# 58

Thanks to Aryan Bhatt, Eric Neyman, and Vivek Hebbar for feedback.

This post gets more math-heavy over time; we convey some intuitions and overall takeaways first, and then get more detailed. Read for as long as you're getting value out of things!

## TLDR

How much should you optimize for a flawed measurement? If you model optimization as selecting for high values of your goal  plus an independent error , then the answer ends up being very sensitive to the distribution of the error : if it’s heavy-tailed you shouldn't optimize too hard, but if it’s light-tailed you can go full speed ahead.

Why the tails come apart by Thrasymachus discusses a sort of "weak Goodhart" effect, where extremal proxy measurements won't have extremal values of your goal (even if they're still pretty good). It implicitly looks at cases similar to a normal distribution.

Scott Garrabrant's taxonomy of Goodhart's Law discusses several ways that the law can manifest. This post is about the "Regressional Goodhart" case.

Scaling Laws for Reward Model Overoptimization (Gao et al., 2022) considers very similar conditioning dynamics in real-world RLHF reward models. In their Appendix A, they show a special case of this phenomenon for light-tailed error, which we'll prove a generalization of in the next post.

Defining and Characterizing Reward Hacking (Skalse et al., 2022) shows that under certain conditions, leaving any terms out of a reward function makes it possible to increase expected proxy return while decreasing expected true return.

How much do you believe your results? by Eric Neyman tackles very similar phenomena to the ones discussed here, particularly in section IV; in this post we're interested in characterizing that sort of behavior and when it occurs. We strongly recommend reading it first if you'd like better intuitions behind some of the math presented here - though our post was written independently, it's something of a sequel to Eric's.

An Arbital page defines Goodhart's Curse and notes

The exact conditions for Goodhart's Curse applying between  and a point estimate or probability distribution over  [a proxy measure that an AI is optimizing], have not yet been written out in a convincing way.

To the extent this post adopts a reasonable frame, we think it makes progress towards this goal.

## Motivation/intuition

Goodhart's Law says

When a measure becomes a target, it ceases to be a good measure.

When I (Drake) first heard about Goodhart's Law, I internalized something like "if you have a goal, and you optimize for a proxy that is less than perfectly correlated with the goal, hard enough optimization for the proxy won't get you what you wanted." This was a useful frame to have in my toolbox, but it wasn't very detailed - I mostly had vague intuitions and some idealized fables from real life.

Much later, I saw some objections to this frame on Goodhart that actually used math.[1] The objection went something like:

Let's try to sketch out an actual formal model here. What's the simplest setup of "two correlated measurements"? We could have a joint normal distribution over two random variables,  and , with zero mean and positive covariance. You actually value , but you measure a proxy . Then we can just do the math: if I optimize really hard for , and give you a random datapoint with  or something, how much  do you expect to get?

If we look at the joint distribution of  and , we'll see a distribution with elliptical contour lines, like so:

Now, the naïve hope is that expected  as a function of observed  would go along the semi-major axis, shown in red below:

But actually we'll get the blue line, passing through the points at which the ellipses are tangent to the -axis.[2]

Importantly, though, we're still getting a line: we get linearly more value  for every additional unit of  we select for! Applying  percentile selection on  isn't going to be as good as  percentile selection on , but it's still going to give us more  than any lower percentile selection on .[3] The proxy is inefficient, but it's not doomed.

Lately, however, I've come to think that this story is a little too rosy. One thing that's going on here is that we're just thinking about a "regressional Goodhart" problem, which is only one of several ways something Goodhart-like can come into play - see Scott Garrabrant's taxonomy. But even in this setting, I think things can be much thornier.

In the story above, we can think of our measurement  as being some multiple of  plus an independent normally-distributed source of error, . When we ask for an outcome with a really high value of , we're asking for a datapoint where  is very high.[4]

Because normal distributions drop off in probability very fast, it gets harder and harder to select for high values of either component: given that a datapoint is at least 4 standard deviations above the mean, the odds that it's at least 5 standard deviations above are less than 1%. So the least-rare outcomes with high  are going to look like a compromise between the noise  and value , where we have a medium amount of each piece (because going to the extremes for either one is disproportionately costly in terms of improbability).

To see this more visually, here are some plots of possible  pairs, restricted to the triangle of values where . Points are brighter if that outcome is more probable, and the black contour lines show regions of equal probability density. On the right, we have the expected value of  as a function of our proxy threshold .

