arXiv subjects

Fengling Chen

Publications and source records attributed to Fengling Chen.

2 recordsLinked to original sources

arXiv2026

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

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.

stat.ML↗

arXiv2026

Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules

Scaling laws have emerged as a unifying lens for understanding and guiding the training of large language models (LLMs). However, existing studies predominantly focus on the final-step loss, leaving open whether the entire loss dynamics obey similar laws and, crucially, how the learning rate schedule (LRS) shapes them. We address these gaps in a controlled theoretical setting by analyzing stochastic gradient descent (SGD) on a power-law kernel regression model. The key insight is a novel intrinsic-time viewpoint, which captures the training progress more faithfully than iteration count. We then establish a Functional Scaling Law (FSL) that captures the full loss trajectory under arbitrary LRSs, with the schedule's influence entering through a simple convolutional functional. We further instantiate the theory for three representative LRSs -- constant, exponential decay, and warmup-stable-decay (WSD) -- and derive explicit scaling relations in both data- and compute-limited regimes. These comparisons explain key empirical phenomena: (i) higher-capacity models are more data- and compute-efficient; (ii) learning-rate decay improves training efficiency; and (iii) WSD-type schedules outperform pure decay. Finally, experiments on LLMs ranging from 0.1B to 1B parameters demonstrate the practical relevance of FSL as a surrogate model for fitting and predicting loss trajectories in large-scale pre-training.

cs.LG↗