Raymond C. Yu

Mingze Wang, Ruoxi Yu, Weinan E et al.

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'' $n$-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 {\em 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 {\em 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 {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses.

Penghao Yu, Haotian Jiang, Zeyu Bao et al.

Transformer has become the dominant architecture for sequence modeling, yet a detailed understanding of how its structural parameters influence expressive power remains limited. In this work, we study the approximation properties of transformers, with particular emphasis on the role of the number of attention heads. Our analysis begins with the introduction of a generalized $D$-retrieval task, which we prove to be dense in the space of continuous functions, thereby providing the basis for our theoretical framework. We then establish both upper and lower bounds on the parameter complexity required for $ε$-approximation. Specifically, we show that transformers with sufficiently many heads admit efficient approximation, whereas with too few heads, the number of parameters must scale at least as $O(1/ε^{cT})$, for some constant $c$ and sequence length $T$. To the best of our knowledge, this constitutes the first rigorous lower bound of this type in a nonlinear and practically relevant setting. We further examine the single-head case and demonstrate that an embedding dimension of order $O(T)$ allows complete memorization of the input, where approximation is entirely achieved by the feed-forward block. Finally, we validate our theoretical findings with experiments on both synthetic data and real-world tasks, illustrating the practical relevance of our results.

Ruoxi Yu, Haotian Jiang, Jingpu Cheng et al.

Transformers have achieved remarkable successes across a wide range of applications, yet the theoretical foundation of their model efficiency remains underexplored. In this work, we investigate how the model parameters -- mainly attention heads and head dimensions -- should be allocated across layers to balance expressivity and efficiency. We first provide mathematical analysis on the role of early layers in information extraction from an approximation perspective, with a theoretical characterization on the trade-off between the number of heads and head dimension under a fixed parameter budget. In addition, we uncover and prove the \emph{saturation} behavior of softmax activations: Continuously increasing head dimensions can lead to diminishing returns in learning errors, particularly for long sequences. Supported by both theory and experiments, this saturation pattern suggests that later layers can operate more efficiently with reduced parameters. Combining these insights, we propose principled strategies for allocating attention heads and dimensions across Transformers' layers, shedding light on theoretically-grounded model efficiency of Transformer-based architectures.