[2411.12135] Exact Risk Curves of signSGD in High-Dimensions: Quantifying Preconditioning and Noise-Compression Effects

Statistics > Machine Learning arXiv:2411.12135 (stat) [Submitted on 19 Nov 2024 (v1), last revised 25 Mar 2026 (this version, v3)] Title:Exact Risk Curves of signSGD in High-Dimensions: Quantifying Preconditioning and Noise-Compression Effects Authors:Ke Liang Xiao, Noah Marshall, Atish Agarwala, Elliot Paquette View a PDF of the paper titled Exact Risk Curves of signSGD in High-Dimensions: Quantifying Preconditioning and Noise-Compression Effects, by Ke Liang Xiao and 3 other authors View PDF HTML (experimental) Abstract:In recent years, signSGD has garnered interest as both a practical optimizer as well as a simple model to understand adaptive optimizers like Adam. Though there is a general consensus that signSGD acts to precondition optimization and reshapes noise, quantitatively understanding these effects in theoretically solvable settings remains difficult. We present an analysis of signSGD in a high dimensional limit, and derive a limiting SDE and ODE to describe the risk. Using this framework we quantify four effects of signSGD: effective learning rate, noise compression, diagonal preconditioning, and gradient noise reshaping. Our analysis is consistent with experimental observations but moves beyond that by quantifying the dependence of these effects on the data and noise distributions. We conclude with a conjecture on how these results might be extended to Adam. Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG) Cite as: arXiv:2411.12135 [stat.ML]   (...