Well, if you are dealing with an adversarial situation against an equal or stronger opponent, the NFL implies that you should plan for the worst case, not a likely or average or median case. Unless I understand it wrong.
That gives the whole thing more credit than it deserves. The NFL theorem really only works with a flat prior and it that case the NFL theorem shows you that you have already lost (every policy does (in expectation) as well as any other). So this prior should actually have 0 influence on your policy. It’s self-defeating if you are the least bit unsure about it, similar to nihilism as a moral code.
Yes, but in worlds where not every sequence {0,1} * is equally likely (eg, your possible worlds have ANY structure) there will be predictors that outperform random predictors (like AIXI for example).
(this is not literally true up to maximum pedantry (eg. there are infinitely measures on all languages where AIXI/solomonoff induction never works, but for all of those see my other comment))
Well… I don’t know about you, but even if I believed that the most likely explanation for my observations was that I am a boltzmann brain, my current beliefs will lead me to effectively act as if I have 0 crecedence in that belief (since these worlds have no implications for my policy). As long as I put 0 value on this frame, I can actually discard it even if I have knightian uncertainty about which is the right prior to use (Logical uncertainty makes this more complicated than it needs to be and I think the basic point still stands. I am basically appealing to pragmatism).
This might not apply to every theorem that has ever been called NFL theorem. I think that what I wrote is true for the stuff that Wolpert shows in this paper.
So it’s about how adversarial inputs can produce maximally wrong answers? Wouldn’t the best policy in that case just be to ignore adversarial inputs and rely entirely on your priors?
Well, if you are dealing with an adversarial situation against an equal or stronger opponent, the NFL implies that you should plan for the worst case, not a likely or average or median case. Unless I understand it wrong.
That gives the whole thing more credit than it deserves. The NFL theorem really only works with a flat prior and it that case the NFL theorem shows you that you have already lost (every policy does (in expectation) as well as any other). So this prior should actually have 0 influence on your policy. It’s self-defeating if you are the least bit unsure about it, similar to nihilism as a moral code.
It works with any prior! “If you assign more than the prior to anything, you must assign less than it to something.”
Yes, but in worlds where not every sequence {0,1} * is equally likely (eg, your possible worlds have ANY structure) there will be predictors that outperform random predictors (like AIXI for example). (this is not literally true up to maximum pedantry (eg. there are infinitely measures on all languages where AIXI/solomonoff induction never works, but for all of those see my other comment))
Well… I don’t know about you, but even if I believed that the most likely explanation for my observations was that I am a boltzmann brain, my current beliefs will lead me to effectively act as if I have 0 crecedence in that belief (since these worlds have no implications for my policy). As long as I put 0 value on this frame, I can actually discard it even if I have knightian uncertainty about which is the right prior to use (Logical uncertainty makes this more complicated than it needs to be and I think the basic point still stands. I am basically appealing to pragmatism).
This might not apply to every theorem that has ever been called NFL theorem. I think that what I wrote is true for the stuff that Wolpert shows in this paper.
Thanks!
So it’s about how adversarial inputs can produce maximally wrong answers? Wouldn’t the best policy in that case just be to ignore adversarial inputs and rely entirely on your priors?