Goodhart’s regulation famously says: “When a measure turns into a goal, it ceases to be a superb measure.” Though initially from economics, it’s one thing we’ve to grapple with at OpenAI when determining find out how to optimize aims 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 must 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 aims that require people to rigorously test issues over. Because of this, 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. But it surely’s essential to maintain observe of how nicely the true goal is being optimized.

On this put up we’ll take a look at a number of the arithmetic behind how we do that. We’ll deal with a setting that’s notably clear to investigate, by which we’ve entry to the true goal. In apply, 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 various methods by which one might 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 best in accordance with the proxy goal.

Though this technique may be very easy, it could possibly truly be aggressive with extra superior strategies similar 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 received to browse many extra web sites. Even making use of best-of-$4$ supplied a major increase 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 examine best-of-$n$ sampling extra formally. Suppose we’ve 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 one way or the other 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 we’ve optimized the true goal.
- The KL divergence $D_{textual content{KL}}left(P^primeparallel Pright)$ measures how a lot optimization we’ve 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 may 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 occasions, measure the true goal on these samples, and common the outcomes. Nonetheless, there’s a higher estimator. If we’ve $Ngeq n$ samples from $P$ general, then we will concurrently contemplate *each attainable subset* of those samples of dimension $n$, weight every pattern by the variety of subsets for which it’s the greatest 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 $N$ (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 lots). 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 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:

Greatest-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 is just 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, similar to summarization, we’ve sometimes been capable of attain a KL of round 10 nats utilizing reinforcement studying earlier than the true goal begins to lower resulting from Goodhart’s regulation. We’d should 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 aims 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 giant KLs.

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