Could the argument be rescued by some additional assumptions?
I agree with, and have previously also stated, the point you make there about “any non-auto-regressive model can be converted into an equivalent auto-regressive model by […]”, but, if one imposes additional restrictions on e.g. computation time, or something like that, I think that construction no longer works.
Well, of course there are some additional assumptions which would rescue the argument, so I guess my real question is whether there’s some combination of extra assumptions which both rescue the argument, and actually result in it being interesting.
If one makes the assumptions that there is a positive common lower bound on the probability of each token being incorrect assuming each previous token is correct, and that if any token is incorrect, then the whole generated text is incorrect, then of course the argument goes through, though the assumption doesn’t necessarily seem very likely.
Then, if we apply the construction, you mentioned to a text generation process with a low enough probability of error, then by the contrapositive, there cannot be an especially high common lower bound on the probability of error per token.
[“edit” prior to posting: I notice that at this point I started using symbols as if I was going to start doing actual algebraic manipulations, but did not actually do any algebraic manipulations which would justify the use of said symbols. I think what I wrote below would be clearer if I had just used words. Unfortunately I don’t want to take the time to rewrite it. I apologize for introducing formalisms without having a good reason to do so.]
If we have the assumption that there is a procedure with error rate and epsilon’(x) will be much smaller than epsilon(x), while t’(x) l(x) >> t(x) (at least, assuming l(x) is somewhat large).
So, that particular construction does not preclude the possibility that there is no algorithm that works auto-regressively and which both has an error rate(for overall generated text) as low as [the error rate for some non-auto-regressive model that runs quickly enough], and which runs quickly enough .
If there are cryptographically secure families of hashing functions (in the sense of, asymptotically in the size of the hash length, while the hash can be computed in polynomial time, finding preimages or collisions cannot be done in polynomial time) it seems that there should probably be functions from strings to strings which can be computed in time bounded above by some polynomial, but which can’t be computed autoregressively in time bounded above by a polynomial of the same degree.
(So like, maybe it can be computed in time 5n^4 when not autoregressive, but needs at least 2n^5 time to do auto regressively)
(I’m imagining something like, “compute a string of the form ‘hash(y), y’ where y is the result of some computation done on the input which takes a polynomial amount of time to compute from the input. So, the easiest way to compute this would be to compute y and the compute hash(y). So, to do this auto-regressively, it would need to compute y again for each token in the hash.)
Of course, a single factor of n might not be that compelling, and appealing to strong hashing functions is probably trying to kill a fly with a sledgehammer(probably there are arguments that work as well without assuming this), but it’s what came to mind.
Perhaps one could do something like this to show that for some problems, any auto-regressive solution that has certain runtime bounds, will have some positive lower bound on the error rate per token?