This post examines simple models of recursive self-improvement, where intelligence is the derivative of knowledge, but intelligence is also some function of knowledge (since knowledge can be applied to improve intelligence). It concludes that growth in intelligence is sublinear so long as returns from knowledge diminish faster than √x; subexponential so long as returns are diminishing at all; exponential precisely when returns are linear; and superexponential (having a singularity at finite time) if returns increase like some polynomial.
In a comment there, I argue that it makes more sense to think in terms of the growth in capabilities, rather than the growth in intelligence; making that shift, it seems like almost any assumption gives you superlinear growth, but the crossover to superexponential is still at the same spot.
This seems like a wonderful exploration of more sophisticated versions of the model discussed in 1960: the year the singularity was cancelled. A quick glance suggests that it doesn’t make the modification I was interested in exploring, but I have not read it thoroughly yet.
Takeoff Speed: Simple Asymptotics in a Toy Model.
This post examines simple models of recursive self-improvement, where intelligence is the derivative of knowledge, but intelligence is also some function of knowledge (since knowledge can be applied to improve intelligence). It concludes that growth in intelligence is sublinear so long as returns from knowledge diminish faster than √x; subexponential so long as returns are diminishing at all; exponential precisely when returns are linear; and superexponential (having a singularity at finite time) if returns increase like some polynomial.
In a comment there, I argue that it makes more sense to think in terms of the growth in capabilities, rather than the growth in intelligence; making that shift, it seems like almost any assumption gives you superlinear growth, but the crossover to superexponential is still at the same spot.
Modeling the Human Trajectory
This seems like a wonderful exploration of more sophisticated versions of the model discussed in 1960: the year the singularity was cancelled. A quick glance suggests that it doesn’t make the modification I was interested in exploring, but I have not read it thoroughly yet.