Preventing Reward Hacking with Occupancy Measure Regularization

Reward hacking occurs when an agent performs very well with respect to a "proxy" reward function (which may be hand-specified or learned), but poorly with respect to the unknown true reward. Since ensuring good alignment between the proxy and true reward is extremely difficult, one approach to prevent reward hacking is optimizing the proxy conservatively. Prior work has particularly focused on enforcing the learned policy to behave similarly to a "safe" policy by penalizing the KL divergence between their action distributions (AD). However, AD regularization doesn't always work well since a small change in action distribution at a single state can lead to potentially calamitous outcomes, while large changes might not be indicative of any dangerous activity. Our insight is that when reward hacking, the agent visits drastically different states from those reached by the safe policy, causing large deviations in state occupancy measure (OM). Thus, we propose regularizing based on the OM divergence between policies instead of AD divergence to prevent reward hacking. We theoretically establish that OM regularization can more effectively avoid large drops in true reward. Then, we empirically demonstrate in a variety of realistic environments that OM divergence is superior to AD divergence for preventing reward hacking by regularizing towards a safe policy. Furthermore, we show that occupancy measure divergence can also regularize learned policies away from reward hacking behavior. Our code and data are available at https://github.com/cassidylaidlaw/orpo