Trying to disambiguate different questions about whether RLHF is “good”

(The worked example in this comment was a joint effort with Eric Neyman and Drake Thomas.)

Here's a toy example in which we get worse Goodharting for RL than for filtering: suppose that our model has three submodules

• A, which tries to produce outputs which are both true and persuasive
• B, which tries to produce outputs which are true, but have no effect on persuasiveness
• C, which tries to produce outputs which are persuasive, but with no effect on truthiness.

Our model has parameters $\alpha ,\beta ,\gamma$ summing to 1 which determine how much to listen to each of these submodules. More specifically, our submodules produce samples $a,b,c$ from the normal distributions $N({\mu }_A,{\sigma }_A^2),N({\mu }_B,{\sigma }_B^2),N({\mu }_C,{\sigma }_C^2)$, respectively, and then our model puts these samples together to produce an output which has truth score 
$T=\alpha a+\beta b$ 
and persuasiveness score 
$P=\alpha a+\gamma c$. 
We'll assume that we're only able to measure persuasiveness, but that we want truthiness.(Some unstated assumptions: $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$ and ${\mu }_A,{\mu }_B,{\mu }_C>0$.)

Our model was trained on data in which truthiness and persuasiveness were positively correlated; this will be reflected in having $\alpha >0$, so that $T$ and $P$ are positively correlated. If this is true, then conditioning on some persuasiveness score $p$ results in getting an output with expected truthiness score
$E[T|P=p]=\frac{p-\alpha {\mu }_A-\gamma {\mu }_C}{1+\left(\gamma {\sigma }_C/\alpha {\sigma }_A\right)}+\alpha {\mu }_A+\beta {\mu }_B$.
Note that this scales linearly with $p$, so that as we ask for more persuasiveness, we get more truthiness on average, as we'd hope.

In contrast, suppose we do RL on our model for high persuasiveness scores; imagine that this doesn't change the submodules A, B, and C much, but does tune the parameters $\alpha ,\beta ,\gamma$. Then:

• if ${\mu }_A>{\mu }_C$ we'll set $(\alpha ,\beta ,\gamma )=(1,0,0)$, i.e. always use the submodule which tries to produce true and persuasive outputs. This will result in average truthiness ${\mu }_A$.
• but if ${\mu }_C>{\mu }_A$ we'll set $(\alpha ,\beta ,\gamma )=(0,0,1)$, i.e. always use the submodule which tries to be persuasive but not true. This will result in average truthiness $0$, much worse than we would get if we had done filtering.

Really this is just a dressed-up version of the classic Goodharting story, where you have a constrained resource ($\alpha +\beta +\gamma =1$) to allocate among various options(=the submodules A,B,C), so you put 100% of your resources into the option which is cheapest in persuasiveness-per-resource; unfortunately, this was not the option which gave the best truth-per-resource.

Some misc comments:

• This example was a bit silly, but I think it captures some pieces of many folks' intuitions around RL and Goodharting: pretrained models have lots of capabilities, which are in some sense competing for the steering wheel: you can't LARP an economist writing an excellent paper and simultaneously LARP a deceptive agent who wants paperclips but finds it instrumentally useful to write economics papers. Whichever capability scores best for your proxy will win out, and with all other possible ways the model could have completed the training task getting no say.
• By thinking of "persuasiveness" as being something which we actually wanted to get, this example also serves as an illustration of how filtering can be uncompetitive: filtering produces outputs whose persuasiveness is distributed as $N(\alpha {\mu }_A+\gamma {\mu }_C,{\alpha }^2{\sigma }_A^2+{\gamma }^2{\sigma }_C^2)$ whereas RL produces a model whose outputs have persuasiveness ${\mu }_C$ on average; if ${\mu }_C$ is large, that means that you'd have to filter roughly the order of $\exp \left({\mu }_C^2\right)$ outputs to get the same persuasiveness as the average output of the RL-optimized model.
• I spent a while confused about how this squares with the baseline-probability-time-Boltzman-factor classification of what RL with a KL penalty will converge to. (The example above didn't have a KL penalty, but adding a small one wouldn't have much much difference.) I think the answer is that the model I described wasn't expressive enough to represent the baseline-probability-time-Boltzman-factor distribution that RL with a KL penalty would optimally converge to. This lack of expressivity seems quite related to the fact our model was a linear combination of three distributions which we modeled as not changing throughout training. That means that this story, which is based on the frame that generative models are a giant pile of capabilities which can be elicited, is in tension with the frame that neural networks are flexible function approximators; I found this pretty interesting.
• All this being said, I'm pretty skeptical that whatever sort of Goodharting is being captured in this example has much to do with the sort of Goodharting we empirically observe in RLHF, since this example doesn't work with best-of-n optimization (whereas extremal Goodharting does occur for best-of-n, as Buck pointed out elsethread).
• Overall, I don't put much stock in this example beyond helping articulate the point that RL amplifies capabilities in proportion to how causally downstream of high-reward outputs they are, whereas filtering only takes into account their correlations with high-reward outputs.

