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 optimize goals which can be troublesome or expensive to measure. It’s typically essential to introduce some **proxy goal** that’s simpler or cheaper to measure, however once 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 wish 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 verify issues over. For that reason, we prepare a mannequin to foretell these human preferences, referred to as a **reward mannequin**, and use the reward mannequin’s predictions as a proxy goal. However it’s necessary to maintain monitor of how effectively the true goal is being optimized.

On this put up we’ll have a look at among the arithmetic behind how we do that. We’ll give attention to a setting that’s significantly clear to research, through which now we have entry to the true goal. In apply, even human preferences can fail to measure what we actually care about, however we’re setting that concern apart on this put up.

## Finest-of-$n$ sampling

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

Though this technique may be very easy, it might truly be aggressive with extra superior methods reminiscent of 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 major enhance to human preferences.

As well as, best-of-$n$ sampling has dependable efficiency and is simple to research 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 area $S$ (such because the set of potential 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 someway 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 effectively 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 detrimental 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 method 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. Nevertheless, there’s a higher estimator. If now we have $Ngeq n$ samples from $P$ total, then we are able to concurrently contemplate *each potential subset* of those samples of measurement $n$, weight every pattern by the variety of subsets for which it’s the finest in accordance with 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 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 formulation that works for any steady likelihood distribution $P$ (i.e., so long as $P$ has no level lots). One would possibly 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 enable 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:

Finest-of-$n$ efficiency for WebGPT 175B

## 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 sometimes use reinforcement studying. Within the settings we’ve studied thus far, reminiscent of summarization, we’ve sometimes been in a position to attain a KL of round 10 nats utilizing reinforcement studying earlier than the true goal begins to lower attributable to 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.

Nevertheless, 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 pressure” method, making it extra information-theoretically environment friendly than reinforcement studying, however much less computationally environment friendly at massive KLs.

*We’re actively learning the scaling properties of proxy goals as a part of our work to align our fashions with human intent and values. Should you’d like to assist us with this analysis, we’re hiring!*