How Transformers Implement Induction Heads: Approximation and Optimization Analysis

Mingze Wang, Ruoxi Yu, Weinan E, Lei Wu

Keywords: Transformer, mechanisms, approximiation, training dynamics, abrupt transition

Abstract:

Transformers exhibit exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remain limited. A recent work (Elhage et al., 2021) identified a "rich" in-context mechanism known as induction head, contrasting with "lazy" $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and optimization analyses of how transformers implement induction heads. In the approximation analysis, we formalize both standard and generalized induction head mechanisms, and examine whether two-layer single- or multi-head transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the optimization analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This setting enables 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.

Primary Area: learning theory

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Submission Number: 9368