How does Gradient Descent Interact with Goodhart?

While at a recent CFAR workshop with Scott, Peter Schmidt-Nielsen and I wrote some code to run experiments of the form that Scott is talking about here. If anyone is interested, the code can be found here , though I'll also try to summarize our results below.

Our methodology was as follows:
1. Generate a real utility function $V:[0,1{]}^{10}\to [0,1]$ by randomly initializing a feed-forward neural network with 3 hidden layers with 10 neurons each and tanh activations, then train it using 5000 steps of gradient descent with a learning rate of 0.1 on a set of 1023 uniformly sampled data points. The reason we pre-train the network on random data is that we found that randomly initialized neural networks tended to be very similar and very smooth such that it was very easy for the proxy network to learn them, whereas networks trained on random data were significantly more variable.
2. Generate a proxy utility function $U:[0,1{]}^{10}\to [0,1]$ by training a randomly initialized neural network with the same architecture as the real network on 50 uniformly sampled points from the real utility using 1000 steps of gradient descent with a learning rate of 0.1.
3. Fix μ to be uniform sampling.
4. Let $\hat{\mu }$ be uniform sampling followed by 50 steps of gradient descent on the proxy network with a learning rate of 0.1.
5. Sample 1000000 points from μ, then optimize those same points according to $\hat{\mu }$. Create buckets of radius 0.01 utilons for all proxy utility values and compute the real utility values for points in that bucket from the μ set and the $\hat{\mu }$ set.
6. Repeat steps 1-5 10 times, then average the final real utility values per bucket and plot them. Furthermore, compute the "Goodhart error" as the real utility for the proxy utility points minus the real utility for the random points plotted against their proxy utility values.

The plot generated by this process is given below:



As can be seen from the plot, the Goodhart error is fairly consistently negative, implying that the gradient descent optimized points are performing worse on the real utility conditional on the proxy utility.

However, using an alternative $\hat{\mu }$ , we were able to reverse the effect. That is, we ran the same experiment, but instead of optimizing the proxy utility to be close to the real utility on the sampled points, we optimized the gradient of the proxy utility to be close to the gradient of the real utility on the sampled points. This resulted in the following graph:


As can be seen from the plot, the Goodhart error flipped and became positive in this case, implying that the gradient optimization did significantly better than the point optimization.

Finally, we also did a couple of checks to ensure the correctness of our methodology.

First, one concern was that our method of bucketing could be biased. To determine the degree of "bucket error" we computed the average proxy utility for each bucket from the μ and $\hat{\mu }$ datasets and took the difference. This should be identically zero, since the buckets are generated based on proxy utility, while any deviation from zero would imply a systematic bias in the buckets. We did find a significant bucket error for large bucket sizes, but for our final bucket size of 0.01, we found a bucket error in the range of 0 - 0.01, which should be negligible.

Second, another thing we did to check our methodology was to generate $\hat{\mu }$ simply by sampling 100 random points, then selecting the one with the highest proxy utility value. This should give exactly the same results as μ, since bucketing conditions on the proxy utility value, and in fact that was what we got.

My answer will address the special case where $V$ is the true risk and $U$ the empirical risk of an ANN for some offline learning task. The question is more general, but I think that this special case is important and instructional.

The questions you ask seem to be closely related to statistical learning theory. Statistical learning theory studies questions such as, (a) "how many samples do I need in my training set, to be sure that the model my algorithm learned generalizes well outside the training set?" and (b) "is the given training set sufficiently representative to warrant generalization?" In the case of AI alignment, we are worried that, a model of human values trained on a limited data set will not generalize well outside this data set. Indeed, many classical failure modes that have been speculated about can be regarded as, human values coincide with a different function inside the training set (e.g. with human approval, or with the electronic reward signal fed into the AI) but not on the entire space.

The classical solutions to questions (a) and (b) for offline learning is Vapnik-Chervonenkis theory and its various offshoots. Specifically, VC dimension determines the answer to question (a) and Rademacher complexity determines the answer to question (b). The problem is, VC theory is known to be inadequate for deep learning. Specifically, the VC dimension of a typical ANN architecture is too large to explain the empirical sample complexity. Moreover, risk minimization for ANNs is known to be NP-complete (even learning the intersection of two half-spaces is already NP-complete), but gradient descent obviously does a "good enough" job in some sense, despite that most known guarantees for gradient descent only apply to convex cost functions.

