  Transformers have recently revolutionized many domains in modern machine
learning and one salient discovery is their remarkable in-context learning
capability, where models can solve an unseen task by utilizing task-specific
prompts without further parameters fine-tuning. This also inspired recent
theoretical studies aiming to understand the in-context learning mechanism of
transformers, which however focused only on linear transformers. In this work,
we take the first step toward studying the learning dynamics of a one-layer
transformer with softmax attention trained via gradient descent in order to
in-context learn linear function classes. We consider a structured data model,
where each token is randomly sampled from a set of feature vectors in either
balanced or imbalanced fashion. For data with balanced features, we establish
the finite-time convergence guarantee with near-zero prediction error by
navigating our analysis over two phases of the training dynamics of the
attention map. More notably, for data with imbalanced features, we show that
the learning dynamics take a stage-wise convergence process, where the
transformer first converges to a near-zero prediction error for the query
tokens of dominant features, and then converges later to a near-zero prediction
error for the query tokens of under-represented features, respectively via one
and four training phases. Our proof features new techniques for analyzing the
competing strengths of two types of attention weights, the change of which
determines different training phases.
