On The Computational Complexity of Self-Attention

TL;DR

This paper proves that the quadratic time complexity of self-attention in transformers is unavoidable under common complexity assumptions, even approximately, and explores approximation methods with trade-offs.

Contribution

It establishes rigorous lower bounds on self-attention complexity and demonstrates feasible approximation techniques with exponential trade-offs.

Findings

Quadratic complexity of self-attention is necessary unless SETH is false.

Approximate self-attention can be achieved in linear time using Taylor series.

Approximation incurs exponential dependence on the polynomial order.

Abstract

Transformer architectures have led to remarkable progress in many state-of-art applications. However, despite their successes, modern transformers rely on the self-attention mechanism, whose time- and space-complexity is quadratic in the length of the input. Several approaches have been proposed to speed up self-attention mechanisms to achieve sub-quadratic running time; however, the large majority of these works are not accompanied by rigorous error guarantees. In this work, we establish lower bounds on the computational complexity of self-attention in a number of scenarios. We prove that the time complexity of self-attention is necessarily quadratic in the input length, unless the Strong Exponential Time Hypothesis (SETH) is false. This argument holds even if the attention computation is performed only approximately, and for a variety of attention mechanisms. As a complement to our lower bounds, we show that it is indeed possible to approximate dot-product self-attention using finite Taylor series in linear-time, at the cost of having an exponential dependence on the polynomial order.