Preconditioned Attention: Enhancing Efficiency in Transformers

arXiv cs.LG / 3/31/2026

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Key Points

The paper argues that standard transformer attention can form ill-conditioned attention matrices with large condition numbers, which hinders efficient gradient-based optimization during training.
It introduces “preconditioned attention,” which adds a conditioning matrix within each attention head to reduce the attention matrix condition number.
Theoretical results show the proposed conditioning improves matrix conditioning, which is expected to make optimization more effective.
Preconditioned attention is designed as a simple drop-in replacement for many existing attention variants in the literature.
Experiments across multiple tasks—including vision (classification, detection, segmentation), long-sequence modeling, and language modeling—validate that the approach improves training efficiency and effectiveness.

Abstract

Central to the success of Transformers is the attention block, which effectively models global dependencies among input tokens associated to a dataset. However, we theoretically demonstrate that standard attention mechanisms in transformers often produce ill-conditioned matrices with large condition numbers. This ill-conditioning is a well-known obstacle for gradient-based optimizers, leading to inefficient training. To address this issue, we introduce preconditioned attention, a novel approach that incorporates a conditioning matrix into each attention head. Our theoretical analysis shows that this method significantly reduces the condition number of attention matrices, resulting in better-conditioned matrices that improve optimization. Conditioned attention serves as a simple drop-in replacement for a wide variety of attention mechanisms in the literature. We validate the effectiveness of preconditioned attention across a diverse set of transformer applications, including image classification, object detection, instance segmentation, long sequence modeling and language modeling.