Asymptotic Risk of Learning-Rate Schedules for Least-Squares Regression with Discretely Distributed Features

Abstract: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 $\Theta(T^{-q})$, 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 length only affects the constant, which is minimized by linear decay ($c = 1$). Finally, we show that the optimal risk depends on the batch size $b$ and horizon $T$ only through $bT$, up to a critical batch size.

Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2610.16518 [cs.LG]

(or arXiv:2610.16518v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.16518 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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