Title: On the Effect of Positional Encoding for In-context Learning in Transformers
Primary area: interpretability and explainable AI
Keywords: Transformer Theory, In-context Learning, Positional Encoding

Abstract: Transformer models have demonstrated a remarkable ability to perform a wide range of tasks through in-context learning (ICL), where the model infers patterns from a small number of example prompts provided during inference. However, empirical studies have shown that the effectiveness of ICL can be significantly influenced by the order in which these prompts are presented. Despite its significance, this phenomenon has been largely unexplored from a theoretical perspective. In this paper, we theoretically investigate how positional encoding (PE) affects the ICL capabilities of Transformer models, particularly in tasks where prompt order plays a crucial role. We examine two distinct cases: linear regression, which represents an order-equivariant task, and dynamic systems, a classic time-series task that is inherently sensitive to the order of input prompts. Theoretically, we evaluated the change in the model output when positional encoding (PE) is incorporated and the prompt order is altered. We proved that the magnitude of this change follows a convergence rate of $\mathcal{O}(k/N)$, where $k$ is the degree of permutation to the original prompt and $N$ is the number of in-context examples. Furthermore, for dynamical systems, we demonstrated that PE enables the Transformer to perform approximate gradient descent (GD) on permuted prompts, thereby ensuring robustness to changes in prompt order. These theoretical findings are experimentally validated.