In complete generality, you could write effective labor as
L=Z(H,AKinf,AKtrain).
That is, effective labor is some function of the number of human researchers we have, the effective inference compute we have (quantity of AIs we can run) and the effective training compute (quality of AIs we trained).
The perfect substitution claim is that once training compute is sufficiently high, then eventually we can spend the inference compute on running some AI that substitutes for human researchers. Mathematically, for some x,
Z(H,AKinf,x)=H+AKinfc
where c is the compute cost to run the system.
So you could think of our analysis as saying, once we have an AI that perfectly substitutes for AI researchers, what happens next?
Now of course, you might expect substantial recursive self-improvement even with an AI system that doesn’t perfectly substitute for AI labor. I think this is a super interesting and important question. I’m trying to think more about this question, but its hard to make progress because its unclear what Z(⋅) looks like. But let me try to gesture at a few things. Let’s fix x at some sub-human level
At the very least, you need some function that goes to infinity as A goes to infinity. For example, if there are certain tasks which must be done in AI research and these tasks can only be done by humans, then these tasks will always bottleneck progress.
If you assume say Cobb-Douglas, i.e.
Z(H,AKinf,x)=Hα(x)AKinfc1−a(x)
where 1−α(x) denotes the share of labor tasks that AI can do, then you’ll pick up another 1−α(x) in the explosion condition i.e.ϕ+λ>1 will become ϕ+(1−α(x))λ>1. This captures the intuition that as the fraction of tasks an AI can do increases, the explosion condition gets easier and easier to hit.
In complete generality, you could write effective labor as
L=Z(H,AKinf,AKtrain).
That is, effective labor is some function of the number of human researchers we have, the effective inference compute we have (quantity of AIs we can run) and the effective training compute (quality of AIs we trained).
The perfect substitution claim is that once training compute is sufficiently high, then eventually we can spend the inference compute on running some AI that substitutes for human researchers. Mathematically, for some x,
Z(H,AKinf,x)=H+AKinfc
where c is the compute cost to run the system.
So you could think of our analysis as saying, once we have an AI that perfectly substitutes for AI researchers, what happens next?
Now of course, you might expect substantial recursive self-improvement even with an AI system that doesn’t perfectly substitute for AI labor. I think this is a super interesting and important question. I’m trying to think more about this question, but its hard to make progress because its unclear what Z(⋅) looks like. But let me try to gesture at a few things. Let’s fix x at some sub-human level
At the very least, you need some function that goes to infinity as A goes to infinity. For example, if there are certain tasks which must be done in AI research and these tasks can only be done by humans, then these tasks will always bottleneck progress.
If you assume say Cobb-Douglas, i.e.
Z(H,AKinf,x)=Hα(x)AKinfc1−a(x)
where 1−α(x) denotes the share of labor tasks that AI can do, then you’ll pick up another 1−α(x) in the explosion condition i.e.ϕ+λ>1 will become ϕ+(1−α(x))λ>1. This captures the intuition that as the fraction of tasks an AI can do increases, the explosion condition gets easier and easier to hit.
Thanks for this Parker. I continue to think this research is insanely cool.
I agree with 1)
On 2), the condition you find makes sense, but aren’t you implicitly assuming an elasticity of substitution of 1 with Cobb-Douglas?
Could be interesting to compare with Aghion et al. 2017. They look at a CES case with imperfect substitution (i.e., humans needed for some tasks).
https://www.nber.org/system/files/working_papers/w23928/w23928.pdf
Yes, definitely. In general, I don’t have a great idea about what Z(⋅) looks like. The Cobb-Douglas case is just an example.