Boltzmann factor to upweight answers that your overseer likes. AFAICT this doesn't generally induce more causal Goodhart problems than best-of-N selection does.

This seems correct insofar as your proxy reward does not have huge upward errors (that you don't remove via some sort of clipping). For example, if there's 1 million normal sentences with reward uniformly distributed between [0, 100] and one adversarial sentence with reward r=10^5, conditioning on reward>99 leads to a 1/10,000 chance of sampling the adversarial sentence, while it's very tricky (if not impossible) to correctly set the KL penalty so you end up optimizing reward without just outputting the adversarial sentence over and over again. 

I don't feel totally satisfied by this argument though, because RL with KL penalty generally seems kind of unprincipled to me. I'd

I don't think it's particularly unprincipled; the KL penalty is just our way of encoding that the base policy is relatively reasonable (in a value-free way), and the model shouldn't deviate from it too much. Similarly, BoN is another way of encoding that the base policy is reasonable, albeit one that's easier to reason about on average.

Another way people regularize their RL is to mix in self-supervised training with RL (for example, they did this for text-davinci-003). I'm pretty sure this is also equivalent to RL w/ KL penalty.  

I am unsure whether other reasonable models of RL will similarly not have the causal Goodhart problem.

There's the impact penalties approach (e.g. AUP or RLSP), which seem more principled than KL penalties in cases where you more information on the space of rewards. 

You can also write do constrained optimization, which is again equivalent to modifying the reward function, but is way easier for humans to reason about and give feedback on.

In the first model I thought through, though, I don't think that you're right: if you train a model with RL with a KL penalty, it will end up with a policy that outputs a distribution over answers which is equivalent to taking the generative distribution and then applying a Boltzmann factor to upweight answers that your overseer likes. AFAICT this doesn't generally induce more causal Goodhart problems than best-of-N selection does.

As far as I can tell, the argument for this assumes that the model can generalise the reward function perfectly, which seems questionable because it only sees a few samples of the reward function (call this claim 1).

One possibility is that the Boltzmann factor upweights answers the model is confident the overseer will like more than answers it's less confident about. This could end up applying pressure to concentrate on high-confidence approved answers, which could break desirable correlations (call this claim 2).

I'm fairly confident in the claim that in general a Bayesian learner will not generalise the reward function perfectly (claim 1); how the reward function generalises will typically depend on the posterior over parameters and there are different posteriors that yield the same distribution over texts (Sam's sibling comment illustrates this point). I've no idea about what's going on with transformers, though - they can generalise text prediction quite well, so maybe they generalise approval quite well too, but that's just a wild guess. Claim 2 is idle speculation that I only mean to illustrate the point about pressure to break desirable correlations.

This is very interesting. I had previously thought the “KL penalty” being used in RLHF was just the local one that’s part of the PPO RL algorithm, but apparently I didn’t read the InstructGPT paper carefully enough.

I feel slightly better about RLHF now, but not much.

It’s true that minimizing KL subject to a constraint of always exceeding a certain reward threshold would theoretically be equivalent to Bayesian conditioning and therefore equivalent to filtering. That could be seen as a lexicographic objective where the binarised reward gets optimised first and then the KL penalty relative to the predictive model would restore the predictive model’s correlations (once the binarised reward is absolutely saturated). Unfortunately, this would be computationally difficult with gradient descent since you would already have mode-collapse before the KL penalty started to act.

In practice (in the InstructGPT paper, at least), we have a linear mixture of the reward and the global KL penalty. Obviously, if the global KL penalty is weighted at zero, it doesn’t help avoid Causal Goodhart, nor if it’s weighted at ${10}^{-10}$. Conversely, if it’s weighted at ${10}^{10}$, the model won’t noticeably respond to human feedback. I think this setup has a linear tradeoff between how much helpfulness you get and how much you avoid Causal Goodhart.

The ELBO argument in the post you linked requires explicitly transforming the reward into a Boltzmann distribution (relative to the prior of the purely predictive model) before using it in the objective function, which seems computationally difficult. That post also suggests some other alternatives to RLHF that are more like cleverly accelerated filtering, such as PPLM, and has a broad conclusion that RL doesn’t seem like the best framework for aligning LMs.

That being said, both of the things I said seem computationally difficult above also seem not-necessarily-impossible and would be research directions I would want to allocate a lot of thought to if I were leaning into RLHF as an alignment strategy.

Extremal Goodhart relies on a feasibility boundary in $U,V$-space that lacks orthogonality, in such a way that maximal $U$ logically implies non-maximal $V$. In the case of useful and human-approved answers, I expect that in fact, there exist maximally human-approved answers that are also maximally useful—even though there are also maximally human-approved answers that are minimally useful! I think the feasible zone here looks pretty orthogonal, pretty close to a Cartesian product, so Extremal Goodhart won't come up in either near-term or long-term applications. Near-term, it's Causal Goodhart [LW · GW] and Regressional Goodhart [LW · GW], and long-term, it might be Adversarial Goodhart [LW · GW].

