Goodhart’s Law states that “any observed statistical regularity will tend to collapse once pressure is placed upon it for control purposes.” However, this is not a single phenomenon. In Goodhart Taxonomy, I proposed that there are (at least) four different mechanisms through which proxy measures break when you optimize for them: Regressional, Extremal, Causal, and Adversarial.

David Manheim has now helped write up my taxonomy as a paper going into more detail on the these mechanisms: “Categorizing variants of Goodhart’s Law.” From the conclusion:

This paper represents an attempt to categorize a class of simple statistical misalignments that occur both in any algorithmic system used for optimization, and in many human systems that rely on metrics for optimization. The dynamics highlighted are hopefully useful to explain many situations of interest in policy design, in machine learning, and in specific questions about AI alignment.

In policy, these dynamics are commonly encountered but too-rarely discussed clearly. In machine learning, these errors include extremal Goodhart effects due to using limited data and choosing overly parsimonious models, errors that occur due to myopic consideration of goals, and mistakes that occur when ignoring causality in a system. Finally, in AI alignment, these issues are fundamental to both aligning systems towards a goal, and assuring that the system’s metrics do not have perverse effects once the system begins optimizing for them.

Let V refer to the true goal, while U refers to a proxy for that goal which was observed to correlate with V and which is being optimized in some way. Then the four subtypes of Goodhart’s Law are as follows:

Regressional Goodhart — When selecting for a proxy measure, you select not only for the true goal, but also for the difference between the proxy and the goal.

• Model: When U is equal to V + X, where X is some noise, a point with a large U value will likely have a large V value, but also a large X value. Thus, when U is large, you can expect V to be predictably smaller than U.
• Example: Height is correlated with basketball ability, and does actually directly help, but the best player is only 6’3″, and a random 7′ person in their 20s would probably not be as good.
  

Extremal Goodhart — Worlds in which the proxy takes an extreme value may be very different from the ordinary worlds in which the correlation between the proxy and the goal was observed.

• Model: Patterns tend to break at simple joints. One simple subset of worlds is those worlds in which U is very large. Thus, a strong correlation between U and V observed for naturally occuring U values may not transfer to worlds in which U is very large. Further, since there may be relatively few naturally occuring worlds in which U is very large, extremely large U may coincide with small V values without breaking the statistical correlation.
• Example: The tallest person on record, Robert Wadlow, was 8’11” (2.72m). He grew to that height because of a pituitary disorder; he would have struggled to play basketball because he “required leg braces to walk and had little feeling in his legs and feet.”
  

Causal Goodhart — When there is a non-causal correlation between the proxy and the goal, intervening on the proxy may fail to intervene on the goal.

• Model: If V causes U (or if V and U are both caused by some third thing), then a correlation between V and U may be observed. However, when you intervene to increase U through some mechanism that does not involve V, you will fail to also increase V.
• Example: Someone who wishes to be taller might observe that height is correlated with basketball skill and decide to start practicing basketball.

Adversarial Goodhart — When you optimize for a proxy, you provide an incentive for adversaries to correlate their goal with your proxy, thus destroying the correlation with your goal.

• Model: Consider an agent A with some different goal W. Since they depend on common resources, W and V are naturally opposed. If you optimize U as a proxy for V, and A knows this, A is incentivized to make large U values coincide with large W values, thus stopping them from coinciding with large V values.
• Example: Aspiring NBA players might just lie about their height.

For more on this topic, see Eliezer Yudkowsky’s write-up, Goodhart’s Curse.