Reinforcement learning (RL) in Partially Observable Markov Decision Processes (POMDPs) is computationally challenging due to the curse of long-range dependence on history. Fortunately, recent studies have shown that a range of POMDPs that satisfy the filter stability assumption can be well approximated by a superstate Markov Decision Process (MDP) based on a truncated history in a finite-memory window, with a value gap that decays with the window length. However, existing finite-time convergence analyses on such superstate MDP surrogates are either limited to over-simplified linear function approximation or rely on an i.i.d. sampling oracle that is violated by POMDP trajectories, both of which severely limit the applicability of these theoretical results in practice. This motivates us to bridge the gap by investigating RL in POMDPs with nonlinear neural value function approximation. Our main contributions are threefold: i) We first establish a new finite-state MDP point-wise approximation error bound, which lays the foundation for our RL algorithm design and analysis in POMDPs; ii) By applying Temporal-Difference (TD) learning to POMDP-generated superstate transitions and targeting the Bellman fixed point of the superstate MDP, we prove a finite-time critic error bound with nonlinear neural value function approximation; and iii) By combining the neural critic function approximation error bound with the policy mirror descent framework, we obtain a finite-time sublinear regret against the optimal POMDP value. To the best of our knowledge, such a finite-time sublinear regret based on neural function approximation is the first in the literature for finite-memory POMDPs. Our numerical experiments also verify our theoretical results.

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