We can see that the most likely outcomes skew towards one side or the other depending on which of  and  has more variance, but because these contour lines are convex, we still expect to see outcomes that have some of each component.

But now let's look at a case where  and  are heavy-tailed, such that each additional unit of  or  requires fewer bits of optimization power.[5] Say that the probability density functions (PDFs) of  and  are proportional to , instead of  like before.[6] Then we'll see something more like

The resulting distribution is symmetric about  and , of course, but unlike in the normal case, that doesn't manifest as " and  will be about the same", but instead as "the outcome will be almost entirely  or almost entirely  with even odds".

In this heavy-tailed regime, though, we care a lot about which of  or  has the edge here. For instance, suppose that optimizing a given amount for  only gets us half as far as it would for  (so e.g. the 99th percentile  value is half as large as the 99th percentile  value). Our plot now looks like

and in the limit for large  we won't get any expected  at all by optimizing for the sum - all that optimization power goes towards producing high  values. We call this catastrophic Goodhart because the end result, in terms of , is as bad as if we hadn't conditioned at all.

(In general, if the right-hand tails of  and  are each on the order of , we'll switch between the two regimes right at  - that's when these contour lines switch from being convex to being concave.)

To help visualize this behavior, let's zoom in closer on a concrete example where we get catastrophic Goodhart.[7] See below for plots of the PDFs of  and :

On the left is a standard plot of the two PDFs; on the right is a plot of their negative logarithms. The right-hand plot makes it apparent that  has heavier right tails, because the green line gets arbitrarily far below the orange line in the limit.

Here is a GIF of the conditional distribution on  as  goes from  up to , with a dashed blue line indicating the conditional expectation:

Note the spike in the conditional PDF around , corresponding to outcomes where  is small and  is large; because of the heavier tails on , this spike gets smaller and smaller with larger . (We recommend staring at this GIF until you feel like you have a good understanding of why it looks the way it does.)

The expected value initially goes up when we apply a little selection pressure to our proxy, but as we optimize harder, that optimization pressure gets shunted more and more into optimization for , and less and less for , even in absolute terms. (This is the same dynamic that Eric Neyman recently discussed in section IV of How much do you believe your results?, put in a slightly different framing.)

In the next post, we're going to prove some results about when this effect happens; this will be pretty technical, so we'll talk a bit about the results in broad strokes here.

## Proof statement

Suppose that  and  are independent real-valued random variables. We'll show, roughly, that if

•  is subexponential (a slightly stronger property than being heavy-tailed).
•  has lighter tails than  by more than a linear factor, meaning that the ratio of the tails of  and the tails of  grows​​ superlinearly.[8]

then .

Less formally, we're saying something like "if it requires relatively little selection pressure on  to get more of  and asymptotically more selection pressure on  to get more of , then applying very strong optimization towards  will not get you even a little bit of optimization towards  - all the optimization power will go towards , where it has the best return on investment."

We'll also show a sort of inverse to this: if  has right tails that are lighter than an exponential (for instance, if  is normal or bounded), then we'll get infinitely much  in the limit no matter what kind of tail distribution  has.

(What if  is heavy-tailed but  has even heavier tails than ? Then we can exchange their places in the first theorem, and conclude that we get zero  in the limit - which means that all of that optimization is going towards .)

In the next post, we'll prove these claims.

## Application to alignment

We might want to use unaligned AI to generate alignment research for us. One model for this is sampling a random document from the space of 10000-bit strings, then conditioning on a high human rating. If evaluation of alignment proposals is substantially easier than generating good alignment proposals, these plans will be useful. If not, we’ll have a hard time getting research out of the AI. This is a crux between John Wentworth and Paul Christiano + Jan Leike that informs their differing approaches to alignment.

We can frame the problem of evaluation in terms of Goodhart’s Law. Let  be the true quality of an alignment plan (say in utility contributed to the future), and  be the human rating, so that  is the human’s rating error. If  and  are independent, and we have access to arbitrarily strong optimization for , then our result implies that to implement an alignment plan better than random…

• … if V is light-tailed, X must not be heavy-tailed.
• … if V is heavy-tailed, X must not be much heavier-tailed than V.

We don’t know whether V is heavy- or light-tailed in real life, so to be safe, we should make X light-tailed. To the extent this model is accurate, a large part of alignment reduces to the problem of finding a classifier with light-tailed errors, which is able to operate in the exceptionally complicated domain of evaluating plans, and is not itself dangerous.

