  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.
