  We study the Safe Reinforcement Learning (SRL) problem using the Constrained
Markov Decision Process (CMDP) formulation in which an agent aims to maximize
the expected total reward subject to a safety constraint on the expected total
value of a utility function. We focus on an episodic setting with the function
approximation where the Markov transition kernels have a linear structure but
do not impose any additional assumptions on the sampling model. Designing SRL
algorithms with provable computational and statistical efficiency is
particularly challenging under this setting because of the need to incorporate
both the safety constraint and the function approximation into the fundamental
exploitation/exploration tradeoff. To this end, we present an
\underline{O}ptimistic \underline{P}rimal-\underline{D}ual Proximal Policy
\underline{OP}timization (OPDOP) algorithm where the value function is
estimated by combining the least-squares policy evaluation and an additional
bonus term for safe exploration. We prove that the proposed algorithm achieves
an $\tilde{O}(d H^{2.5}\sqrt{T})$ regret and an $\tilde{O}(d H^{2.5}\sqrt{T})$
constraint violation, where $d$ is the dimension of the feature mapping, $H$ is
the horizon of each episode, and $T$ is the total number of steps. These bounds
hold when the reward/utility functions are fixed but the feedback after each
episode is bandit. Our bounds depend on the capacity of the state-action space
only through the dimension of the feature mapping and thus our results hold
even when the number of states goes to infinity. To the best of our knowledge,
we provide the first provably efficient online policy optimization algorithm
for CMDP with safe exploration in the function approximation setting.
