Rafael Harth comments on Conditions for Mesa-Optimization

[Rafael Harth]

As one moves to more and more diverse environments—that is, as $N$ increases—this model suggests that $x$ will dominate $\frac{P-f\left(x\right)}{N}$, implying that mesa-optimization will become more and more favorable.

I believe this sentence should say ”$x$ will dominate $\frac{P-f\left(x\right)}{N}+x$, implying [...]”. Same for the sentence in the paper.

[evhub]

I suppose it depends on what you mean by “dominate.” Mathematically, the point here is just that

$$\underset{N\to \infty }{lim}\frac{P-f\left(x\right)}{N}+x=x$$

which I would describe as the $x$ term “dominating” the $\frac{P-f\left(x\right)}{N}$ term.

[Rafael Harth]

Ah, I see. I was thinking of ‘dominate’ as narrowly meaning ‘determines the total value of the term’, but I agree that the usage above works perfectly well.