  The problem of phase retrieval has many applications in the field of optical
imaging. Motivated by imaging experiments with biological specimens, we
primarily consider the setting of low-dose illumination where Poisson noise
plays the dominant role. In this paper, we discuss gradient descent algorithms
based on different loss functions adapted to data affected by Poisson noise, in
particular in the low-dose regime. Starting from the maximum log-likelihood
function for the Poisson distribution, we investigate different regularizations
and approximations of the problem to design an algorithm that meets the
requirements that are faced in applications. In the course of this, we focus on
low-count measurements. For all suggested loss functions, we study the
convergence of the respective gradient descent algorithms to stationary points
and find constant step sizes that guarantee descent of the loss in each
iteration. Numerical experiments in the low-dose regime are performed to
corroborate the theoretical observations.
