It will be interesting to see how far this quadratic algorithm carries in practice. Even the longest documents can only have hundreds of thousands of tokens, right?
The latter is definitely the cheapest option; updates are trivial.
GPT-4 seems to show that linear algebra definitely can do the job, but training is so expensive and the model gets so huge and inflexible.
It seems like having fixed format vectors of knowledge that the model can use-- denser and more precise than just incorporating tool results as tokens like OpenAI's plugin approach-- is a path forward towards extensibility and online learning.
There may be a middle ground between these two approaches though. If every query used the same prompt prefix (because you only update the codebase + docs occasionally) then you could put it into the model once and cache the keys and values from the attention heads. I wonder if OpenAI does this with whatever prefix they use for ChatGPT?