Frank-Wolfe Works for Non-Lipschitz Continuous Gradient Objectives: Scalable Poisson Phase Retrieval

TL;DR

This paper demonstrates that the Frank-Wolfe algorithm can effectively solve a Poisson phase retrieval problem with a non-Lipschitz continuous gradient, providing theoretical convergence guarantees and numerical validation.

Contribution

It proves the convergence of the Frank-Wolfe algorithm for a non-Lipschitz objective in Poisson phase retrieval, extending its applicability beyond Lipschitz continuous cases.

Findings

Frank-Wolfe algorithm achieves O(1/t) convergence rate.

Theoretical guarantees are established for non-Lipschitz objectives.

Numerical results validate the theoretical findings.

Abstract

We study a phase retrieval problem in the Poisson noise model. Motivated by the PhaseLift approach, we approximate the maximum-likelihood estimator by solving a convex program with a nuclear norm constraint. While the Frank-Wolfe algorithm, together with the Lanczos method, can efficiently deal with nuclear norm constraints, our objective function does not have a Lipschitz continuous gradient, and hence existing convergence guarantees for the Frank-Wolfe algorithm do not apply. In this paper, we show that the Frank-Wolfe algorithm works for the Poisson phase retrieval problem, and has a global convergence rate of O(1/t), where t is the iteration counter. We provide rigorous theoretical guarantee and illustrating numerical results.