Scaling Law for Stochastic Gradient Descent in Quadratically Parameterized Linear Regression

• URL: http://arxiv.org/abs/2502.09106v1
• Date: Thu, 13 Feb 2025 09:29:04 GMT
• Title: Scaling Law for Stochastic Gradient Descent in Quadratically Parameterized Linear Regression
• Authors: Shihong Ding, Haihan Zhang, Hanzhen Zhao, Cong Fang,
• Abstract summary: In machine learning, the scaling law describes how the model performance improves with the model and data size scaling up. This paper studies the scaling law over a linear regression with the model being quadratically parameterized. As a result, in the canonical linear regression, we provide explicit separations for curves between generalization with and without feature learning, and the information-theoretical lower bound that is to parametrization method and the algorithm.
• Score: 5.801904710149222
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• Abstract: In machine learning, the scaling law describes how the model performance improves with the model and data size scaling up. From a learning theory perspective, this class of results establishes upper and lower generalization bounds for a specific learning algorithm. Here, the exact algorithm running using a specific model parameterization often offers a crucial implicit regularization effect, leading to good generalization. To characterize the scaling law, previous theoretical studies mainly focus on linear models, whereas, feature learning, a notable process that contributes to the remarkable empirical success of neural networks, is regretfully vacant. This paper studies the scaling law over a linear regression with the model being quadratically parameterized. We consider infinitely dimensional data and slope ground truth, both signals exhibiting certain power-law decay rates. We study convergence rates for Stochastic Gradient Descent and demonstrate the learning rates for variables will automatically adapt to the ground truth. As a result, in the canonical linear regression, we provide explicit separations for generalization curves between SGD with and without feature learning, and the information-theoretical lower bound that is agnostic to parametrization method and the algorithm. Our analysis for decaying ground truth provides a new characterization for the learning dynamic of the model.

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