Explaining these paradoxes is an active field of research, and the problem is not solved. That said, there have been some results in recent years that IMO provide some very important insights. One type of result is, algorithms different from gradient descent that provably learn ANNs under some assumptions. Goel and Klivans 2017 show how to learn an ANN with two non-linear layers. They bypass the no-go results by assuming either learning a sufficiently smooth function, or, learning a binary classification but with a sufficient margin between the positive examples and the negative examples. Zhang et al 2017 learn ANNs with any number of layers, assuming a margin and $l_1$ regularization. Another type of result is Allen-Zhu, Li and Liang 2018 (although it wasn't peer reviewed yet, AFAIK, and I certainly haven't verified all the proofs EDIT: it was published in NIPS 2019). They examine a rather realistic algorithm (ReLU response, stochastic gradient descent with something similar to drop-outs) but limited to three layers. Intriguingly, they prove that the algorithm successfully achieves improper learning of a different class of functions, which have the form of somewhat smaller ANNs with smooth response. The "smoothness" serves the same role as in the previous results, to avoid the NP-completeness results. The "improperness" solves the apparent paradox of high VC dimension: it is indeed infeasible to learn a function that looks like the actual neural network, instead the neural network is just a means to learn a somewhat simpler function.

I am especially excited about the improper learning result. Assuming this result can be generalized to any number of layers (and I don't see any theoretical reason why it can't), the next question is understanding the expressiveness of the class that is learned. That is, it is well known that ANNs can approximate any Boolean circuit, but the "smooth" ANNs that are actually learned are somewhat weaker. Understanding this class of functions better might provide deep insights regarding the power and limitation of deep learning.

I think it depends almost entirely on the shape of V and W.

In order to do gradient descent, you need a function which is continuous and differentiable. So W can't be noise in the traditional regression sense (independent and identically distributed for each individual observation), because that's not going to be differentiable.

If W has lots of narrow, spiky local maxima with broad bases, then gradient descent is likely to find those local maxima, while random sampling rarely hits them. In this case, fake wins are likely to outnumber real wins in the gradient descent group, but not the random sampling group.

More generally, if U = V + W, then dU/dx = dV/dx + dW/dx. If V's gradient is typically bigger than W's gradient, gradient descent will mostly pay attention to V; the reverse is true if W's gradient is typically bigger.

But even if W's gradient typically exceeds V's gradient, U's gradient will still correlate with V's, assuming dV/dx and dW/dx are uncorrelated. (cov(dU, dV) = cov(dV+dW, dV) = cov(dV, dV) + cov(dW, dV) = cov(dV, dV).)

So I'd expect that if you change your experiment so instead of looking at the results in some band, you instead take the best n results from each group, the best n results of the gradient descent group will be better on average. Another intuition pump: Let's consider the spiky W scenario again. If V is constant everywhere, gradient descent will basically find us the nearest local maximum in W, which essentially adds random movement. But if V is a plane with a constant slope, and the random initialization is near two different local maxima in W, gradient descent will be biased towards the local maximum in W which is higher up on the plane of V. The very best points will tend to be those that are both on top of a spike in W and high up on the plane of V.

I think this is a more general point which applies regardless of the optimization algorithm you're using: If your proxy consists of something you're trying to maximize plus unrelated noise that's roughly constant in magnitude, you're still best off maximizing the heck out of that proxy, because the very highest value of the proxy will tend to be a point where the noise is high and the thing you're trying to maximize is also high.

"Constant unrelated noise" is an important assumption. For example, if you're dealing with a machine learning model, noise is likely to be higher for inputs off of the training distribution, so the top n points might be points far off the training distribution chosen mainly on the basis of noise. (Goodhart's Law arguably reduces to the problem of distributional shift.) I guess then the question is what the analogous region of input space is for approval. Where does the correspondence between approval and human value tend to break down?

(Note: Although W can't be i.i.d., W's gradient could be faked so it is. I think this corresponds to perturbed gradient descent, which apparently helps performance on V too.)

If we want to think about reasonably realistic Goodhart issues, random functions on $R^n$ seem like the wrong setting. John Maxwell put it nicely in his answer:

If your proxy consists of something you're trying to maximize plus unrelated noise that's roughly constant in magnitude, you're still best off maximizing the heck out of that proxy, because the very highest value of the proxy will tend to be a point where the noise is high and the thing you're trying to maximize is also high.

