  This paper explores the realm of infinite horizon average reward Constrained
Markov Decision Processes (CMDP). To the best of our knowledge, this work is
the first to delve into the regret and constraint violation analysis of average
reward CMDPs with a general policy parametrization. To address this challenge,
we propose a primal dual based policy gradient algorithm that adeptly manages
the constraints while ensuring a low regret guarantee toward achieving a global
optimal policy. In particular, we demonstrate that our proposed algorithm
achieves $\tilde{\mathcal{O}}({T}^{4/5})$ objective regret and
$\tilde{\mathcal{O}}({T}^{4/5})$ constraint violation bounds.