This model makes two really strong assumptions: that optimization is like conditioning, and that  and  are independent. These are violated in real life:

• Optimization is not simply conditioning; SGD has too many inductive biases for us to list here, and (Gao et al., 2022) found that for a given level of optimization, RL uses far more KL distance from the prior than best-of-n sampling.
•  and  will not be independent. Among other reasons, we expect that more complicated or optimized plans are more likely to have large impacts on the world (thus having higher variance of ), and harder to evaluate (thus having higher variance of ). However, in some cases, really good plans might be easier to evaluate; for example, formalized proofs can be efficiently checked.

There's also a sort of implicit assumption in even using a framing that thinks about things as ; the world might be better thought of as naturally containing  tuples (with  our proxy measurement), and  could be a sort of unnatural construction that doesn't make sense to single out in the real world. (We do think this framing is relatively natural, but won't get into justifications here.)

Despite these caveats, some takeaways we endorse:

• Optimization for imperfect proxies is sometimes fine and sometimes doomed, depending on your distribution.
• Goodhart's law is subtle - even within a given framing of a problem, what happens when you optimize can be very sensitive to the exact numerical details of your measurements.
• In particular, reaching for a normally-distributed toy model by default can be super misleading for thinking about a lot of real-world dynamics, because the tails are much lighter than most things in a way that affects the qualitative takeaways.
• In an alignment plan involving generation and evaluation, you should either (a) have reason to believe that your classifier's errors are light-tailed, (b) have a reason why training an AI on human (or AI) feedback will be importantly different from conditioning on high feedback scores, or (c) have a story for why non-independence works in your favor.

## Exercises

1. Show that when  and  are independent and . Conclude that . This means that given independence, optimization always produces a plan that is no worse than random.
2. When independence is violated, an optimized plan can be worse than random, even if your evaluator is unbiased. Construct a joint distribution  for  and  such that , and  for any , but .

Answers to exercises are at the end of the next post.

1. ^

Thanks to Eric Neyman for first making this observation clear to me.

2. ^

One way to see this intuitively is to consider the shear transformation replacing  by , where  is a constant such that the resulting random variable is uncorrelated with . In that situation we'd have a constant expectation of , so adding the  component back in should give us a linear expectation.

3. ^

To be precise,

4. ^

Technically we could have , but we can just rescale  until the  coefficient is 1 without changing anything.

5. ^

Most heavy-tailed distributions are also long-tailed, which means that  for all . So the optimization needed to get from the event " is at least " to " is at least " becomes arbitrarily small for large .

6. ^

Note that this effect doesn't depend on the behavior of  or  right around zero, just on their right tails.

7. ^

We'll suppose that  has a PDF proportional to  and  has a PDF proportional to , where  is an odd function that quickly asymptotes to , so  has tails like  for large  in either direction but is smooth around

8. ^

We'll use something slightly stronger than this; we'd like 's tails to be larger by a factor of . More precise details in the next post.

New Comment

For what it's worth, it seems much more likely to me for catastrophic Goodhart to happen because the noise isn't independent from the thing we care about, rather than the noise being independent but heavy tailed.

I finally got around to reading this sequence, and I really like the ideas behind these methods. This feels like someone actually trying to figure out exactly how fragile human values are. It's especially exciting because it seems like it hooks right into an existing, normal field of academia (thus making it easier to leverage their resources toward alignment).

I do have one major issue with how the takeaway is communicated, starting with the term "catastrophic". I would only use that word when the outcome of the optimization is really bad, much worse that "average" in some sense. That's in line with the idea that the AI will "use the atoms for something else", and not just leave us alone to optimize its own thing. But the theorems in this sequence don't seem to be about that;

We call this catastrophic Goodhart because the end result, in terms of , is as bad as if we hadn't conditioned at all.

Being as bad as if you hadn't optimized at all isn't very bad; it's where we started from!

I think this has almost the opposite takeaway from the intended one. I can imagine someone (say, OpenAI) reading these results and thinking something like, great! They just proved that in the worst case scenario, we do no harm. Full speed ahead!

(Of course, putting a bunch of optimization power into something and then getting no result would still be a waste of the resources put into it, which is presumably not built into . But that's still not very bad.)

That said, my intuition says that these same techniques could also suss out the cases where optimizing for  pessimizes for , in the previously mentioned use-our-atoms sense.

We considered that "catastrophic" might have that connotation, but we couldn't think of a better name and I still feel okay about it. Our intention with "catastrophic" was to echo the standard ML term of "catastrophic forgetting", not a global catastrophe. In catastrophic forgetting the model completely forgets how to do task A after it is trained on task B, it doesn't do A much worse than random. So we think that "catastrophic Goodhart" gives the correct idea to people who come from ML.

