Integer multiplication x * y can be trivially done in O(k): k = log₂(min(x, y)). This is because we can do addition in constant time, adding all bits in parallel.
replyBy combining many more adding units, we can do (fixed-size) multiplication in constant time, too: https://en.wikipedia.org/wiki/Dadda_multiplier
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replyDoesn't that have to do with how many bits you allow in the actual calculation in physical reality?
replyWell, for multiplication complexity is defined in terms of on the number of digits/bits digits directly. For attention, complexity is defined on terms of the number of input vectors which are all at fixed precision. I don't understand what happens to the method proposed in the paper at higher precision (since I don't understand the paper), but in reality in doesn't matter since there is no value in anything over float16 for machine learning.
replyMultiplication has some properties like being cumulative. If we assume the sequence has any specific properties then we no longer have a general sequence model.
replyI think you meant commutative.
replyAttention also has some specific properties.
And sometimes results are just unexpected. Did you know that anything a Turing machine can do in t tome steps, a different Turing machine can do in O(sqrt(t log t)) memory cells? https://news.ycombinator.com/item?id=44055347