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

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.

Although note that this argument works only with the CES in compute formulation. For the CES in frontier experiments, you would have the $\frac{A{K}_{res}}{A{K}_{train}}$ so the A cancels out.

Yep, as you say in your footnote, you can choose to freeze the frontier, so you train models of a fixed capability using less and less compute (at least for a while). 

However, if $\sigma <1$, then a software-only intelligence explosion occurs only if $\varphi >1$. But if this condition held, we could get an intelligence explosion with constant, human-only research input. While not impossible, we find this condition fairly implausible. 

Hmm, I think a software-only intelligence explosion is plausible even if  $\sigma <1$, but without the implication that you can do it with human-only research input.

The basic idea is that when you double the efficiency of software, you can now:

• Run twice as many experiments
• Have twice as much cognitive labour

So both the inputs to software R&D double.

I think this corresponds to:

dA = A^phi F(A K_res, A K_inf)

And then you only need phi > 0 to get an intelligence explosion. Not phi > 1.

This is really an explosion in the efficiency at which you can run AI algorithms, but you could do that for a while and then quickly use your massive workforce to develop superintelligence, or start training your ultra-efficient algorithms using way more compute.

Thanks for this!

Let me try and summarise what I think is the high-level dynamic driving the result, and you can correct me if I'm confused.

CES in compute.

Compute has become cheaper while wages have stayed ~constant. The economic model then implies that:

• If compute and labour were complements, then labs would spend a greater fraction of their research budgets on labour. (This prevents labour from becoming a bottleneck as compute becomes cheaper.)

Labs aren't doing this, suggesting that compute and labour are substitutes. 

CES in frontier experiments.

Frontier experiments have become more expensive while wages have stayed ~constant. The economic model then implies that:

• If compute and labour were complements, then labs would spend a greater fraction of their research budgets on compute. (This relieves the key bottleneck of expensive frontier experiments.)

Labs are indeed doing this, suggesting that compute and labour are indeed complements. 

(Though your 'Research compute per employee' data shows they're not doing that much since 2018, so the argument against the intelligence explosion is weaker here than I'd have expected.)

The $\sigma >1$ condition is exactly what Epoch and Forethought consider when they analyze whether the returns to research are high enough for a singularity.^[5]

Though we initially consider this, we then adjust for compute as an input to R&D and so end up considering the sigma=1 condition. It's under that condition that I think it's more likely than not that the condition for a software-only intelligence explosion holds

I like the vividness of the comparisons!

A few points against this being nearly as crazy as the comparisons suggest:

• GPT-2030 may learn much less sample efficiently, and much less compute efficiently, than humans. In fact, this is pretty likely. Ball-parking, humans do 1e24 FLOP before they're 30, which is ~20X less than GPT-4. And we learn languages/maths from way fewer data points. So the actual rate at which GPT-2030 itself gets smarter will be lower than the rates implied. 
  • This is a sense of "learn" as in "improves its own understanding". There's another sense which is "produces knowledge for the rest of the world to use, eg science papers" where I think your comparisons are right. 
• Learning may be bottlenecked by serial thinking time past a certain point, after which adding more parallel copies won't help. This could make the conclusion much less extreme.
• Learning may also be bottlenecked by experiments in the real world, which may not immediately get much faster.

Thanks, this is a great comment.

The first and second examples seems pretty good, and useful reference points.

The third example don't seem like they are nearly as useful though. What's particularly unusual about this case is that there are two useful inputs to AI R&D -- cognitive labour and compute for experiments -- and the former will rise very rapidly but the other will not. In particular, I imagine CS departments also saw compute inputs growing in that time. And I imagine some of the developments discussed (eg proofs about algorithms) only have cognitive labour as an input.

The second example (quant finance), I suppose the 'data' input to doing this work stayed constant while the cognitive effort rose. So it works as an example. Though it may be a field with an unusual superabundance of data, unlike ML. 

The first example involves a kind of 'data overhang' that the cognitive labour quickly eats up. Perhaps in a similar way AGI will "eat up" all the insights that are implicit in existing data from ML experiments.

What i think all the examples currently lack is a measure of how the pace of overall progress changed that isn't completely made up. Could be interested to list out the achievements in each time period and ask some experts what they think. There an interesting empirical project here I think. 

All the examples also lack anything like the scale to which cognitive labour will increase with AGI. This makes comparison even harder. (Though if we can get 3X speed-ups from mild influxes of cognitive labour, that makes 10X speed ups more plausible.)

I tried to edit the paragraph (though LW won't let me) to: 

I think we don't know what perspective is right,  we haven't had many examples where a huge amount of cognitive labour has been dumped on a scientific field and  other inputs to progress have remained constant and we've accurately measured how much overall progress in that field accelerates. (Edit: though this comment suggests some interesting examples.) 

•  I think utilitarianism is often a natural generalization of "I care about the experience of XYZ, it seems arbitrary/dumb/bad to draw the boundary narrowly, so I should extend this further" (This is how I get to utilitarianism.) I think the AI optimization looks considerably worse than this by default.

Why is this different between AIs and humans? Do you expect AIs to care less about experience than humans, maybe bc humans get reward during life-time learning about AIs don't get reward during in context learning?