Lower Bounds for Learning in Revealing POMDPs

This paper studies the fundamental limits of reinforcement learning (RL) in the challenging *partially observable* setting. While it is well-established that learning in Partially Observable Markov Decision Processes (POMDPs) requires exponentially many samples in the worst case, a surge of recent work shows that polynomial sample complexities are achievable under the *revealing condition*---A natural condition that requires the observables to reveal some information about the unobserved latent states. However, the fundamental limits for learning in revealing POMDPs are much less understood, with existing lower bounds being rather preliminary and having substantial gaps from the current best upper bounds. We establish strong PAC and regret lower bounds for learning in revealing POMDPs. Our lower bounds scale polynomially in all relevant problem parameters in a multiplicative fashion, and achieve significantly smaller gaps against the current best upper bounds, providing a solid starting point for future studies. In particular, for *multi-step* revealing POMDPs, we show that (1) the latent state-space dependence is at least $\Omega(S^{1.5})$ in the PAC sample complexity, which is notably harder than the $\widetilde{\Theta}(S)$ scaling for fully-observable MDPs; (2) Any polynomial sublinear regret is at least $\Omega(T^{2/3})$, suggesting its fundamental difference from the *single-step* case where $\widetilde{\mathcal{O}}(\sqrt{T})$ regret is achievable. Technically, our hard instance construction adapts techniques in *distribution testing*, which is new to the RL literature and may be of independent interest. We also complement our results with new sharp regret upper bounds for *strongly B-stable PSRs*, which include single-step revealing POMDPs as a special case.