Circuit Complexity Bounds for RoPE-based Transformer Architecture

• URL: http://arxiv.org/abs/2411.07602v1
• Date: Tue, 12 Nov 2024 07:24:41 GMT
• Title: Circuit Complexity Bounds for RoPE-based Transformer Architecture
• Authors: Bo Chen, Xiaoyu Li, Yingyu Liang, Jiangxuan Long, Zhenmei Shi, Zhao Song,
• Abstract summary: Empirical evidence suggests that $mathsfRoPE$-based Transformer architectures demonstrate greater generalization capabilities. We show that unless $mathsfTC0 = mathsfNC1$, a $mathsfRoPE$-based Transformer with $mathrmpoly(n)$-precision, $O(1)$ layers, hidden dimension $d leq O(n)$ cannot solve the arithmetic problem.
• Score: 25.2590541420499
• License:
• Abstract: Characterizing the express power of the Transformer architecture is critical to understanding its capacity limits and scaling law. Recent works provide the circuit complexity bounds to Transformer-like architecture. On the other hand, Rotary Position Embedding ($\mathsf{RoPE}$) has emerged as a crucial technique in modern large language models, offering superior performance in capturing positional information compared to traditional position embeddings, which shows great potential in application prospects, particularly for the long context scenario. Empirical evidence also suggests that $\mathsf{RoPE}$-based Transformer architectures demonstrate greater generalization capabilities compared to conventional Transformer models. In this work, we establish a tighter circuit complexity bound for Transformers with $\mathsf{RoPE}$ attention. Our key contribution is that we show that unless $\mathsf{TC}^0 = \mathsf{NC}^1$, a $\mathsf{RoPE}$-based Transformer with $\mathrm{poly}(n)$-precision, $O(1)$ layers, hidden dimension $d \leq O(n)$ cannot solve the arithmetic problem or the Boolean formula value problem. This result significantly demonstrates the fundamental limitation of the expressivity of the $\mathsf{RoPE}$-based Transformer architecture, although it achieves giant empirical success. Our theoretical framework not only establishes tighter complexity bounds but also may instruct further work on the $\mathsf{RoPE}$-based Transformer.

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