How Transformers Implement Induction Heads: Approximation and Optimization Analysis

Transformers have demonstrated 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 how 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 {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses.

Modeling Electromagnetic Navigation Systems for Medical Applications using Random Forests and Artificial Neural Networks

Electromagnetic Navigation Systems (eMNS) can be used to control a variety of multiscale devices within the human body for remote surgery. Accurate modeling of the magnetic fields generated by the electromagnets of an eMNS is crucial for the precise control of these devices. Existing methods assume a linear behavior of these systems, leading to significant modeling errors within nonlinear regions exhibited at higher magnetic fields. In this paper, we use a random forest (RF) and an artificial neural network (ANN) to model the nonlinear behavior of the magnetic fields generated by an eMNS. Both machine learning methods outperformed the state-of-the-art linear multipole electromagnet method (LMEM). The RF and the ANN model reduced the root mean squared error of the LMEM when predicting the field magnitude by around 40% and 80%, respectively, over the entire current range of the eMNS. At high current regions, especially between 30 and 35 A, the field-magnitude RMSE improvement of the ANN model over the LMEM was over 35 mT. This study demonstrates the feasibility of using machine learning methods to model an eMNS for medical applications, and its ability to account for complex nonlinear behavior at high currents. The use of machine learning thus shows promise for improving surgical procedures that use magnetic navigation.