Error-Free Linear Attention is a Free Lunch: Exact Solution from Continuous-Time Dynamics
Abstract
Error-Free Linear Attention (EFLA) is a stable, parallelizable, and theoretically sound linear-time attention mechanism that outperforms DeltaNet in language modeling and downstream tasks.
Linear-time attention and State Space Models (SSMs) promise to solve the quadratic cost bottleneck in long-context language models employing softmax attention. We introduce Error-Free Linear Attention (EFLA), a numerically stable, fully parallelism and generalized formulation of the delta rule. Specifically, we formulate the online learning update as a continuous-time dynamical system and prove that its exact solution is not only attainable but also computable in linear time with full parallelism. By leveraging the rank-1 structure of the dynamics matrix, we directly derive the exact closed-form solution effectively corresponding to the infinite-order Runge-Kutta method. This attention mechanism is theoretically free from error accumulation, perfectly capturing the continuous dynamics while preserving the linear-time complexity. Through an extensive suite of experiments, we show that EFLA enables robust performance in noisy environments, achieving lower language modeling perplexity and superior downstream benchmark performance than DeltaNet without introducing additional parameters. Our work provides a new theoretical foundation for building high-fidelity, scalable linear-time attention models.
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Error-Free Linear Attention is a Free Lunch!
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I'm trying to understand the causality here: Does DeltaNet just happen to look like an Euler solution? Or is the premise that Linear Attention should be defined as an ODE solution in the first place, and we have simply been using the Euler method to approximate it (resulting in DeltaNet)?
I am not deeply familiar with the fundamental theory of linear attention, so I might be incorrect here.
https://kexue.fm/archives/11486
you can refer this, Jianlin Su has provided a very accurate comment on our paper
That's an important question. In EFLA, we showed that linear attention can be considered as an ODE and its natural Euler solution is equal to deltanet equation. If you notice (and also covered in a famous Chinese blog), this derivation brings L2 norm naturally in the equation.
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