The natural question is then: why didn't we study circumstances in which optimizing for a proxy gives you  utility in the limit? Because it isn't true under the assumptions we are making. We wanted to study regressional Goodhart, and this naturally led us to the independence assumption. Previous work like Zhuang et al and Skalse et al has already formalized the extremal Goodhart / "use the atoms for something else" argument that optimizing for one goal would be bad for another goal, and we thought the more interesting part was showing that bad outcomes are possible even when error and utility are independent. Under the independence assumption, it isn't possible to get less than 0 utility.

To get  utility in the frame where proxy = error + utility, you would need to assume something about the dependence between error and utility, and we couldn't think of a simple assumption to make that didn't have too many moving parts. I think extremal Goodhart is overall more important, but it's not what we were trying to model.

Lastly, I think you're imagining "average" outcome as a random policy, which is an agent incapable of doing significant harm. The utility of the universe is still positive because you can go about your life. But in a different frame, random is really bad. Right now we pretrain models and then apply RLHF (and hopefully soon, better alignment techniques). If our alignment techniques produce no more utility than the prior, this means the model is no more aligned than the base model, which is a bad outcome for OpenAI. Superintelligent models might be arbitrarily capable of doing things, so the prior might be better thought of as irreversibly putting the world in a random state, which is a global catastrophe.

Another piece of related work: Simon Zhuang, Dylan Hadfield-Mennel: Consequences of Misaligned AI.
The authors assume a model where the state of the world is characterized by multiple "features". There are two key assumptions: (1) our utility is (strictly) increasing in each feature, so -- by definition -- features are things we care about (I imagine money, QUALYs, chocolate). (2) We have a limited budget, and any increase in any of the features always has a non-zero cost. The paper shows that: (A) if you are only allowed to tell your optimiser about a strict subset of the features, all of the non-specified features get thrown under the buss. (B) However, if you can optimise things gradually, then you can alternate which features you focus on, and somehow things will end up being pretty okay.

Personal note: Because of the assumption (2), I find the result (A) extremely unsurprising, and perhaps misleading. Yes, it is true that at the Pareto-frontier of resource allocation, there is no space for positive-sum interactions (ie, getting better on some axis must hurt us on some other axis). But the assumption (2) instead claims that positive-sum interactions are literally never possible. This is clearly untrue in the real-world, about things we care about.

That said, I find the result (B) quite interesting, and I don't mean to hate on the paper :-).

have a reason why training an AI on human (or AI) feedback will be importantly different from conditioning on high feedback scores

I think these are generally not the same, and I object to any implied privileging of this hypothesis. But above you say that SGD has a ton of inductive biases in general, so why do you seem to endorse a takeaway like (my words) "you need to have a reason why SGD has the relevant inductive biases"?

SGD has inductive biases, but we'd have to actually engineer them to get high  rather than high  when only trained on . In the Gao et al paper, optimization and overoptimization happened at the same relative rate in RL as in conditioning, so I think the null hypothesis is that training does about as well as conditioning. I'm pretty excited about work that improves on that paper to get higher gold reward while only having access to the proxy reward model.

I think the point still holds in mainline shard theory world, which in my understanding is using reward shaping + interp to get an agent composed of shards that value proxies that more often correlate with high  rather than higher , where we are selecting on something other than . When the AI ultimately outputs a plan for alignment, why would it inherently value having the accurate plan, rather than inherently value misleading humans? I think we agree that it's because SGD has inductive biases and we understand them well enough to do directionally better than conditioning at constructing an AI that does what we want.

Great post! I especially enjoyed the intuitive visualizations for how the heavy-tailed distributions affect the degree of overoptimization of X.

As a possibly interesting connection, your set of criteria for an alignment plan can also be thought of as criteria for selecting a model specification that approximates the ideal specification well, especially trying to ensure that the approximation error is light-tailed.

This model makes two really strong assumptions: that optimization is like conditioning, and that  and  are independent.

[...]

There's also a sort of implicit assumption in even using a framing that thinks about things as ; the world might be better thought of as naturally containing  tuples (with  our proxy measurement), and  could be a sort of unnatural construction that doesn't make sense to single out in the real world. (We do think this framing is relatively natural, but won't get into justifications here.)

