Scaling Law Phenomena Across Regression Paradigms: Multiple and Kernel Approaches
Yifang Chen, Xuyang Guo, Xiaoyu Li, Yingyu Liang, Zhenmei Shi, Zhao Song
TL;DR
The paper extends scaling law insights from large transformer-based models to more expressive regression paradigms by analyzing multiple and kernel regression under sketching. Under Gaussian-feature assumptions, well-specified models, and power-law eigenvalue decay, it derives scaling laws that decompose generalization error into irreducible, approximation, and excess components, with explicit bounds that depend on sketch dimension $m$, effective sample size $N_{eff}$, and learning rate. It provides precise approximation-error bounds (upper and lower) for sketched regression, tight bias bounds, and excess-error analyses that together yield a cohesive scaling law framework for sketched multi- and kernel-regression. The results shed light on how sketching can reduce computation while preserving predictive performance, and they connect to broader implications for generalization in large-scale and multi-task learning contexts such as LLMs.
Abstract
Recently, Large Language Models (LLMs) have achieved remarkable success. A key factor behind this success is the scaling law observed by OpenAI. Specifically, for models with Transformer architecture, the test loss exhibits a power-law relationship with model size, dataset size, and the amount of computation used in training, demonstrating trends that span more than seven orders of magnitude. This scaling law challenges traditional machine learning wisdom, notably the Oscar Scissors principle, which suggests that an overparametrized algorithm will overfit the training datasets, resulting in poor test performance. Recent research has also identified the scaling law in simpler machine learning contexts, such as linear regression. However, fully explaining the scaling law in large practical models remains an elusive goal. In this work, we advance our understanding by demonstrating that the scaling law phenomenon extends to multiple regression and kernel regression settings, which are significantly more expressive and powerful than linear methods. Our analysis provides deeper insights into the scaling law, potentially enhancing our understanding of LLMs.