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Goodhart’s regulation famously says: “When a measure turns into a goal, it ceases to be an excellent measure.” Though initially from economics, it’s one thing now we have to grapple with at OpenAI when determining the best way to optimize goals which can be tough or pricey to measure. It’s typically essential to introduce some proxy goal that’s simpler or cheaper to measure, however after we do that, we have to be cautious to not optimize it an excessive amount of.
For instance, as a part of our work to align fashions like GPT-3 with human intent and values, we want to optimize issues like “How useful is that this response?”, or “How factually correct is that this declare?”. These are complicated goals that require people to fastidiously test issues over. For that reason, we practice a mannequin to foretell these human preferences, often known as a reward mannequin, and use the reward mannequin’s predictions as a proxy goal. However it’s vital to maintain observe of how properly the true goal is being optimized.
On this put up we’ll have a look at a few of the arithmetic behind how we do that. We’ll concentrate on a setting that’s notably clear to investigate, during which now we have entry to the true goal. In follow, even human preferences can fail to measure what we actually care about, however we’re setting that problem apart on this put up.
Greatest-of-$n$ sampling
There are lots of methods during which one may optimize the proxy goal, however maybe the best is best-of-$n$ sampling, also referred to as rejection sampling or reranking. We merely pattern $n$ occasions and take the one which scores the very best based on the proxy goal.
Though this technique may be very easy, it may well really be aggressive with extra superior strategies comparable to reinforcement studying, albeit at the price of extra inference-time compute. For instance, in WebGPT, our best-of-$64$ mannequin outperformed our reinforcement studying mannequin, maybe partially as a result of the best-of-$64$ mannequin bought to browse many extra web sites. Even making use of best-of-$4$ supplied a big enhance to human preferences.
As well as, best-of-$n$ sampling has dependable efficiency and is simple to investigate mathematically, making it well-suited to empirical research of Goodhart’s regulation and associated phenomena.
The arithmetic of best-of-$n$ sampling
Let’s research best-of-$n$ sampling extra formally. Suppose now we have some pattern house $S$ (such because the set of doable question-answer pairs), some likelihood distribution $P$ over $S$, a real goal (or “reward”) $R_{textual content{true}}:Stomathbb R$, and a proxy goal $R_{textual content{proxy}}:Stomathbb R$. Let’s say that we by some means optimize $R_{textual content{proxy}}$ and thereby acquire some new distribution $P^prime$. Then:
- The expectation $mathbb E_{x^primesim P^prime}left[R_{text{true}}left(x^primeright)right]$ measures how properly now we have optimized the true goal.
- The KL divergence $D_{textual content{KL}}left(P^primeparallel Pright)$ measures how a lot optimization now we have executed. For instance, if $P^prime$ is obtained by taking the primary pattern from $P$ that lies in some subset $S^primesubseteq S$, then this KL divergence is simply the destructive log likelihood {that a} pattern from $P$ lies in $S^prime$.
It seems that within the case of best-of-$n$ sampling, each of those portions could be estimated effectively utilizing samples from $P$.
Let’s have a look at the expectation first. The naive strategy is to make use of a Monte Carlo estimator: run best-of-$n$ sampling many occasions, measure the true goal on these samples, and common the outcomes. Nonetheless, there’s a higher estimator. If now we have $Ngeq n$ samples from $P$ total, then we are able to concurrently take into account each doable subset of those samples of measurement $n$, weight every pattern by the variety of subsets for which it’s the finest based on the proxy goal, after which take the weighted common true goal rating. This weight is simply the binomial coefficient $binom{k-1}{n-1}$, the place $okay$ is the rank of the pattern beneath the proxy goal, from $1$ (worst) as much as $N$ (finest). In addition to utilizing samples extra effectively, this additionally permits us to reuse samples for various values of $n$.
As for the KL divergence, surprisingly, this seems to have a precise components that works for any steady likelihood distribution $P$ (i.e., so long as $P$ has no level plenty). One may naively guess that the reply is $log n$, since best-of-$n$ is doing one thing like taking the highest $frac 1n$ of the distribution, and that is roughly right: the precise reply is $log n-frac{n-1}n$.
Collectively, these estimators permit us to simply analyze how the true goal varies with the quantity of optimization utilized to the proxy goal.
Right here’s a real-life instance from WebGPT:
Greatest-of-$n$ efficiency for WebGPT 175B
Greatest-of-$n$ efficiency for WebGPT, with shaded areas representing $pm 1$ commonplace error, and the KL axis following a sq. root scale. Right here, the unique distribution ($P$) is given by the 175B mannequin skilled utilizing conduct cloning, the proxy goal used to compute best-of-$n$ ($R_{textual content{proxy}}$) is given by the coaching reward mannequin, and we take into account three putatively “true” goals ($R_{textual content{true}}$): the coaching reward mannequin itself, a validation reward mannequin skilled on held-out knowledge, and precise human preferences. There is not a lot over-optimization of the proxy goal, however we’d anticipate there to be at increased KLs.
Going past best-of-$n$ sampling
The principle limitation of best-of-$n$ sampling is that the KL divergence grows logarithmically with $n$, so it’s only appropriate for making use of a small quantity of optimization.
To use extra optimization, we usually use reinforcement studying. Within the settings we’ve studied up to now, comparable to summarization, we’ve usually been in a position to attain a KL of round 10 nats utilizing reinforcement studying earlier than the true goal begins to lower because of Goodhart’s regulation. We’d should take $n$ to be round 60,000 to succeed in this KL utilizing best-of-$n$, and we hope to have the ability to attain a lot bigger KLs than this with enhancements to our reward modeling and reinforcement studying practices.
Nonetheless, not all nats are equal. Empirically, for small KL budgets, best-of-$n$ higher optimizes each the proxy and the true goals than reinforcement studying. Intuitively, best-of-$n$ is the “brute power” strategy, making it extra information-theoretically environment friendly than reinforcement studying, however much less computationally environment friendly at massive KLs.
We’re actively finding out the scaling properties of proxy goals as a part of our work to align our fashions with human intent and values. In the event you’d like to assist us with this analysis, we’re hiring!
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