Lifted Bellman Linear Programming for Offline Reinforcement Learning

Hyukjun Yang, Jongchan Park, Narim Jeong, Donghwan Lee

Offline reinforcement learning (RL) typically trains a critic by minimizing a regression loss against bootstrapped value targets stabilized by target networks with exponential moving average (EMA) updates. Multi-step targets incorporate behavior-policy actions and therefore require off-policy correction. We instead impose in-sample Bellman optimality on the critic through inequality constraints. We formulate the Lifted Bellman Linear Program (LBLP), which lifts the linear programming characterization of Bellman optimality to the joint $(Q,V)$ space so that every constraint involves only state-action pairs in the dataset. Its unique minimizer is the in-sample optimal pair, and constraints along $K$-step segments of dataset trajectories leave this minimizer unchanged for any rollout policy and horizon. Under deterministic dynamics, this minimizer lies between the best dataset return and the optimal value. Relaxing the constraints into hinge penalties recovers the same solution above a finite penalty coefficient in the tabular case. Approximate Lifted Bellman Unconstrained Minimization (ALBUM) implements this relaxation with neural networks and detaches the $K$-step rollout targets…