Optimal Learning-Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decay

We study optimal learning-rate schedules (LRSs) under the functional scaling law (FSL) framework introduced in [Li et al. (2025)](https://arxiv.org/abs/2509.19189), which accurately models the loss dynamics of both linear regression and large language model (LLM) pre-training. Within FSL, loss dynamics are governed by two exponents: a source exponent $s>0$ controlling the rate of signal learning, and a capacity exponent $\beta>1$ determining the rate of noise forgetting. Focusing on a fixed training horizon $N$, we derive the optimal LRSs and reveal a sharp phase transition. In the easy-task regime $s \ge 1 - 1/\beta$, the optimal schedule follows a power decay to zero, $\eta^*(z) = \eta_{\mathrm{peak}}(1 - z/N)^{2\beta - 1}$, where the peak learning rate scales as $\eta_{\mathrm{peak}} \eqsim N^{-\nu}$ for an explicit exponent $\nu = \nu(s,\beta)$. In contrast, in the hard-task regime $s < 1 - 1/\beta$, the optimal LRS exhibits a warmup--stable--decay (WSD) [(Hu et al., 2024)](https://arxiv.org/abs/2404.06395) structure: it maintains the largest admissible learning rate for most of training and decays only near the end, with the decay phase occupying a vanishing fraction of the horizon. We further analyze optimal shape-fixed schedules, where only the peak learning rate is tuned--a strategy widely adopted in practice--and characterize their strengths and intrinsic limitations. This yields a principled evaluation of commonly used schedules such as cosine and linear decay. Finally, we apply the power-decay LRS to one-pass stochastic gradient descent (SGD) for kernel regression and show the last iterate attains the exact minimax-optimal rate, eliminating the logarithmic suboptimality present in prior analyses. Numerical experiments corroborate our theoretical predictions.