradient and stochastic gradient descent methods for the minimization of amplitude-base squared loss. In particular, this includes extended Ptychographic Iterative Engine as a special case of stochastic gradient descent. We show that all methods converge to a critical point at a sublinear rate with a proper choice of step sizes. We also discuss possibilities for larger step sizes.

We consider recovery from ptychographic measurements also known as Short-Time Fourier Transform phase retrieval. We study Amplitude Flow, Error Reduction and Ptychographic Iterative Engine algorithms and derive their convergence speed. In addition, Block Phase Retrieval algorithm and its modifications are studied. Furthermore, we consider blind ptychography and devise a version of Amplitude Flow with guaranteed convergence. Finally, some results are extended to polychromatic ptychography.

In this paper, we focus on Stochastic Amplitude Flow (SAF) for phase retrieval, a stochastic gradient descent for the amplitude-based squared loss. While the convergence to a critical point of (nonstochastic) Amplitude Flow is well-understood, SAF is a much less studied algorithm. We close this gap by deriving the convergence guarantees for SAF based on the contributions for Amplitude Flow and analysis for stochastic gradient descent. These results are then applied to two more algorithms, which can be seen as instances of SAF. The first is an extension of the Kaczmarz method for phase retrieval. The second is Ptychographic Iterative Engine, which is a popular algorithm for ptychography, a special case of phase retrieval with the short-time Fourier transform. Keywords: phase retrieval, Amplitude Flow, stochastic gradient descent, ptychography, Ptychographic Iterative Engine, Kaczmarz method.