  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
$\widetilde{O}(K^{\frac{5}{6}}A^{\frac{1}{2}}d\ln(1+M)/(1-\gamma)^2)$ 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 $\gamma$ 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 $\Omega(\frac{\gamma^2}{1-\gamma} \sqrt{d A K})$
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.
