Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning
This addresses the challenge of efficient and accurate matrix estimation for low-rank RL problems, offering incremental improvements in performance guarantees.
The paper tackles the problem of low-rank matrix estimation in reinforcement learning, where each entry of the matrix (e.g., rewards or transitions) must be accurately estimated despite correlated data, and shows that spectral-based methods achieve nearly-minimal entry-wise error, enabling state-of-the-art algorithms for low-rank bandits and MDPs.
We study matrix estimation problems arising in reinforcement learning (RL) with low-rank structure. In low-rank bandits, the matrix to be recovered specifies the expected arm rewards, and for low-rank Markov Decision Processes (MDPs), it may for example characterize the transition kernel of the MDP. In both cases, each entry of the matrix carries important information, and we seek estimation methods with low entry-wise error. Importantly, these methods further need to accommodate for inherent correlations in the available data (e.g. for MDPs, the data consists of system trajectories). We investigate the performance of simple spectral-based matrix estimation approaches: we show that they efficiently recover the singular subspaces of the matrix and exhibit nearly-minimal entry-wise error. These new results on low-rank matrix estimation make it possible to devise reinforcement learning algorithms that fully exploit the underlying low-rank structure. We provide two examples of such algorithms: a regret minimization algorithm for low-rank bandit problems, and a best policy identification algorithm for reward-free RL in low-rank MDPs. Both algorithms yield state-of-the-art performance guarantees.