This paper presents a theoretical analysis of the performance of transformers with softmax attention in in-context learning (ICL) for linear regression tasks. The study focuses on comparing the performance of single-head and multi-head attention mechanisms. The main findings are: 1. **Theoretical Analysis**: The paper derives exact prediction risks for both single-head and multi-head attention, demonstrating that multi-head attention with a substantial embedding dimension performs better than single-head attention. The prediction loss for multi-head attention is in \(O(1/D)\), where \(D\) is the number of in-context examples, with a smaller multiplicative constant compared to single-head attention. 2. **Scenarios**: The analysis is extended to various scenarios, including noisy labels, local examples, correlated features, and prior knowledge. Multi-head attention is generally preferred over single-head attention in these scenarios, with some interesting behaviors observed, such as the effectiveness of strong prior knowledge and the impact of local examples on prediction performance. 3. **Experimental Validation**: Simulations and experiments are conducted to validate the theoretical findings. Results show that multi-head attention consistently outperforms single-head attention in terms of prediction loss, even under different conditions. 4. **Conclusion**: The study provides a comprehensive understanding of the impact of single- and multi-head attention on ICL performance, highlighting the advantages of multi-head attention. The findings offer practical guidance for selecting the appropriate attention mechanism in real-world applications, emphasizing the importance of a larger embedding dimension relative to the number of heads. The paper contributes to the theoretical understanding of ICL and provides insights into the design and optimization of transformer architectures for linear regression tasks.This paper presents a theoretical analysis of the performance of transformers with softmax attention in in-context learning (ICL) for linear regression tasks. The study focuses on comparing the performance of single-head and multi-head attention mechanisms. The main findings are: 1. **Theoretical Analysis**: The paper derives exact prediction risks for both single-head and multi-head attention, demonstrating that multi-head attention with a substantial embedding dimension performs better than single-head attention. The prediction loss for multi-head attention is in \(O(1/D)\), where \(D\) is the number of in-context examples, with a smaller multiplicative constant compared to single-head attention. 2. **Scenarios**: The analysis is extended to various scenarios, including noisy labels, local examples, correlated features, and prior knowledge. Multi-head attention is generally preferred over single-head attention in these scenarios, with some interesting behaviors observed, such as the effectiveness of strong prior knowledge and the impact of local examples on prediction performance. 3. **Experimental Validation**: Simulations and experiments are conducted to validate the theoretical findings. Results show that multi-head attention consistently outperforms single-head attention in terms of prediction loss, even under different conditions. 4. **Conclusion**: The study provides a comprehensive understanding of the impact of single- and multi-head attention on ICL performance, highlighting the advantages of multi-head attention. The findings offer practical guidance for selecting the appropriate attention mechanism in real-world applications, emphasizing the importance of a larger embedding dimension relative to the number of heads. The paper contributes to the theoretical understanding of ICL and provides insights into the design and optimization of transformer architectures for linear regression tasks.