Stochastic Algorithms for Large-Scale Composite Optimization: the Case of Single-Shot X-FEL Imaging

TL;DR

This paper extends stochastic optimization analysis to large-scale nonconvex problems, demonstrating its application to single-shot X-FEL imaging for molecular electron density reconstruction, with promising numerical results.

Contribution

It introduces a theoretical framework for stochastic algorithms in nonconvex composite optimization, applied to X-FEL imaging, facilitating broader algorithmic development.

Findings

Successful application of stochastic gradient descent to X-FEL imaging

Numerical proof of concept on synthetic data

Framework applicable to various machine learning problems

Abstract

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.