Extremal is not a special case of regressional, but you cannot seperate them completely because regressional is always there. I think the tails come apart is in the right place. (but I didn’t reread the post when I made this)
If you sample a bunch of points from a multivariate normal without the large circular boundary in my example, the points will roughly form an ellipse, and the tails come apart thing will still happen. This would be Regerssional Goodhart. When you add the circular boundary, something weird happens where now you optimization is not just failing to find the best point, but actively working against you. If you optimize weakly for the proxy, you will get a large true value, but when you optimize very strongly you will end up with a low true value.
Extremal is not a special case of regressional, but you cannot seperate them completely because regressional is always there. I think the tails come apart is in the right place. (but I didn’t reread the post when I made this)
If you sample a bunch of points from a multivariate normal without the large circular boundary in my example, the points will roughly form an ellipse, and the tails come apart thing will still happen. This would be Regerssional Goodhart. When you add the circular boundary, something weird happens where now you optimization is not just failing to find the best point, but actively working against you. If you optimize weakly for the proxy, you will get a large true value, but when you optimize very strongly you will end up with a low true value.