Extremal Goodhart might come into play if, for example, there are some truths about what's useful that humans simply cannot be convinced of. In that case, I am fine with answers that pretend those things aren't true, because I think the scope of that extremal tradeoff phenomenon will be small enough to cope with for the purpose of ending the acute risk period. (I would not trust it in the setting of "ambitious value learning that we defer the whole lightcone to.")

For the record, I'm not very optimistic about filtering as an alignment scheme either, but in the setting of "let's have some near-term assistance with alignment research", I think Causal Goodhart [LW · GW] is a huge problem for RLHF that is not a problem for equally powerful filtering. Regressional Goodhart will be a problem in any case, but it might be manageable given a training distribution of human origin.

I agree with the general point, but I'll note that at equal proxy reward model scores, the RL policy has significantly more KL divergence with the base policy. 

In RLHF there are at least three different (stochastic) reward functions:

1. the learned value network
2. the “human clicks 👍/👎” process, and
3. the “what if we asked a whole human research group and they had unlimited time and assistance to deliberate about this one answer” process.

I think the first two correspond to what that paper calls “proxy” and “gold” but I am instead concerned with the ways in which 2 is a proxy for 3.

Briefly, the alternative optimisation target I would suggest is performance at achieving intelligible, formally specified goals within a purely predictive model/simulation of the real world.

Humans could then look at what happens in the simulations and say "gee, that doesn't look good," and specify better goals instead, and the policy won't experience gradient pressure to make those evaluations systematically wrong.

This isn't the place where I want to make a case for the "competitiveness" or tractability of that kind of approach, but what I want to claim here is that it is an example of an alignment paradigm that does leverage machine learning (both to make a realistic model of the world and to optimise policies for acting within that model) but does not directly use human approval (or an opaque model thereof) as an optimisation target in the kind of way that seems problematic about RLHF.

You can weaken the premises a lot:

1. Instead of having a single "level of performance", you can have different levels of performance for "ending the acute risk period" vs "treacherous turn". This allows you to get rid of the "higher" part of premise 3, which does seem pretty sketchy to me. It also allows you to talk about ${\lambda }_A$ factors for particular tasks on particular models rather than for "overall performance" in premise 4, which seems much more realistic (since ${\lambda }_A$ could vary for different tasks or different models).
2. You can get rid of Premise 5 entirely, if you have Premise 6 say "at least as likely" rather than "more likely".

I'd rewrite as:

Premise 1: There is some quantity called a "level of performance at a given task".

Definition 1: Let $P_S$ be the level of performance at whatever you want the model to do (e.g. ending the acute risk period), and $P_S^{\ast }$, is the level of performance you want.

Definition 2: Let $P_T$ be the level of performance at executing a treacherous turn, and $P_T^{\ast }$ be the level of performance required to be successful.

Definition 3: Consider a setting where we apply an alignment strategy $A$ to a particular model. Suppose that the model has latent capabilities sufficient to achieve performance $(P_S,P_T)$. Then, the resulting model must have actual capabilities $({\lambda }_S^A{P}_S,{\lambda }_T^A{P}_T)$ for some factors ${\lambda }_S^A,{\lambda }_T^A\in [0,\infty ]$. (If you assume that $A$ can only elicit capabilities rather than creating new capabilities, then you have ${\lambda }_{\ast }^A\le 1$.)

Lemma 1: Consider two alignment strategies $A$ and $B$ applied to a model M, where you are uncertain about the latent capabilities $(P_S,P_T)$, but you know that ${\lambda }_S^A>{\lambda }_S^B$, and ${\lambda }_T^A={\lambda }_T^B$. Then, it is at least as likely that ${\lambda }_S^A{P}_S\ge P_S^{\ast }\wedge {\lambda }_T^A{P}_T<P_T^{\ast }$ than that ${\lambda }_S^B{P}_S\ge P_S^{\ast }\wedge {\lambda }_T^B{P}_T<P_T^{\ast }$.

Premise 2: ${\lambda }_S^{RLHF}>{\lambda }_S^{prompt}$

Premise 3: ${\lambda }_T^{RLHF}={\lambda }_T^{prompt}$ . This corresponds to Buck's point, which I agree with, here:

I don’t think RLHF seems worse than these other ways you could try to get your model to use its capabilities to be helpful. (Many LessWrong commenters disagree with me here but I haven’t heard them give any arguments that I find compelling.)

Conclusion: RLHF is at least as likely to allow you to [reach the desired level of performance at the task while avoiding a treacherous turn] as prompt engineering.

(If you consider the task "end the acute risk period", then this leads to the same conclusion as in your argument, though that bakes in an assumption that we will use a single AI model to end the acute risk period, which I don't agree with.)

Critiquing the premises of this new argument:

Premise 1 is a simplification but seems like a reasonable one.

Premises 2 and 3 are not actually correct: the more accurate thing to say here is that this is true in expectation given our uncertainty about the underlying parameters. Unfortunately lemma 1 stops being a theorem if you have distributions over ${\lambda }_T^A$ and ${\lambda }_T^B$. Still, I think the fact that this argument if you replace distributions by their expectations should still be a strong argument in favor of the conclusion absent some reason to disbelieve it.