It seems to me that the discussion on "Extremal Goodhart" is a bunch of good examples of what would be, following your ideas, "Nonlinear Goodhart". It's rather obvious that someone who is 270cm tall has an unusual body that could go either way for basketball, but it's less clear what happens to someone who is 215 cm (when the NBA average is 198cm).
We basically observe correlations "locally", in a neighborhood of the current values. Therefore we think of them as essentially linear, because every smooth function is "arbitarily close to linear in an arbitrarily close vicinity" (the first-order Taylor expansion).
So let's say, for example, that we see basketball players with the following heights and (abstract, think of Elo scores) "power ratings"
170cm - 0.8
180cm - 1.0
190cm - 1.2
200cm - 1.5
Note that this is already not linear, but gives salience to a lower threshold above which marginal increases in height give large increases in power. But because very tall people are scarce, we don't know clearly whether already at 220cm power ratings are still increasing, let alone if they are increasing at smaller increments. Combined with Adversarial Goodhart, this is the stuff of asset bubbles.
I distinguish this from Regression Goodhart because the chief operating principle there is that we're confused by noise -- Goodhart was after all a central banker in the 60s, an era in which macroeconomists only had noisy quarterly datasets going back to the late 40s.
It seems to me that the discussion on "Extremal Goodhart" is a bunch of good examples of what would be, following your ideas, "Nonlinear Goodhart". It's rather obvious that someone who is 270cm tall has an unusual body that could go either way for basketball, but it's less clear what happens to someone who is 215 cm (when the NBA average is 198cm).
We basically observe correlations "locally", in a neighborhood of the current values. Therefore we think of them as essentially linear, because every smooth function is "arbitarily close to linear in an arbitrarily close vicinity" (the first-order Taylor expansion).
So let's say, for example, that we see basketball players with the following heights and (abstract, think of Elo scores) "power ratings"
170cm - 0.8
180cm - 1.0
190cm - 1.2
200cm - 1.5
Note that this is already not linear, but gives salience to a lower threshold above which marginal increases in height give large increases in power. But because very tall people are scarce, we don't know clearly whether already at 220cm power ratings are still increasing, let alone if they are increasing at smaller increments. Combined with Adversarial Goodhart, this is the stuff of asset bubbles.
I distinguish this from Regression Goodhart because the chief operating principle there is that we're confused by noise -- Goodhart was after all a central banker in the 60s, an era in which macroeconomists only had noisy quarterly datasets going back to the late 40s.