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.
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.