Learning with Synthetic Data via SGD in High-Dimensional Linear Regression

• URL: http://arxiv.org/abs/2609.09572v1
• Date: Wed, 09 Sep 2026 00:53:32 GMT
• Title: Learning with Synthetic Data via SGD in High-Dimensional Linear Regression
• Abstract summary: We study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift.<n>Mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage.
• Score: 29.71683151194804
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Synthetic data has become a promising way to scale model training beyond limited human-generated data but it may also induce strong model collapse (Dohmatob et al., 2024), where any fixed fraction of synthetic data prevents model performance from improving under data scaling, leaving a non-vanishing excess risk floor. In this paper, we study how synthetic data affects the generalization of one-pass SGD in high-dimensional linear regression with model shift. We establish finite-sample risk bounds for mixed and two-stage training, separating standard bias and variance from source-mismatch effects, namely fluctuation and persistent drift under mixing and filtered initialization bias under two-stage. These bounds reveal a sharp contrast: mixed training induces strong model collapse, while two-stage training avoids the floor by using synthetic data only in the first stage, showing that collapse is not inevitable under a simple data curriculum. Under a random sketch model, we further obtain scaling laws for both protocols, with tight results for mixed training in the optimization-saturated regime. These laws show that larger models may amplify synthetic-induced degradation under mixing, and quantify how high-quality synthetic pretraining may reduce bias in two-stage training. Finally, we establish an exact finite-sample necessary-and-sufficient condition for two-stage training to strictly outperform real-only training under the same real-data budget and identical real-stage updates. Overall, our results highlight that synthetic data is neither inherently harmful nor beneficial; its effect depends critically on both its quality and the training protocol used to incorporate it.

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