alt Hacker NewsWell, reliably but still with a chance of failure - in the same way that you can have a program which is provably correct but can still run into real world issues like being killed, but yes I would say that "almost surely" is a pretty large jump from "more than likely" (50%+1) where I'd say LLM output generally lives these days.
> a chance of failure
Well, technically, no chance of failure. The chance of failure is absolute zero. Not close to zero, absolute zero. There will be no failure if the assumptions of the model are correct.
The real catch here is in the assumptions.
How long do you have before you need to have a solution? An hour, a year, a century? Too bad, almost sure convergence only provides a guarantee if you wait an infinite amount of time.
And then there's the question of the probability space you assume. (The sigma algebra.) Which things do you assume to have probability zero from the start and is that realistic?