Getting some precise bounds is helpful! I do want to mention, though, that uniform bounds seem like a very unrealistic sort of bound to have for many cases. A more realistic bound assumption might be that the average error is low under some distribution (such as a distribution of suggested plans humans might think up), since we expect the error to be very high in a few cases such as wireheading (where the proxy is able to be entirely fooled). In than case we don’t get nice bounds on goodhart for maximizers, since the maximizer may well choose such states, but I believe quantilizers do better.
Getting some precise bounds is helpful! I do want to mention, though, that uniform bounds seem like a very unrealistic sort of bound to have for many cases. A more realistic bound assumption might be that the average error is low under some distribution (such as a distribution of suggested plans humans might think up), since we expect the error to be very high in a few cases such as wireheading (where the proxy is able to be entirely fooled). In than case we don’t get nice bounds on goodhart for maximizers, since the maximizer may well choose such states, but I believe quantilizers do better.