That intuition is easy to formalize: we have our "true" objective $u\left(x\right)$ that we want to maximize, but we can only observe $u$ plus some (differentiable) systematic error $ϵ\left(x\right)$. Assuming we don't have any useful knowledge about that error, the expected value given our information $E\left[u\right(x\left)|\right(u+ϵ\left)\right(x\left)\right]$ will still be maximized when $(u+ϵ)\left(x\right)$ is maximized. There is no Goodhart.

I'd think about it on a causal DAG instead. In practice, the way Goodhart usually pops up is that we have some deep, complicated causal DAG which determines some output we really want to optimize. We notice that some node in the middle of that DAG is highly predictive of happy outputs, so we optimize for that thing as a proxy. If our proxy were a bottleneck in the DAG - i.e. it's on every possible path from inputs to output - then that would work just fine. But in practice, there are other nodes in parallel to the proxy which also matter for the output. By optimizing for the proxy, we accept trade-offs which harm nodes in parallel to it, which potentially adds up to net-harmful effect on the output.

For example, there's the old story about soviet nail factories evaluated on number of nails made, and producing huge numbers of tiny useless nails. We really want to optimize something like the total economic value of nails produced. There's some complicated causal network leading from the factory's inputs to the economic value of its outputs. If we pick a specific cross-section of that network, we might find that economic value is mediated by number of nails, size, strength, and so forth. If we then choose number of nails as a proxy, then the factories trade off number of nails against any other nodes in that cross-section. But we'll also see optimization pressure in the right direction for any nodes which effect number of nails without effecting any of those other variables.

So that at least gives us a workable formalization, but we haven't really answered the question yet. I'm gonna chew on it some more; hopefully this formulation will be helpful to others.

In the rocket example, procedures A and B can both be optimized either by random sampling or by local search. A is optimizing some hand-coded rocket specifications, while B is optimizing a complicated human approval model.

The problem with A is that it relies on human hand-coding. If we put in the wrong specifications, and the output is extremely optimized, there are two possible cases: we recognize that this rocket wouldn't work and we don't approve it, or we think that it looks good but are probably wrong, and the rocket doesn't work.

On the upside, if we successfully hand-coded in how a rocket should be, it will output working rockets.

The problem with B is that it's simply the wrong thing to optimize if you want a working rocket. And because it's modeling the environment and trying to find an output that makes the environment-model do something specific, you'll get bad agent-like behavior.

Let's go back to take a closer look at case A. Suppose you have the wrong rocket specifications, but they're "pretty close" in some sense. Maybe the most spec-friendly rocket doesn't function, but the top 0.01% of designs by the program are mostly in the top 1% of rockets ranked by your approval.

The programmed goal is proxy #1. Then you look at some of the sampled (either randomly or through local search) top 0.01% designs for something you think will fly. Your approval is proxy #2. Your goal is the rocket working well.

What you're really hoping for in designing this system is that even if proxy #1 and proxy #2 are both misaligned, their overlap or product is more aligned - more likely to produce an actual working rocket - than either alone.

This makes sense, especially under the model of proxies as "true value + noise," but to the extent that model is violated maybe this doesn't work out.

This is another way of seeing what's wrong with case B. Case B just purely optimizes proxy #2, when the whole point of having human approval is to try to combine human approval with some different proxy to get better results.

As for local search vs. random sampling, this is a question about the landscape of your optimized proxy, and how this compares to the true value - neither way is going to be better literally 100% of the time.

If we imagine local optimization like water flowing downhill in the U.S., given a random starting point, the water is much more likely to end up at the mouth of the Mississippi river than it is to end up in Death Valley, even though Death Valley is below sea level. The Mississippi just has a broad network of similar states that lead into it via local optimization, whereas Death Valley is a "surprising" optimum. Under random sampling, you're equally likely to find equal areas of the mouth of the Mississippi or Death Valley.

Applying this to rockets, I would actually expect local search to produce much safer results in case B. Working rockets probably have broad basins of similar almost-working rockets that feed into them in configuation-space, whereas the rocket that spells out a message to the experimenter is quite a bit more fragile to perturbations.

(Even if rockets are so complicated and finnicky that we expect almost-working rockets to be rarer than convincing messages to the experimenter, we still might think that the gradient landscape makes gradient descent relatively better.)

In case A, I would expect much less difference between locally optimizing proxy #1 and sampling until it was satisfied. The difference for human approval came because we specifically didn't want to find the unstable, surprising maxima of human approval. And maybe the same is true of our hand-coded rocket specifications, but I would expect this to be less important.