Algorithm and Hardness for Dynamic Attention Maintenance in Large Language Models
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Large language models (LLMs) have made fundamental changes in human life. The attention scheme is one of the key components over all the LLMs, such as BERT, GPT-1, Transformers, GPT-2, 3, 3.5 and 4. Inspired by previous theoretical study of static version of the attention multiplication problem [Zandieh, Han, Daliri, and Karbasi arXiv 2023, Alman and Song arXiv 2023]. In this work, we formally define a dynamic version of attention matrix multiplication problem. There are matrices Q,K, V ∈ℝ^n × d, they represent query, key and value in LLMs. In each iteration we update one entry in K or V. In the query stage, we receive (i,j) ∈ [n] × [d] as input, and want to answer (D^-1 A V)_i,j, where A:=exp(QK^⊤) ∈ℝ^n × n is a square matrix and D := diag(A 1_n) ∈ℝ^n × n is a diagonal matrix. Here 1_n denote a length-n vector that all the entries are ones. We provide two results: an algorithm and a conditional lower bound. ∙ On one hand, inspired by the lazy update idea from [Demetrescu and Italiano FOCS 2000, Sankowski FOCS 2004, Cohen, Lee and Song STOC 2019, Brand SODA 2020], we provide a data-structure that uses O(n^ω(1,1,τ)-τ) amortized update time, and O(n^1+τ) worst-case query time. ∙ On the other hand, show that unless the hinted matrix vector multiplication conjecture [Brand, Nanongkai and Saranurak FOCS 2019] is false, there is no algorithm that can use both O(n^ω(1,1,τ) - τ- Ω(1)) amortized update time, and O(n^1+τ-Ω(1)) worst query time. In conclusion, our algorithmic result is conditionally optimal unless hinted matrix vector multiplication conjecture is false.
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