Many reinforcement learning (RL) environments in practice feature enormous state spaces that may be described compactly by a “factored” structure, that may be modeled by Factored Markov Decision Processes (FMDPs). We present the first polynomial time algorithm for RL in Factored State MDPs (generalizing FMDPs) that neither relies on an oracle planner nor requires a linear transition model; it only requires a linear value function with a suitable local basis with respect to the factorization, permitting efficient variable elimination. With this assumption, we can solve this family of Factored State MDPs in polynomial time by constructing an efficient separation oracle for convex optimization. Importantly, and in contrast to prior work on FMDPs, we do not assume that the transitions on various factors areconditionally independent.