  We study reinforcement learning for partially observed Markov decision
processes (POMDPs) with infinite observation and state spaces, which remains
less investigated theoretically. To this end, we make the first attempt at
bridging partial observability and function approximation for a class of POMDPs
with a linear structure. In detail, we propose a reinforcement learning
algorithm (Optimistic Exploration via Adversarial Integral Equation or
OP-TENET) that attains an $\epsilon$-optimal policy within $O(1/\epsilon^2)$
episodes. In particular, the sample complexity scales polynomially in the
intrinsic dimension of the linear structure and is independent of the size of
the observation and state spaces.
  The sample efficiency of OP-TENET is enabled by a sequence of ingredients:
(i) a Bellman operator with finite memory, which represents the value function
in a recursive manner, (ii) the identification and estimation of such an
operator via an adversarial integral equation, which features a smoothed
discriminator tailored to the linear structure, and (iii) the exploration of
the observation and state spaces via optimism, which is based on quantifying
the uncertainty in the adversarial integral equation.
