Learning Adversarial Low-rank Markov Decision Processes with Unknown Transition and Full-information Feedback
This addresses the challenge of reinforcement learning with nonlinear function approximation and adversarial losses, showing low-rank MDPs are statistically harder than linear MDPs.
The paper tackles the problem of learning low-rank Markov Decision Processes with adversarially changing losses in a full-information feedback setting, proposing the POLO algorithm that achieves a sublinear regret bound of O~(K^{5/6}A^{1/2}d ln(1+M)/(1-γ)^2) and proving a matching lower bound of Ω(γ^2/(1-γ)√(dAK)).
In this work, we study the low-rank MDPs with adversarially changed losses in the full-information feedback setting. In particular, the unknown transition probability kernel admits a low-rank matrix decomposition \citep{REPUCB22}, and the loss functions may change adversarially but are revealed to the learner at the end of each episode. We propose a policy optimization-based algorithm POLO, and we prove that it attains the $\widetilde{O}(K^{\frac{5}{6}}A^{\frac{1}{2}}d\ln(1+M)/(1-γ)^2)$ regret guarantee, where $d$ is rank of the transition kernel (and hence the dimension of the unknown representations), $A$ is the cardinality of the action space, $M$ is the cardinality of the model class, and $γ$ is the discounted factor. Notably, our algorithm is oracle-efficient and has a regret guarantee with no dependence on the size of potentially arbitrarily large state space. Furthermore, we also prove an $Ω(\frac{γ^2}{1-γ} \sqrt{d A K})$ regret lower bound for this problem, showing that low-rank MDPs are statistically more difficult to learn than linear MDPs in the regret minimization setting. To the best of our knowledge, we present the first algorithm that interleaves representation learning, exploration, and exploitation to achieve the sublinear regret guarantee for RL with nonlinear function approximation and adversarial losses.