Misspecified Nonconvex Statistical Optimization for Phase Retrieval
This addresses a fundamental limitation in optimization theory for researchers in statistics and machine learning, though it is incremental as it builds on existing methods.
The paper tackles the problem of model misspecification in nonconvex statistical optimization for high-dimensional sparse phase retrieval, proposing a variant of the thresholded Wirtinger flow algorithm that linearly converges to an estimator with optimal statistical accuracy for unknown link functions.
Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.