Organizations: Center for Machine Learning Research, Peking University · School of Mathematical Sciences, Peking University · AI for Science Institute, Beijing
We study optimal learning rate (LR) schedules under the functional scaling law (FSL) framework (Li et al., 2025), which decomposes training dynamics into signal learning and noise forgetting. In power-law kernel regression, these two components are governed by a source exponent s>0 and a capacity exponent q>1, respectively, with smaller s corresponding to harder tasks. For a fixed training horizon N, we characterize the schedules that minimize the final-step loss under a stability constraint and reveal a sharp phase transition. In the easy-task regime s>1−1/q, the optimal schedule follows power decay from the beginning of training; in the hard-task regime s<1−1/q, it becomes warmup-stable-decay (WSD)-like (Hu et al., 2024), staying at the largest admissible LR for most of training before a final decay. In both regimes, the decay exponent is 2q−1: task difficulty determines when to decay, while model capacity determines how to decay. Beyond the exact optimum, we study fractional schedules, whose shape is defined over relative training progress. We show that precise tuning of the decay shape is often unnecessary: a broad class of profiles attains the optimal convergence rate, while overly slow terminal decay leads to schedule-induced capacity saturation. Finally, for one-pass SGD in kernel regression, FSL-motivated power-decay schedules achieve optimal last-iterate rates. Experiments support the theoretical predictions and the task-dependent transition between early and delayed decay.
We study scaling laws of signSGD under a power-law random features (PLRF) model that accounts for both feature and target decay. We analyze the population risk of a linear model trained with one-pass signSGD on Gaussian-sketched features. We express the risk as a function of model size, training steps, learning rate, and the feature and target decay parameters. Comparing against the SGD risk analyzed by Paquette et al. (2024), we identify a drift-normalization effect and a noise-reshaping effect unique to signSGD. We then obtain compute-optimal scaling laws under the optimal choice of learning rate. Our analysis shows that the noise-reshaping effect can make the compute-optimal slope of signSGD steeper than that of SGD in regimes where noise is dominant. Finally, we observe that the widely used warmup-stable-decay (WSD) schedule further reduces the noise term and sharpens the compute-optimal slope, when feature decay is fast but target decay is slow.
Standard learning rate schedules such as cosine annealing are tied to a fixed training horizon, limiting their ability to accommodate post hoc horizon extension. Warmup-stable-decay (WSD) partially addresses this issue by maintaining a long constant-rate phase before a short linear cooldown, allowing training to resume from a pre-decay checkpoint. However, its peak learning rate is still tuned based on the original training horizon and can become suboptimal when training is extended. Motivated by stochastic convex optimization, we propose WSqD (Warmup with Square-root base and linear Decay), a learning rate schedule that replaces WSD's constant stable phase with a shifted inverse-square-root base while retaining the final linear cooldown. In the stochastic convex setting, WSqD provably attains the minimax-optimal O(1/T) last-iterate convergence rate. Importantly, its base learning rate schedule is horizon-independent, and the training horizon is needed only to determine when to begin the final cooldown. Empirically, on language-model pretraining using the SlimPajama corpus, WSqD matches or outperforms carefully tuned WSD and other baselines across multiple training horizons while reusing a single peak learning rate.
The cooldown phase of a warmup-stable-decay (WSD) learning-rate schedule, now a default in large-model pretraining, lowers the final training loss in some settings and does nothing in others. We give a provable account of which case obtains, and it turns on two properties together: the structure of the gradient noise and whether the optimizer normalizes its update. On a strongly convex objective with multiplicative (gradient-proportional) noise, stochastic gradient descent contracts geometrically at a constant learning rate, so cooldown has nothing to improve. Under the same objective and noise, sign-based and normalized methods, the standard surrogates for adaptive optimizers, settle on a noise floor of order η2 and reach the minimizer only as the learning rate is driven to zero; any additive noise then reinstates a floor for every method. The mechanism is elementary: an SGD step shrinks in proportion to the gradient and so anneals itself, whereas a normalized step keeps unit scale and cannot. We solve the signSGD stationary law on the quadratic exactly and obtain the floor constant in closed form, prove a local form of the dissociation under (L0,L1)-smoothness, extend the floor to normalized SGD in dimension d>1 by a scale-invariance argument, and establish robustness to momentum and heavy-tailed noise. Simulation confirms every prediction, and we demonstrate the resulting noise-regime diagnostic on a real classification task with directly measured gradient noise. The mechanism explains whether cooldown helps; the interior cooldown fraction used at scale lies outside stationary landscape-and-noise geometry.