Revisiting scaling laws for reward optimization

Ali Aouad, Aymane El Gadarri, Vivek F. Farias

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 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. Informed by this, we conduct extensive empirical evaluations using a real-world annotation setup, whereby…