Reinforcement Learning with Multi-Step Transition Predictions: A Bellman Theory

Multi-step predictions of future transition realizations can improve reinforcement learning, but their value remains difficult to quantify. Directly including these predictions in the state can lead to exponential state-space growth, making direct dynamic programming computationally expensive. We study discounted finite MDPs under a block-feedback protocol: at the start of each K-period block, the decision maker observes $K$-step probabilistic transition predictions for a subset of actions and adapts decisions to observed states within the block. We show that the optimal value satisfies a Bayesian Bellman equation on the original state space. This representation reveals that predictive information creates value by recursively changing the local order of maximization and expectation in the Bellman expansion. We formalize this effect through a Bellman--Jensen analysis and quantify the effects of prediction horizon, error, action coverage, and within-block feedback. We propose block-feedback Bayesian offline learning with online adaptation (BF--BOLA), a model-based algorithm for learning near-optimal policies, and establish upper and lower sample-complexity bounds that match up to logarithmic factors in the stated regimes. The results show that predictions can reduce transition learning to uncovered rows and that learning a near-optimal policy can require fewer prediction samples than estimating the Bayesian value function.