Spectral Gradient Descent Mitigates Anisotropy-Driven Misalignment: A Case Study in Phase Retrieval

TL;DR

This paper demonstrates that spectral gradient descent improves alignment and accelerates convergence in phase retrieval problems with anisotropic inputs by mitigating variance-induced misalignment effects.

Contribution

It provides a dynamical analysis showing how spectral gradient methods counteract anisotropy-driven misalignment, a novel insight into their empirical success.

Findings

Spectral gradient descent prevents spike amplification in anisotropic phase retrieval.

GD suffers from variance-induced misalignment, slowing convergence.

Numerical experiments validate the theoretical analysis across broader covariances.

Abstract

Spectral gradient methods, such as the Muon optimizer, modify gradient updates by preserving directional information while discarding scale, and have shown strong empirical performance in deep learning. We investigate the mechanisms underlying these gains through a dynamical analysis of a nonlinear phase retrieval model with anisotropic Gaussian inputs, equivalent to training a two-layer neural network with the quadratic activation and fixed second-layer weights. Focusing on a spiked covariance setting where the dominant variance direction is orthogonal to the signal, we show that gradient descent (GD) suffers from a variance-induced misalignment: during the early escaping stage, the high-variance but uninformative spike direction is multiplicatively amplified, degrading alignment with the true signal under strong anisotropy. In contrast, spectral gradient descent (SpecGD) removes this spike amplification effect, leading to stable alignment and accelerated noise contraction. Numerical experiments confirm the theory and show that these phenomena persist under broader anisotropic covariances.