RoPE Attention Can Be Trained in Almost Linear Time
Yang Cao, Jiayan Huo, Yingyu Liang, Zhenmei Shi, Zhao Song
TL;DR
This paper introduces an almost linear time algorithm for the backward pass of RoPE-based attention in Transformers, significantly improving efficiency while establishing theoretical bounds for such computations.
Contribution
It presents the first almost linear time algorithm for backward RoPE attention computations under bounded entries, combining polynomial methods and FFT techniques.
Findings
Developed an almost linear time backward algorithm for RoPE attention.
Proved the necessity of bounded entries for subquadratic performance based on SETH.
Enhanced understanding of the computational complexity of RoPE mechanisms.
Abstract
The Rotary Position Embedding (RoPE) mechanism has become a powerful enhancement to the Transformer architecture, which enables models to capture token relationships when encoding positional information. However, the RoPE mechanisms make the computations of attention mechanisms more complicated, which makes efficient algorithms challenging. Earlier research introduced almost linear time algorithms for the forward computation under specific parameter settings of bounded entries (i.e., in time where is the number of input tokens), but has not addressed backward computation. In this work, we develop the first almost linear time algorithm for backward computations in the RoPE-based attention under bounded entries. Our approach builds on recent advancements in fast RoPE attention computations, utilizing a novel combination of the polynomial method and the Fast Fourier…
Peer Reviews
Decision·Submitted to ICLR 2026
1. The paper addresses a critical gap in the literature by providing the first almost linear-time algorithm for backward computations in RoPE-based attention. Previous work primarily focused on forward computations. This is highly significant for the efficient training and optimization of LLMs that incorporate RoPE. 2. The paper offers a comprehensive theoretical treatment, including: Formulation of closed-form gradients for RoPE attention (Lemma 4.1). Detailed time complexity analysis for exact
1. The mathematical notation and dense theoretical arguments might make the paper challenging for readers not deeply familiar with computational complexity theory, tensor algebra, and advanced matrix operations. While necessary for rigor, it could limit accessibility to a broader ML audience. 2. The paper is purely theoretical. While the theoretical contributions are strong, the absence of experimental results or empirical validation on actual LLM training tasks is a notable weakness. Demonstrat
1. Technical Solution (for the Defined Problem): The paper is technically sound in solving the specific problem it sets for itself. The "Generalized RoPE" problem it addresses ($A_{i,j} = exp(Q_{i,*} W_{i-j} K_{j,*}^\top)$) is indeed a non-low-rank, Toeplitz-like problem. Adapting the complex Polynomial + FFT machinery from the forward pass to the even more complex backward pass is a non-trivial technical achievement. 2. Theoretical Completeness: For the generalized problem they define, the aut
1. Fundamental Mismatch with Practical RoPE Formulation: The paper's entire premise and claim to novelty appear to be based on a problem definition that does not match the RoPE implementation used in practice (e.g., in Llama, as defined by Su et al., 2024). The Paper's Problem: The authors solve a "Generalization of... ROPE" (Def 3.1) where the matrix $W_{i-j}$ is a general, sparse, non-decomposable matrix that depends on the relative position $i-j$. This structure is indeed non-low-rank and re
1. This is the first work to achieve almost linear time gradient computation for RoPE attention, and the gradient computation for RoPE is substantially more complex than standard attention due to position-dependent rotations. 2. The lower bound result (Theorem 6.1) demonstrates the necessity of bounded entries, showing the assumptions are not just sufficient but required
The paper is purely theoretical with no experimental results demonstrating, e.g., actual runtime improvements; approximation quality on practical model sizes; memory consumption compared to standard implementations
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Taxonomy
TopicsSparse and Compressive Sensing Techniques · Blind Source Separation Techniques · Neural Networks and Reservoir Computing
MethodsAttention Is All You Need · Linear Layer · Byte Pair Encoding · Absolute Position Encodings · Dense Connections · Multi-Head Attention · Position-Wise Feed-Forward Layer · Label Smoothing · Residual Connection · Adam