  We study the dynamics of gradient flow for training a multi-head softmax
attention model for in-context learning of multi-task linear regression. We
establish the global convergence of gradient flow under suitable choices of
initialization. In addition, we prove that an interesting "task allocation"
phenomenon emerges during the gradient flow dynamics, where each attention head
focuses on solving a single task of the multi-task model. Specifically, we
prove that the gradient flow dynamics can be split into three phases -- a
warm-up phase where the loss decreases rather slowly and the attention heads
gradually build up their inclination towards individual tasks, an emergence
phase where each head selects a single task and the loss rapidly decreases, and
a convergence phase where the attention parameters converge to a limit.
Furthermore, we prove the optimality of gradient flow in the sense that the
limiting model learned by gradient flow is on par with the best possible
multi-head softmax attention model up to a constant factor. Our analysis also
delineates a strict separation in terms of the prediction accuracy of ICL
between single-head and multi-head attention models. The key technique for our
convergence analysis is to map the gradient flow dynamics in the parameter
space to a set of ordinary differential equations in the spectral domain, where
the relative magnitudes of the semi-singular values of the attention weights
determines task allocation. To our best knowledge, our work provides the first
convergence result for the multi-head softmax attention model.
