  We consider reinforcement learning (RL) in Markov Decision Processes in which
an agent repeatedly interacts with an environment that is modeled by a
controlled Markov process. At each time step $t$, it earns a reward, and also
incurs a cost-vector consisting of $M$ costs. We design model-based RL
algorithms that maximize the cumulative reward earned over a time horizon of
$T$ time-steps, while simultaneously ensuring that the average values of the
$M$ cost expenditures are bounded by agent-specified thresholds
$c^{ub}_i,i=1,2,\ldots,M$.
  In order to measure the performance of a reinforcement learning algorithm
that satisfies the average cost constraints, we define an $M+1$ dimensional
regret vector that is composed of its reward regret, and $M$ cost regrets. The
reward regret measures the sub-optimality in the cumulative reward, while the
$i$-th component of the cost regret vector is the difference between its $i$-th
cumulative cost expense and the expected cost expenditures $Tc^{ub}_i$.
  We prove that the expected value of the regret vector of UCRL-CMDP, is
upper-bounded as $\tilde{O}\left(T^{2\slash 3}\right)$, where $T$ is the time
horizon. We further show how to reduce the regret of a desired subset of the
$M$ costs, at the expense of increasing the regrets of rewards and the
remaining costs. To the best of our knowledge, ours is the only work that
considers non-episodic RL under average cost constraints, and derive algorithms
that can~\emph{tune the regret vector} according to the agent's requirements on
its cost regrets.
