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

• URL: http://arxiv.org/abs/2509.19189v3
• Date: Mon, 03 Nov 2025 13:29:04 GMT
• Title: Functional Scaling Laws in Kernel Regression: Loss Dynamics and Learning Rate Schedules
• Authors: Binghui Li, Fengling Chen, Zixun Huang, Lean Wang, Lei Wu,
• Abstract summary: Scaling laws have emerged as a unifying lens for understanding and guiding the training of large language models.<n>We establish a Functional Scaling Law that captures the full loss trajectory under arbitrary LRSs.<n>We derive explicit scaling relations in both data- and compute-limited regimes.
• Score: 9.332823269318842
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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