Catastrophic Goodhart: regularizing RLHF with KL divergence does not mitigate heavy-tailed reward misspecification

TL;DR

This paper demonstrates that KL regularization in reinforcement learning from human feedback does not prevent reward hacking when reward errors are heavy-tailed, leading to catastrophic outcomes.

Contribution

It introduces the concept of catastrophic Goodhart, showing that heavy-tailed reward errors can cause policies to exploit reward misspecification despite KL regularization.

Findings

Reward errors are light-tailed in measured models.

Heavy-tailed reward errors are common in real-world applications.

KL regularization fails to prevent reward hacking with heavy-tailed errors.

Abstract

When applying reinforcement learning from human feedback (RLHF), the reward is learned from data and, therefore, always has some error. It is common to mitigate this by regularizing the policy with KL divergence from a base model, with the hope that balancing reward with regularization will achieve desirable outcomes despite this reward misspecification. We show that when the reward function has light-tailed error, optimal policies under less restrictive KL penalties achieve arbitrarily high utility. However, if error is heavy-tailed, some policies obtain arbitrarily high reward despite achieving no more utility than the base model--a phenomenon we call catastrophic Goodhart. We adapt a discrete optimization method to measure the tails of reward models, finding that they are consistent with light-tailed error. However, the pervasiveness of heavy-tailed distributions in many real-world applications indicates that future sources of RL reward could have heavy-tailed error, increasing the likelihood of reward hacking even with KL regularization.