Regret Is Freedom Lost

How much of a reinforcement learning agent's regret comes from its memory, and how much from its policy? Its memory \(M\) represents histories by memory states. A policy must make the same prediction for the same task whenever histories share a memory state, so merging can incur irreducible loss. Together with avoidable errors in decisions, this is weighted regret. I prove that least normalised regret equals \(K_\rho(M)\), the weight deleted to leave one correct prediction per memory state and task. Policies that predict the same past can diverge when predicting the future. Freedom of function measures how many futures remain open to a policy, while simplicity measures its encoding. In encodings where freedom and simplicity are identical, \(K_\rho(M)\) is weighted freedom lost. Under the corresponding independent prior over unseen futures, \(e^{-K_\rho(M)}\) gives the probability of generalisation. Freedom selects trained memories using unlabelled target histories. On unseen tasks, it cut regret by 69% on Repeat Previous against choosing by regret on the trained task. On a delayed cue, it cut regret by 58%. There, even random selection beat choosing by regret on the trained task. This operationalises freedom of function for reinforcement learning, opening doors for future research.