  We present a theoretical analysis of the performance of transformer with
softmax attention in in-context learning with linear regression tasks. While
the existing literature predominantly focuses on the convergence of
transformers with single-/multi-head attention, our research centers on
comparing their performance. We conduct an exact theoretical analysis to
demonstrate that multi-head attention with a substantial embedding dimension
performs better than single-head attention. When the number of in-context
examples D increases, the prediction loss using single-/multi-head attention is
in O(1/D), and the one for multi-head attention has a smaller multiplicative
constant. In addition to the simplest data distribution setting, we consider
more scenarios, e.g., noisy labels, local examples, correlated features, and
prior knowledge. We observe that, in general, multi-head attention is preferred
over single-head attention. Our results verify the effectiveness of the design
of multi-head attention in the transformer architecture.