Will you get into justifications in the next post? Because otherwise the following advice, which I consider literally correct:

• In an alignment plan involving generation and evaluation, you should either have reason to believe that your classifier's errors are light-tailed, or have a story for why inductive bias and/or non-independence work in your favor.

in practice reduces just to the part "have a story for why inductive bias and/or non-independence work in your favor", because I currently think Normality + additivity + independence are bad assumptions, and I see that as almost a null advice.

I think that Normality + additivity + independence come out together if you have a complex system subject to small perturbations, because you can write any dynamic as linear relationships over many variables. This gets you the three perks with:

• Normality: complex system means many variables with nontrivial roles, and so the linearization tends to produce Normal distributions, it behaves like a sum with not too much concentrated weights.
• Additivity: due to the small perturbations that allow you to linearize any relationship as approximation.
• Independence: a linear system should be easy enough to analyze that you expect, if you spend effort, to get to a situation where the error is independent, and all the rest has been accounted for in some way.

Since we want to study the situation in which we apply a lot of optimization pressure, I think this scenario gets thrown out the window.

So:

1. Do you have a more general reason to expect these assumptions? Possibly each one or subsets separately? First raw ideas that come to my mind:
1. Normality because the number of variables involved grows in a balanced way with nonlinearity such that you get Normality
2. Additivity because scenario we can realistically study are limited enough that the kind of errors you can make stay the same, and we have to deliberately put ourselves in that situation
3. Independence because a human manages to get as much information as possible until some hard boundary of chaos
2. Do you have some clever trick such that it is always possible to always see the problem in this light? I expect not because utilities can only be affinely transformed.

Example: here

But now let's look at a case where  and  are heavier-tailed. Say that the probability density functions (PDFs) of  and  are proportional to , instead of  like before.

my gut instinct tells me to look at elliptical distributions like , which will not show this specific split-tail behavior. My gut instinct is not particularly justified, but seems to be making weaker assumptions.

I'm not sure what you mean formally by these assumptions, but I don't think we're making all of them. Certainly we aren't assuming things are normally distributed - the post is in large part about how things change when we stop assuming normality! I also don't think we're making any assumptions with respect to additivity;  is more of a notational or definitional choice, though as we've noted in the post it's a framing that one could think doesn't carve reality at the joints. (Perhaps you meant something different by additivity, though - feel free to clarify if I've misunderstood.)

Independence is absolutely a strong assumption here, and I'm interested in further explorations of how things play out in different non-independent regimes - in particular we'd be excited about theorems that could classify these dynamics under a moderately large space of non-independent distributions. But I do expect that there are pretty similar-looking results where the independence assumption is substantially relaxed. If that's false, that would be interesting!

Late edit: Just a note that Thomas has now published a new post in the sequence addressing things from a non-independence POV.

I wasn't saying you made all those assumption, I was trying to imagine an empirical scenario to get your assumptions, and the first thing to come to my mind produced even stricter ones.

I do realize now that I messed up my comment when I wrote

in practice reduces just to the part "have a story for why inductive bias and/or non-independence work in your favor", because I currently think Normality + additivity + independence are bad assumptions, and I see that as almost a null advice.

Here there should not be Normality, just additivity and independence, in the sense of . Sorry.

But I do expect that there are pretty similar-looking results where the independence assumption is substantially relaxed.

I do agree you could probably obtain similar-looking results with relaxed versions of the assumptions.

However, the same way  seems quite specific to me, and you would need to make a convincing case that this is what you get in some realistic cases to make your theorem look useful, I expect this will continue to apply for whatever relaxed condition you can find that allows you to make a theorem.

Example: if you said "I made a version of the theorem assuming there exists  such that  for  in some class of functions", I'd still ask "and in what realistic situations does such a setup arise, and why?"

An example of the sort of strengthening I wouldn't be surprised to see is something like "If  is not too badly behaved in the following ways, and for all  we have [some light-tailedness condition] on the conditional distribution , then catastrophic Goodhart doesn't happen." This seems relaxed enough that you could actually encounter it in practice.

Suppose that we are selecting for  where V is true utility and X is error. If our estimator is unbiased ( for all v) and X is light-tailed conditional on any value of V, do we have ?

No; here is a counterexample. Suppose that , and  when , otherwise . Then I think .

This is worrying because in the case where  and  independently, we do get infinite V. Merely making the error *smaller* for large values of V causes catastrophe. This suggests that success caused by light-tailed error when V has even lighter tails than X is fragile, and that these successes are “for the wrong reason”: they require a commensurate overestimate of the value when V is high as when V is low.

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