Tom_Davidson comments on Estimating the Substitutability between Compute and Cognitive Labor in AI Research

[Tom_Davidson]

Agree that i wouldn’t particularly expect the efficiency curves to be the same.

But if the phi>0 for both types of efficiency, then I think this argument will still go through.

To put it in math, there would be two types of AI software technology, one for experimental efficiency and one for cognitive labour efficiency: A_exp and A_cog. The equations are then:

dA_exp = A_exp^phi_exp F(A_exp K_res, A_cog K_inf)

dA_cog = A_cog^phi_cog F(A_exp K_res, A_cog K_inf)

And then I think you’ll find that, even with sigma < 1, it explodes when phi_exp>0 and phi_cog>0.

[Parker_Whitfill]

I spent a bit of time thinking about this today.

Lets adopt the notation in your comment and suppose that $F(\cdot )$ is the same across research sectors, with common $\lambda$. Let’s also suppose common $\sigma <1$.

Then we get blow up in $A_{cog}$ iff

$\{\begin{array}{cc}{\varphi }_{cog}+\lambda >1& \text{if}{\varphi }_{cog}\le {\varphi }_{exp}\\ max\{{\varphi }_{cog},{\varphi }_{exp}+\lambda \}>1& \text{if}{\varphi }_{cog}>{\varphi }_{exp}\end{array}$

The intution for this result is that when $\sigma <1$, you are bottlenecked by your slower growing sector.

If the slower growing sector is cognitive labor, then asympotically $F\propto A_{cog}$, and we get $\dot{A}\propto A_{cog}^{{\varphi }_{cog}}{A}_{cog}^{\lambda }$ so we have blow-up iff ${\varphi }_{cog}+\lambda >1$.

If the slower growing sector is experimental compute, then there are two cases. If experimental compute is blowing up on its own, then so is cogntive labor because by assumption cognitive labor is growing faster. If experimental compute is not blowing up on its own then asympotically $F\propto A_{exp}$ and we get ${\dot{A}}_{cog}\propto A_{cog}^{{\varphi }_{cog}}{A}_{exp}^{\lambda }$. Here we get a blow-up iff ${\varphi }_{cog}>1$.^[1]

In contrast, if $\sigma >1$ then F is approximately the fastest growing sector. You get blow-up in both sectors if either sector blows up. Therefore, you get blow-up iff $max\{{\varphi }_{cog}+\lambda ,{\varphi }_{exp}+\lambda \}>1$.

So if you accept this framing, complements vs substitutes only matters if some sectors are blowing up but not others. If all sectors have the returns to research high enough, then we get an intelligence explosion no matter what. This is an update for me, thanks!

1. ^

   I’m only analyzing blow-up conditions here. You could get e.g. double exponential growth here by having ${\varphi }_{cog}=1$ and ${\varphi }_{exp}+\lambda =1$.

[Tom_Davidson]

Nice!

I think that condition is equivalent to saying that A_cog explodes iff either

• phi_cog + lambda > 1 and phi_exp + lambda > 1, or

• phi_cog > 1

Where the second possibility is the unrealistic one where it could explode with just human input