I’ve always been annoyed that I haven’t seen that much more specific quantitative modelling of RSI since I always found the chapter on the specifics of what might happen if we have diminishing returns in Superintelligence quite interesting.
As you’ve so kindly provided some general characteristics of the model here I thought I could expand on this analysis a bit through the perspective of Amdahl’s law.
I think people underestimate the difficulty that different types of serial time issues might potentially induce into a RSI process, especially given current architectures.
I had Claude do some calculations on this based on the approximate ideas. The biggest point is that serial time needs to be diminishing by a constant factor each time you loop for it to be super exponential. The collapsed section has the description and some pictures at the end of it.
Amdahl’s law calculation (Fable):
Amdahl’s law: if a fraction s of a job has to happen in order, then no matter how many workers you add, the job takes at least s times as long as it did. The ceiling on speedup is 1/s, full stop.
Your generation time is exactly such a job. A singularity needs it to go to zero. The way you’d shrink it is parallelism, more copies of the AI working on the next model, and Amdahl says that only takes you down to the serial part.
How much parallelism the loop can use. With serial fraction s and N parallel copies, the loop speeds up by 1/(s + (1−s)/N). At N = 1/s the serial and parallel parts take equal time and you’re at half the ceiling. At N = 10/s you’re at 90%. Past that, more copies do nothing for this loop. So a loop with a 10% serial part can use about a hundred copies’ worth of parallelism and no more; with 1%, a thousand.
What this does to the curve. In your terms, super-exponential means beating every exponential, which needs the generation time to go to zero. With a fixed s it can’t: the loop time falls while N climbs to about 10/s, then sits at s·T. So what looks like a super-exponential is a ramp lasting ten to twenty generations, after which growth is a plain exponential with doubling time about 1.7·s·T (for a 50% gain per generation). Parallelism buys you the ramp and nothing else.
To keep a super-exponential going you need the serial fraction itself to shrink every generation, and the parallelism to keep pace with it: whenever the serial part halves, you need twice the copies to feel it, so N ≳ 1/s at every step. Both, every generation. For a singularity the serial fraction must also shrink fast enough that the loop times add up to a finite total, which means by a roughly constant factor per generation rather than one step at a time.
That means every serial process in the loop has to be made parallel: a training run is a chain of steps, a reasoning trace is a chain of tokens, an experiment has to finish before you learn from it, a result has to be checked before it’s built on. And even then there’s a catch. Parallelising something is itself work with serial parts: you have to figure out how to split the job and then coordinate the pieces. So there’s a floor under the floor. Some serial time always seems to have to be there.
The general shape, then: Amdahl puts a lower bound on the generation time, the bound is set by whatever serial thing is still in the loop, and even small serial things bite because the cap is 1/s. If a tenth of the loop is stuck in order, you get 10× and no more. I’d guess today’s serial fraction is a lot bigger than a tenth once you count the training run.
I’ve always been annoyed that I haven’t seen that much more specific quantitative modelling of RSI since I always found the chapter on the specifics of what might happen if we have diminishing returns in Superintelligence quite interesting.
As you’ve so kindly provided some general characteristics of the model here I thought I could expand on this analysis a bit through the perspective of Amdahl’s law.
I think people underestimate the difficulty that different types of serial time issues might potentially induce into a RSI process, especially given current architectures.
I had Claude do some calculations on this based on the approximate ideas. The biggest point is that serial time needs to be diminishing by a constant factor each time you loop for it to be super exponential. The collapsed section has the description and some pictures at the end of it.
Amdahl’s law calculation (Fable):
Amdahl’s law: if a fraction s of a job has to happen in order, then no matter how many workers you add, the job takes at least s times as long as it did. The ceiling on speedup is 1/s, full stop.
Your generation time is exactly such a job. A singularity needs it to go to zero. The way you’d shrink it is parallelism, more copies of the AI working on the next model, and Amdahl says that only takes you down to the serial part.
How much parallelism the loop can use. With serial fraction s and N parallel copies, the loop speeds up by 1/(s + (1−s)/N). At N = 1/s the serial and parallel parts take equal time and you’re at half the ceiling. At N = 10/s you’re at 90%. Past that, more copies do nothing for this loop. So a loop with a 10% serial part can use about a hundred copies’ worth of parallelism and no more; with 1%, a thousand.
What this does to the curve. In your terms, super-exponential means beating every exponential, which needs the generation time to go to zero. With a fixed s it can’t: the loop time falls while N climbs to about 10/s, then sits at s·T. So what looks like a super-exponential is a ramp lasting ten to twenty generations, after which growth is a plain exponential with doubling time about 1.7·s·T (for a 50% gain per generation). Parallelism buys you the ramp and nothing else.
To keep a super-exponential going you need the serial fraction itself to shrink every generation, and the parallelism to keep pace with it: whenever the serial part halves, you need twice the copies to feel it, so N ≳ 1/s at every step. Both, every generation. For a singularity the serial fraction must also shrink fast enough that the loop times add up to a finite total, which means by a roughly constant factor per generation rather than one step at a time.
That means every serial process in the loop has to be made parallel: a training run is a chain of steps, a reasoning trace is a chain of tokens, an experiment has to finish before you learn from it, a result has to be checked before it’s built on. And even then there’s a catch. Parallelising something is itself work with serial parts: you have to figure out how to split the job and then coordinate the pieces. So there’s a floor under the floor. Some serial time always seems to have to be there.
The general shape, then: Amdahl puts a lower bound on the generation time, the bound is set by whatever serial thing is still in the loop, and even small serial things bite because the cap is 1/s. If a tenth of the loop is stuck in order, you get 10× and no more. I’d guess today’s serial fraction is a lot bigger than a tenth once you count the training run.
First picture:
Second Picture: