Generalized Linear Markov Decision Process

Offline reinforcement learning for longitudinal studies often faces two linked challenges: rewards may be binary or bounded, and reward observations may be available only for a subset of trajectories or time points even when the corresponding state-action-next-state histories are available. Linear Markov decision process methods are tractable because Bellman backups remain linear, but they require linear rewards and do not indicate how transition-only observations should be used. We introduce GRASP-MDP, Generalized Reward And Semi-supervised Pessimism for Markov Decision Processes, a reward-transition separated framework addressing both issues through one Bellman decomposition. It preserves linear transition dynamics while modeling reward means through generalized linear models. Although this breaks the usual linear Bellman form, the backup remains explicit as a nonlinear reward component plus a linear continuation component, yielding a nonlinear-plus-linear Bellman-complete class. In the resulting recursion, observed rewards estimate the generalized reward component, while all available transitions estimate the continuation component. The resulting pessimistic value iteration controls the two estimation errors separately without reward imputation. Finite-sample guarantees show that transition-only observations reduce transition-estimation error while reward uncertainty remains governed by observed rewards. Simulations and a multiple sclerosis electronic health record application illustrate the empirical benefit of retaining transition-only observations.