Problems vary on three axes: uâ(R), R, u_total. Youâre expressing this in the basis uâ(R), R, u_total. The ITN framework uses the basis I, T, N = u_total, uâ(R) * R * 1/â u_total, 1/âR. The basis is arbitrary: we could just as easily use some crazy basis like X, Y, Z = u_total^2, R^5, sqrt(uâ(R)). But we want to use a basis thatâs useful in practice, which means variables with intuitive meanings, hence ITN.
But why is tractability roughly constant with neglectedness in practice? Equivalently, why are there logarithmic returns to many problems? I donât think itâs related to your (1) or (2) because those are about comparing different problems, whereas the mystery is the relationship between uâ(R) * R * 1/â u_total and 1/âR for a given problem. One model that suggests log returns is if we have to surpass some unknown resource threshold r* (the âdifficultyâ) to solve the problem, and r* ranges over many orders of magnitude with an approximately log-uniform distribution [1]. Owen C-B has empirical evidence and some theoretical justification for why this might happen in practice. When it does, my post then derives that tractability doesnât vary dramatically between (most) problems.
Note that sometimes we know r* or have a different prior, and then our problem stops being logarithmic, like in the second section of the post. This is exactly when tractability can vary dramatically between problems. In your AI alignment subproblem example, we know that alignment takes 20 years, which means a strong update away from the logarithmic prior.
[1]: log-uniform distributions over the reals donât exist, so I mean something like âdoesnât vary by more than ~1.5x every doubling in the fat part of the distributionâ.
The entire time Iâve been thinking about this, Iâve been thinking of utility curves as logarithmic, so you donât have to sell me on that. I think my original comment here is another way of understanding why tractability perhaps doesnât vary much between problems, not within a problem.
Problems vary on three axes: uâ(R), R, u_total. Youâre expressing this in the basis uâ(R), R, u_total. The ITN framework uses the basis I, T, N = u_total, uâ(R) * R * 1/â u_total, 1/âR. The basis is arbitrary: we could just as easily use some crazy basis like X, Y, Z = u_total^2, R^5, sqrt(uâ(R)). But we want to use a basis thatâs useful in practice, which means variables with intuitive meanings, hence ITN.
But why is tractability roughly constant with neglectedness in practice? Equivalently, why are there logarithmic returns to many problems? I donât think itâs related to your (1) or (2) because those are about comparing different problems, whereas the mystery is the relationship between uâ(R) * R * 1/â u_total and 1/âR for a given problem. One model that suggests log returns is if we have to surpass some unknown resource threshold r* (the âdifficultyâ) to solve the problem, and r* ranges over many orders of magnitude with an approximately log-uniform distribution [1]. Owen C-B has empirical evidence and some theoretical justification for why this might happen in practice. When it does, my post then derives that tractability doesnât vary dramatically between (most) problems.
Note that sometimes we know r* or have a different prior, and then our problem stops being logarithmic, like in the second section of the post. This is exactly when tractability can vary dramatically between problems. In your AI alignment subproblem example, we know that alignment takes 20 years, which means a strong update away from the logarithmic prior.
[1]: log-uniform distributions over the reals donât exist, so I mean something like âdoesnât vary by more than ~1.5x every doubling in the fat part of the distributionâ.
The entire time Iâve been thinking about this, Iâve been thinking of utility curves as logarithmic, so you donât have to sell me on that. I think my original comment here is another way of understanding why tractability perhaps doesnât vary much between problems, not within a problem.
I donât see why logarithmic utility iff tractability doesnât change with neglectedness.