Scaling laws for optimization against reward models in AI alignment have pinned down how performance depends on optimization effort---measured by a KL-divergence budget relative to a reference policy.
Beyond a certain budget, over-optimization (or reward hacking) can arise: because we optimize against a proxy reward model (distinct from true rewards), performance can plateau or degrade.
Naturally, the proxy reward's accuracy depends on how much preference data (often in the form of pairwise comparisons) was used to train it.
However, existing research does not cleanly identify how performance jointly scales with the amount of training data and the divergence budget.
Our Main Contribution
Our main contribution is to provide an empirically accurate and theoretically grounded scaling law in such context.
Performance roughly scales as $Θ(\sqrt{\min\{\log(M),K\}})$, where $M$ is the number of comparisons in training data and $K$ is the policy's divergence budget.
We develop an information-theoretic model to establish this upper bound and prove it is tightly achievable through a constructive procedure.
Empirical Evaluations
Informed by this, we conduct extensive empirical evaluations using a real-world annotation setup, whereby a large 70B gold reward model generates feedback data and proxy reward models are trained from less capable models (0.6B to 4B).
Our scaling law provides an excellent fit (R2 from 97% to 99%), outperforms alternative specifications, and remains robust across model sizes, noise, and optimization procedures (best-of-$N$ or policy tilting).
Evidence and Analogy
Our evidence suggests that reward optimization is analogous to a surprisingly simple selection task: choosing from a sequence of IID Gaussian random variables using noisy preference feedback.