How Many Pretraining Tasks Are Needed for In-Context Learning of Linear Regression?

TL;DR

This paper provides a theoretical analysis of in-context learning in transformers, showing that a small number of pretraining tasks suffice for near-optimal linear regression performance, aligning with Bayesian methods.

Contribution

It establishes a statistical task complexity bound for pretraining linear attention models and proves their near Bayes optimality in linear regression tasks.

Findings

Effective pretraining requires few independent tasks.

Pretrained models achieve nearly Bayes optimal risk.

Theoretical insights complement prior empirical studies.

Abstract

Transformers pretrained on diverse tasks exhibit remarkable in-context learning (ICL) capabilities, enabling them to solve unseen tasks solely based on input contexts without adjusting model parameters. In this paper, we study ICL in one of its simplest setups: pretraining a linearly parameterized single-layer linear attention model for linear regression with a Gaussian prior. We establish a statistical task complexity bound for the attention model pretraining, showing that effective pretraining only requires a small number of independent tasks. Furthermore, we prove that the pretrained model closely matches the Bayes optimal algorithm, i.e., optimally tuned ridge regression, by achieving nearly Bayes optimal risk on unseen tasks under a fixed context length. These theoretical findings complement prior experimental research and shed light on the statistical foundations of ICL.