Title: How Transformers Get Rich: Approximation and Dynamics Analysis
Primary area: learning theory
Keywords: Transformer, training dynamics analysis, approximiation analysis, abrupt transition

Abstract: Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remains limited. A recent work (Elhage et al., 2021) identified a "rich" in-context mechanism known as induction head, contrasting with "lazy" 
-gram models that overlook long-range dependencies. In this work, we provide both approximation and dynamics analyses of how transformers implement induction heads. In the *approximation* analysis, we formalize both standard and generalized induction head mechanisms, and examine how transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the *dynamics* analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This controlled setting allows us to precisely characterize the entire training process and uncover an abrupt transition from lazy (4-gram) to rich (induction head) mechanisms as training progresses. The theoretical insights are validated experimentally in both synthetic and real-world settings.