- log(n) + log(log(n)) + … seems to describe well the current rate of scientific progress, at least in high-energy physics
I’m going to commit pedantry: nesting enough logarithms eventually gives an undefined term (unless n’s complex!). So where Eliezer says “the sequence log(w) + log(log(w)) + log(log(log(w))) will converge very quickly” (p. 4), that seems wrong, although I see what he’s getting at.
It really bothers me that he calls it a sequence instead of a series (maybe he means the sequence of partial sums?), and that it’s not written correctly.
The series doesn’t converge because log(w) doesn’t have a fixed point at zero.
It makes sense if you replace log(w) with log^+(w) = max{ log(w), 0 }, which is sometimes written as log(w) in computer science papers where the behavior on (0, 1] is irrelevant.
I suppose that amounts to assuming there’s some threshold of cognitive work under which no gains in performance can be made, which seems reasonable.
Since this apparently bothers people, I’ll try to fix it at some point. A more faithful statement would be that we start by investing work w, get a return w2 ~ log(w), reinvest it to get a new return log(w + w2) - log(w) = log ((w+w2)/w). Even more faithful to the same spirit of later arguments would be that we have y’ ~ log(y) which is going to give you basically the same growth as y’ = constant, i.e., whatever rate of work output you had at the beginning, it’s not going to increase significantly as a result of reinvesting all that work.
I’m not sure how to write either more faithful version so that the concept is immediately clear to the reader who does not pause to do differential equations in their head (even if simple ones).
Well, suppose cognitive power (in the sense of amount of cognitive work put unit time) is a function of total effort invested so far, like P=1-e^(-w). Then it’s obvious that while dP/dw= e^(-w) is always positive, it rapidly decreases to basically zero, and total cognitive power converges to some theoretical maximum.
This is in the context of reinvesting dividends of cognitive work, assuming it takes exponentially greater investments to produce linearly greater returns. For example, maybe we get a return of log(X) cognitive work per time with what we have now, and to get returns of log(X+k) per time we need to have invested X+k cognitive work. What does it look like to reinvest all of our dividends? After dt, we have invested X+log(X) and our new return is log(X+log(X)). After 2dt, we have invested X+log(X)+log(X+log(X)), etc.
The corrected paragraph would then look like:
Therefore, an AI trying to invest an amount of cognitive work w to improve its own performance will get returns that go as log(w), or if further reinvested, an additional log(1+log(w)/w), and the sequence log(w)+log(1+log(w)/w)+log(1+log(w+log(w))/(w+log(w))) will converge very quickly.
Except then it’s not at all clear that the series converges quickly. Let’s check… we could say the capital over time is f(t), with f(0)=w, and the derivative at t is f’(t)=log(f(t)). Then our capital over time is f(t)=li^(-1)(t+li(w)). This makes our capital / log-capital approximately linear, so our capital is superlinear, but not exponential.
I’m going to commit pedantry: nesting enough logarithms eventually gives an undefined term (unless n’s complex!). So where Eliezer says “the sequence log(w) + log(log(w)) + log(log(log(w))) will converge very quickly” (p. 4), that seems wrong, although I see what he’s getting at.
It really bothers me that he calls it a sequence instead of a series (maybe he means the sequence of partial sums?), and that it’s not written correctly.
The series doesn’t converge because log(w) doesn’t have a fixed point at zero.
It makes sense if you replace log(w) with log^+(w) = max{ log(w), 0 }, which is sometimes written as log(w) in computer science papers where the behavior on (0, 1] is irrelevant.
I suppose that amounts to assuming there’s some threshold of cognitive work under which no gains in performance can be made, which seems reasonable.
Now fixed, I hope.
Oh yes. That makes far more sense. Thanks for fixing it.
Since this apparently bothers people, I’ll try to fix it at some point. A more faithful statement would be that we start by investing work w, get a return w2 ~ log(w), reinvest it to get a new return log(w + w2) - log(w) = log ((w+w2)/w). Even more faithful to the same spirit of later arguments would be that we have y’ ~ log(y) which is going to give you basically the same growth as y’ = constant, i.e., whatever rate of work output you had at the beginning, it’s not going to increase significantly as a result of reinvesting all that work.
I’m not sure how to write either more faithful version so that the concept is immediately clear to the reader who does not pause to do differential equations in their head (even if simple ones).
Well, suppose cognitive power (in the sense of amount of cognitive work put unit time) is a function of total effort invested so far, like P=1-e^(-w). Then it’s obvious that while dP/dw= e^(-w) is always positive, it rapidly decreases to basically zero, and total cognitive power converges to some theoretical maximum.
This is in the context of reinvesting dividends of cognitive work, assuming it takes exponentially greater investments to produce linearly greater returns. For example, maybe we get a return of log(X) cognitive work per time with what we have now, and to get returns of log(X+k) per time we need to have invested X+k cognitive work. What does it look like to reinvest all of our dividends? After dt, we have invested X+log(X) and our new return is log(X+log(X)). After 2dt, we have invested X+log(X)+log(X+log(X)), etc.
The corrected paragraph would then look like:
Except then it’s not at all clear that the series converges quickly. Let’s check… we could say the capital over time is f(t), with f(0)=w, and the derivative at t is f’(t)=log(f(t)). Then our capital over time is f(t)=li^(-1)(t+li(w)). This makes our capital / log-capital approximately linear, so our capital is superlinear, but not exponential.
I wonder if he meant w + (w+log(w)) + (w+log(w)+log(w+log(w))) + …