  In the optimization of dynamic systems, the variables typically have
constraints. Such problems can be modeled as a Constrained Markov Decision
Process (CMDP). This paper considers the peak Constrained Markov Decision
Process (PCMDP), where the agent chooses the policy to maximize total reward in
the finite horizon as well as satisfy constraints at each epoch with
probability 1. We propose a model-free algorithm that converts PCMDP problem to
an unconstrained problem and a Q-learning based approach is applied. We define
the concept of probably approximately correct (PAC) to the proposed PCMDP
problem. The proposed algorithm is proved to achieve an $(\epsilon,p)$-PAC
policy when the episode $K\geq\Omega(\frac{I^2H^6SA\ell}{\epsilon^2})$, where
$S$ and $A$ are the number of states and actions, respectively. $H$ is the
number of epochs per episode. $I$ is the number of constraint functions, and
$\ell=\log(\frac{SAT}{p})$. We note that this is the first result on PAC kind
of analysis for PCMDP with peak constraints, where the transition dynamics are
not known apriori. We demonstrate the proposed algorithm on an energy
harvesting problem and a single machine scheduling problem, where it performs
close to the theoretical upper bound of the studied optimization problem.
