How Sparse Attention Approximates Exact Attention? Your Attention is Naturally $n^C$-Sparse
This provides foundational insights for efficient LLM deployment by theoretically justifying sparse attention methods, though it is incremental as it builds on existing techniques.
The paper tackles the lack of theoretical understanding of when sparse attention approximates exact attention in LLMs, showing that attention is inherently n^C-sparse, allowing approximation with only the largest Ω(n^C) entries for decreasing loss, and that stable o(log(n))-sparse attention is infeasible due to persistent error.
Sparse Attention is a technique that approximates standard attention computation with sub-quadratic complexity. This is achieved by selectively ignoring smaller entries in the attention matrix during the softmax function computation. Variations of this technique, such as pruning KV cache, sparsity-based fast attention, and Sparse Transformer, have been extensively utilized for efficient Large Language Models (LLMs) deployment. Despite its widespread use, a theoretical understanding of the conditions under which sparse attention performs on par with traditional attention remains elusive. This work aims to $\textbf{bridge this gap by examining the inherent sparsity of standard attention processes}$. Our theoretical framework reveals several brand-new key insights: $\bullet$ Attention is $n^{C}$-sparse, implying that considering only the largest $Ω(n^{C})$ entries out of all $n$ entries is sufficient for sparse attention to approximate the exact attention matrix with decreasing loss. Here, $n$ represents the input length and $C \in (0, 1)$ is a constant. $\bullet$ Stable $o(\log(n))$-sparse attention, which approximates attention computation with $\log(n)$ or fewer entries, may not be feasible since the error will persist at a minimum of $O(1)$. $\bullet$ An adaptive strategy ($α\cdot n^C, α\in \mathbb{R}$) for the window size of efficient attention methods rather than a fixed one is guaranteed to perform more accurately and efficiently in a task for inference on flexible context lengths.