Reward Hacking and Goodhart's Law

Any optimizer strong enough to maximize a proxy reward will find the gap between the proxy and the thing you actually wanted. Gao et al. (ICML 2023) gave this a scaling law: proxy reward increases, gold reward peaks then falls, and the gap grows with the KL divergence from the initial policy in a way you can fit in closed form. Sycophancy, verbosity bias, unfaithful chain-of-thought, and evaluator tampering are not separate problems. They are the same problem in different costumes.