← 研踪 PaperTrailarXiv:2611.09682 · v1
The Multiscale Single-Index Model: An MSE Analysis of Online SGD Dynamics
首次提交 2026-11-07 00:30
版本更新 2026-11-07 00:30
雷达观测 2026-11-09 13:22
摘要
We consider the Multiscale Single-Index Model (MSIM), first introduced by Oymak and Soltanolkotabi (2021), as a stylized model for hierarchical feature learning with scale separation. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, giving a tractable setting in which to study how deep architectures learn local multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for the two hidden-layer case ($K=2$), we analyze joint online spherical SGD under MSE loss, which is a canonical regression objective, in discrete time and show that it achieves weak recovery from random initialization in $O(d\log^2 d)$ samples, followed by strong recovery to accuracy $\varepsilon$ in $O(d\log(1/\varepsilon))$ additional samples. The main technical challenge arises from the non-Gaussian learned features observed by the second layer. We address this challenge through a quantitative Gaussian comparison using delocalization, combined with sharp trajectory-level control of the coupled SGD dynamics. This work is a companion to Bruna (2026, arXiv:2607.03347), which studies recovery under a correlation objective for general $K$.
作者备注：113 pages, 4 figures. NeurIPS 2026
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v1 · 2026-11-07 00:30 · 最新观测版本
The Multiscale Single-Index Model: An MSE Analysis of Online SGD Dynamics
We consider the Multiscale Single-Index Model (MSIM), first introduced by Oymak and Soltanolkotabi (2021), as a stylized model for hierarchical feature learning with scale separation. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, giving a tractable setting in which to study how deep architectures learn local multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for the two hidden-layer case ($K=2$), we analyze joint online spherical SGD under MSE loss, which is a canonical regression objective, in discrete time and show that it achieves weak recovery from random initialization in $O(d\log^2 d)$ samples, followed by strong recovery to accuracy $\varepsilon$ in $O(d\log(1/\varepsilon))$ additional samples. The main technical challenge arises from the non-Gaussian learned features observed by the second layer. We address this challenge through a quantitative Gaussian comparison using delocalization, combined with sharp trajectory-level control of the coupled SGD dynamics. This work is a companion to Bruna (2026, arXiv:2607.03347), which studies recovery under a correlation objective for general $K$.
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