Overcoming the Long Horizon Barrier for Sample-Efficient Reinforcement Learning with Latent Low-Rank Structure

The practicality of reinforcement learning algorithms has been limited due to poor scaling with respect to the problem size, as the sample complexity of learning an ε-optimal policy is Ω(|S||A|H/ ε2) over worst case instances of an MDP with state space S, action space A, and horizon H. We consider a class of MDPs for which the associated optimal Q* function is low rank, where the latent features are unknown. While one would hope to achieve linear sample complexity in |S| and |A| due to the low rank structure, we show that without imposing further assumptions beyond low rank of Q*, if one is constrained to estimate the Q function using only observations from a subset of entries, there is a worst case instance in which one must incur a sample complexity exponential in the horizon H to learn a near optimal policy. We subsequently show that under stronger low rank structural assumptions, given access to a generative model, Low Rank Monte Carlo Policy Iteration (LR-MCPI) and Low Rank Empirical Value Iteration (LR-EVI) achieve the desired sample complexity of Õ((|S|+|A|)poly (d,H)/ε2) for a rank d setting, which is minimax optimal with respect to the scaling of |S|, |A|, and ε. In contrast to literature on linear and low-rank MDPs, we do not require a known feature mapping, our algorithm is computationally simple, and our results hold for long time horizons. Our results provide insights on the minimal low-rank structural assumptions required on the MDP with respect to the transition kernel versus the optimal action-value function.