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 methods to optimize targets which might be tough or pricey to measure. It’s usually essential to introduce some **proxy goal** that’s simpler or cheaper to measure, however after we do that, we should 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 wish to optimize issues like “How useful is that this response?”, or “How factually correct is that this declare?”. These are advanced targets that require people to rigorously verify issues over. For that reason, we practice a mannequin to foretell these human preferences, generally known as a **reward mannequin**, and use the reward mannequin’s predictions as a proxy goal. However it’s necessary to maintain observe of how nicely the true goal is being optimized.

On this submit we’ll take a look at a few of the arithmetic behind how we do that. We’ll concentrate on a setting that’s significantly clear to investigate, through which now we have entry to the true goal. In observe, even human preferences can fail to measure what we actually care about, however we’re setting that situation apart on this submit.

## Greatest-of-$n$ sampling

There are numerous methods through which one may optimize the proxy goal, however maybe the only is **best-of-$n$ sampling**, often known as **rejection sampling** or **reranking**. We merely pattern $n$ instances and take the one which scores the best in response to the proxy goal.

Though this technique could be very easy, it could actually truly 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 partly 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 easy 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 examine best-of-$n$ sampling extra formally. Suppose now we have some pattern house $S$ (such because the set of attainable 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 in some way optimize $R_{textual content{proxy}}$ and thereby receive some new distribution $P^prime$. Then:

- The expectation $mathbb E_{x^primesim P^prime}left[R_{text{true}}left(x^primeright)right]$ measures how nicely 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 completed. 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 unfavourable 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 might be estimated effectively utilizing samples from $P$.

Let’s take a look at the expectation first. The naive strategy is to make use of a Monte Carlo estimator: run best-of-$n$ sampling many instances, 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 think about *each attainable subset* of those samples of dimension $n$, weight every pattern by the variety of subsets for which it’s the greatest in response to 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 $ok$ is the rank of the pattern underneath the proxy goal, from $1$ (worst) as much as $N$ (greatest). 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 an actual formulation 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 appropriate: 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

## Going past best-of-$n$ sampling

The primary 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 to date, 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 as a consequence of Goodhart’s regulation. We’d need to take $n$ to be round 60,000 to achieve 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 targets than reinforcement studying. Intuitively, best-of-$n$ is the “brute pressure” 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 targets as a part of our work to align our fashions with human intent and values. If you happen to’d like to assist us with this analysis, we’re hiring!*