Subquadratic Algorithms and Hardness for Attention with Any Temperature

Despite the popularity of the Transformer architecture, the standard algorithm for computing Attention suffers from quadratic time complexity in context length $n$. Alman and Song showed that when the head dimension $d = \Theta(\log n)$, subquadratic Attention is possible if and only if the inputs have small entries bounded by $B = o(\sqrt{\log n})$ in absolute values, under the Strong Exponential Time Hypothesis ($\mathsf{SETH}$). Equivalently, subquadratic Attention is possible if and only if the softmax is applied with high temperature for $d=\Theta(\log n)$. Running times of these algorithms depend exponentially on $B$ and thus they do not lead to even a polynomial-time algorithm outside the specific range of $B$. This naturally leads to the question: when can Attention be computed efficiently without strong assumptions on temperature? Are there fast attention algorithms that scale polylogarithmically with entry size $B$? In this work, we resolve this question and characterize when fast Attention for arbitrary temperatures is possible. First, for all constant $d = O(1)$, we give the first subquadratic $\tilde{O}(n^{2 - 1/d} \cdot \mathrm{polylog}(B))$ time algorithm for Attention with large $B$. Our result holds even for matrices with large head dimension if they have low rank. Combined with a reduction from Gradient Computation to Attention, we obtain a subquadratic algorithm for the full LLM training process. Furthermore, we show that any substantial improvement on our algorithm is unlikely. In particular, we show that even when $d = 2^{\Theta(\log^* n)}$, Attention requires $n^{2 - o(1)}$ time under $\mathsf{SETH}$. Finally, in the regime where $d = \mathrm{poly}(n)$, the standard algorithm requires $O(n^{2} d)$ time while previous lower bounds only ruled out algorithms with truly subquadratic time in $n$. We close this gap and show that the standard algorithm is optimal under popular fine-grained complexity assumptions.