Abstract: Large language models (LLMs) are known for their exceptional performance in natural language processing, making them highly effective in many human life-related tasks. The attention mechanism in the Transformer architecture is a critical component of LLMs, as it allows the model to selectively focus on specific input parts. The softmax unit, which is a key part of the attention mechanism, normalizes the attention scores. Hence, the performance of LLMs in various NLP tasks depends significantly on the crucial role played by the attention mechanism with the softmax unit.In-context learning is one of the celebrated abilities of recent LLMs. Without further parameter updates, Transformers can learn to predict based on few in-context examples. However, the reason why Transformers becomes in-context learners is not well understood.Recently, in-context learning has been studied from a mathematical perspective with simplified linear self-attention without softmax unit. Based on a linear regression formulation $\min_x\| Ax - b \|_2$, existing works show linear Transformers' capability of learning linear functions in context. The capability of Transformers with softmax unit approaching full Transformers, however, remains unexplored.In this work, we study the in-context learning based on a softmax regression formulation $\min_{x} \| \langle \exp(Ax), {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2$. We show the upper bounds of the data transformations induced by a single self-attention layer with softmax unit and by gradient-descent on a $\ell_2$ regression loss for softmax prediction function.Our theoretical results imply that when training self-attention-only Transformers for fundamental regression tasks, the models learned by gradient-descent and Transformers show great similarity.