Asymptotic Risk of Learning-Rate Schedules for Least-Squares Regression with Discretely Distributed Features
We study the last iterate of stochastic gradient descent (SGD) on streaming least squares with discretely distributed features (Slock's model), which decouples the dynamics across eigendirections. Under power-law source and capacity conditions, we derive the exact finite-time risk and asymptotic equivalents of constant and Warmup-Stable-Decay (WSD) schedules. With a tuned base learning-rate, the resulting scaling law decays as $Θ(T^{-q})$, where $T$ is the horizon, and $q$ is independent of the cooldown length $c \in (0,1]$ but is strictly smaller for the constant schedule: any cooldown of fixed relative length $c > 0$, however short, improves the risk by a polynomial factor, while its le...
| Authors | Noé Colomban, Francis Bach, Fabian Schaipp |
|---|---|
| Primary category | cs.LG |
| Published | 12 October 2026 |
| Last updated at source | 12 October 2026 |
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