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We present a very simple algorithm for attention that requires $O(1)$ memorywith respect to sequence length and an extension to self-attention thatrequires $O(\log n)$ memory. This is in contrast with the frequently statedbelief that self-attention requires $O(n^2)$ memory. While the time complexityis still $O(n^2)$, device memory rather than compute capability is often thelimiting factor on modern accelerators. Thus, reducing the memory requirementsof attention allows processing of longer sequences than might otherwise befeasible. We provide a practical implementation for accelerators that requires$O(\sqrt{n})$ memory, is numerically stable, and is within a few percent of theruntime of the standard implementation of attention. We also demonstrate how todifferentiate the function while remaining memory-efficient. For sequencelength 16384, the memory overhead of self-attention is reduced by 59X forinference and by 32X for differentiation.

Self-attention Does Not Need $O(n^2)$ Memory