Matrix Estimation for Offline Reinforcement Learning with Low-Rank
  Structure
Xumei Xi Christina Lee Yu Yudong Chen
We consider offline Reinforcement Learning (RL), where the agent does not
interact with the environment and must rely on offline data collected using a
behavior policy. Previous works provide policy evaluation guarantees when the
target policy to be evaluated is covered by the behavior policy, that is,
state-action pairs visited by the target policy must also be visited by the
behavior policy. We show that when the MDP has a latent low-rank structure,
this coverage condition can be relaxed. Building on the connection to weighted
matrix completion with non-uniform observations, we propose an offline policy
evaluation algorithm that leverages the low-rank structure to estimate the
values of uncovered state-action pairs. Our algorithm does not require a known
feature representation, and our finite-sample error bound involves a novel
discrepancy measure quantifying the discrepancy between the behavior and target
policies in the spectral space. We provide concrete examples where our
algorithm achieves accurate estimation while existing coverage conditions are
not satisfied. Building on the above evaluation algorithm, we further design an
offline policy optimization algorithm and provide non-asymptotic performance
guarantees.