I thought about this a bit more.
tldr: Under a simple model and reasonable assumptions, if we automate AI R&D and compute growth stays constant then the pace of AI software progress is 3-5X faster. This means the pace of overall AI progress would be 2-3X faster.
Assume AI software R&D is Cobb Douglas:
g_S = L^alpha E^beta S^-1/r
L = cognitive labour
E = experimental compute
r governs ideas getting harder to find as you ramp up both cognitive labour and experimental compute. (Note I’ve often confusingly defined r as the returns when you just ramp up cognitive labour—really I should call that r_cog = r * alpha.)
In this system, before automating away humans, it turns out that:
if L grows exponentially at g_L then (holding E fixed) g_S = g_L * alpha * r
if E grows exponentially at g_E then (holding L fixed) g_S = g_E * beta * r
When you automate AI R&D, L = S. In this situation, it turns out that:
if L grows exponentially at g_L (eg due to exogenously increasing compute), then:
g_S = g_L * alpha * r / (1 - alpha * r)And if r * alpha > 1 we get a software-only intelligence explosion!
if E grows exponentially at g_E (eg due to exogenously increasing compute), S grows at:
g_S = g_E * beta * r / (1 - alpha * r)
In other words, even absent an SIE, automating AI R&D boosts the standard growth rates by a factor of 1 / (1 - alpha * r) due to the fizzling feedback loop of “better software → better AI researchers → better software”.
This model allows us to ballpark how much faster overall AI progress would be in a regime with full automation but no SIE.
That regime causes two changes:
g_L gets faster. Firstly, compute growth is somewhat faster than the growth of human AI researchers. Secondly, L is superlinear in compute bc you can run faster and smarter models with more compute.
All growth rates are boosted by a factor of 1 / (1 - alpha * r). Ryan used an estimate of r_cog = r * alpha = 0.7. So this is a boost by a factor of 1 / (1 − 0.7) = ~3.
Concretely, let’s make the following assumptions:
g_L = g_E today (conservatively assume compute and human researchers have grown at the same pace)
alpha = beta = 0.5
r = 1.4 (so that we recover Ryan’s r * alpha = 0.7)
After automating AI R&D, L grows twice as fast as compute: g_L = 2 * g_E
Then today the total pace of software progress is :
g_S = progress due to growing labour + progress due to growing compute
= g_E * alpha * r + g_E * beta * r
= g_E * r
And after AI R&D automation g_L is 2X faster and everything gets boosted by a factor of 3. The pace of software progress is:
g_S = progress due to growing labour + progress due to growing compute
= (2 * g_E * alpha * r + g_E * beta * r)*3
= 4.5 * g_E * r
So that’s 4.5x faster software progress, holding the rate of compute growth constant! If software and compute contribute equally to AI progress, that’s a bit under 3x faster total AI progress.
I think not too surprising, given we’re using an r*alpha value close to 1.
If we used r*alpha =0.5, our boost factor shrinks to 2 and we’d get 3x faster software progress and 2x faster total AI progress.
Cool! I think the UK could play a leading role here, convening a middle powers coalition and leading on its strategy