(posting this so ideas from our chat can be public)
Ex ante, the tractability range is narrower than 2 orders of magnitude unless you have really strong evidence. Say you’re a high school student presented with a problem of unknown difficulty, and you’ve already spent 100 hours on it without success. What’s the probability that you solve it in the next doubling?
Obviously less than 100%
Probably more than 1%, even if it looks really hard—you might find some trick that solves it!
And you have to have a pretty strong indication that it’s hard (e.g. using concepts you’ve tried and failed to understand) to even put your probability below 3%.
There can be evidence that it’s really hard (<0.1%), maybe for problems like “compute tan(10^123) to 9 decimal places” or “solve this problem that John Conway failed to solve”. This means you’ve updated away from your ignorance prior (which spans many orders of magnitude) and now know the true structure of the problem, or something.
If I’ve spent 100 hours on a (math?) problem without success as a high school student and can’t get hints or learn new material (or have already tried those and failed), then I don’t think less than 1% to solving it in the next 100 hours is unreasonable. I’d probably already have exhausted all the tools I know of by then. Of course, this depends on the person.
The time and resources you (or others) spent on a problem without success (or substantial progress) are evidence for its intractability.
I was going to come back to this and write a comment saying why I either agree or disagree and why, but I keep flipping back and forth.
I now think there are some classes of problems for which I could easily get under 1%, and some for which I can’t, and this partially depends on whether I can learn new material (if I can, I think I’d need to exhaust every promising-looking paper). The question is which is the better reference class for real problems.
You could argue that not learning new material is the better model, because we can’t get external help in real life. But on the other hand, the large action space of real life feels more similar to me to a situation in which we can learn new material—the intuition that the high school student will just “get stuck” seems less strong with an entire academic subfield working on alignment, say.
(posting this so ideas from our chat can be public)
Ex ante, the tractability range is narrower than 2 orders of magnitude unless you have really strong evidence. Say you’re a high school student presented with a problem of unknown difficulty, and you’ve already spent 100 hours on it without success. What’s the probability that you solve it in the next doubling?
Obviously less than 100%
Probably more than 1%, even if it looks really hard—you might find some trick that solves it!
And you have to have a pretty strong indication that it’s hard (e.g. using concepts you’ve tried and failed to understand) to even put your probability below 3%.
There can be evidence that it’s really hard (<0.1%), maybe for problems like “compute tan(10^123) to 9 decimal places” or “solve this problem that John Conway failed to solve”. This means you’ve updated away from your ignorance prior (which spans many orders of magnitude) and now know the true structure of the problem, or something.
If I’ve spent 100 hours on a (math?) problem without success as a high school student and can’t get hints or learn new material (or have already tried those and failed), then I don’t think less than 1% to solving it in the next 100 hours is unreasonable. I’d probably already have exhausted all the tools I know of by then. Of course, this depends on the person.
The time and resources you (or others) spent on a problem without success (or substantial progress) are evidence for its intractability.
I was going to come back to this and write a comment saying why I either agree or disagree and why, but I keep flipping back and forth.
I now think there are some classes of problems for which I could easily get under 1%, and some for which I can’t, and this partially depends on whether I can learn new material (if I can, I think I’d need to exhaust every promising-looking paper). The question is which is the better reference class for real problems.
You could argue that not learning new material is the better model, because we can’t get external help in real life. But on the other hand, the large action space of real life feels more similar to me to a situation in which we can learn new material—the intuition that the high school student will just “get stuck” seems less strong with an entire academic subfield working on alignment, say.