Hacker News Reader

Comment by voidmain | Hacker News Reader

The argument of this very long and technical paper seems to boil down to this:

Solomonoff induction is (in the limit) as good as any computable prediction method, which sounds pretty "universal". But it is not itself a computable prediction method (it's just semi-computable). And if you are satisfied with semi-computability in a predictor, then you should demand instead that a "universal" predictor always be as good as any semi- computable predictor, and it can't meet that more demanding requirement. And so on in infinite regress.

I find this kind of underwhelming. Solomonoff induction represents, in practice, an unattainable ideal that prediction systems can aspire to approximate. This argument shows that if you somehow could reach the ideal, there is an infinite regress of super-ideals beyond it. So what?

View on HN

I think you did a better job than all other commenters here (who seem to be mostly free-associating on the title and abstract), but your comment still doesn't quite capture the contribution of the paper or the reasonable takeaways from it.

The key is section 4.3.3, which draws a distinction between measures and predictors. The Solomonoff lower-semicomputable semimeasure is universal for all lower-semicomputable semimeasures, not just for some narrower class. But if we forget the idea of lower-semicomputable semimeasures and treat Solomonoff as a member of a wider class - limit-computable predictors - it turns out to be not universal for that class. In fact, moving from measures to predictors is so lossy that no predictor can be universal for its own class, by a simple diagonal argument. To me that means Solomonoff's approach with measures is more nuanced and informative than the approach with predictors.

In my admittedly very cursory skim of the paper, I gave it the benefit of the doubt on this point, because this seemed convincing:

"for the purpose of prediction, we are, of course, not so much interested in the probabilities issued by the measure functions, as in the conditional probabilities that give the corresponding predictors’ outputs."

Do you disagree?

(Also, my original summary is wrong at least in that I should have said limit computable where I said semi computable!)

I guess the question is whether we want universality over all limit-computable predictors, or only those derived from lower-semicomputable semimeasures. Since the latter class contains its own universal element and also every computable predictor, maybe it's okay? But yeah, I can see how you can have a different takeaway. Sorry for being a bit harsh in the previous comment.

No need to apologize, I thought your comment was perfectly civil. It can be hard to read tone on the internet!

Reply on news.ycombinator.com