  Constrained Markov Decision Processes (CMDPs) formalize sequential
decision-making problems whose objective is to minimize a cost function while
satisfying constraints on various cost functions. In this paper, we consider
the setting of episodic fixed-horizon CMDPs. We propose an online algorithm
which leverages the linear programming formulation of finite-horizon CMDP for
repeated optimistic planning to provide a probably approximately correct (PAC)
guarantee on the number of episodes needed to ensure an $\epsilon$-optimal
policy, i.e., with resulting objective value within $\epsilon$ of the optimal
value and satisfying the constraints within $\epsilon$-tolerance, with
probability at least $1-\delta$. The number of episodes needed is shown to be
of the order
$\tilde{\mathcal{O}}\big(\frac{|S||A|C^{2}H^{2}}{\epsilon^{2}}\log\frac{1}{\delta}\big)$,
where $C$ is the upper bound on the number of possible successor states for a
state-action pair. Therefore, if $C \ll |S|$, the number of episodes needed
have a linear dependence on the state and action space sizes $|S|$ and $|A|$,
respectively, and quadratic dependence on the time horizon $H$.
