This paper investigates infinite-horizon average reward Constrained Markov Decision Processes (CMDPs) under general parametrized policies with smooth and bounded policy gradients. We propose a Primal-Dual Natural Actor-Critic algorithm that adeptly manages constraints while ensuring a high convergence rate. In particular, our algorithm achieves global convergence and constraint violation rates of $\tilde{\mathcal{O}}(1/\sqrt{T})$ over a horizon of length $T$ when the mixing time, $\tau_{\mathrm{mix}}$, is known to the learner. In absence of knowledge of $\tau_{\mathrm{mix}}$, the achievable rates change to $\tilde{\mathcal{O}}(1/T^{0.5-\epsilon})$ provided that $T \geq \tilde{\mathcal{O}}\left(\tau_{\mathrm{mix}}^{2/\epsilon}\right)$. Our results match the theoretical lower bound for Markov Decision Processes and establish a new benchmark in the theoretical exploration of average reward CMDPs.

In this paper, we consider an infinite horizon average reward Markov Decision Process (MDP). Distinguishing itself from existing works within this context, our approach harnesses the power of the general policy gradient-based algorithm, liberating it from the constraints of assuming a linear MDP structure. We propose a vanilla policy gradient-based algorithm and show its global convergence property. We then prove that the proposed algorithm has O(T^3/4) regret. Remarkably, this paper marks a pioneering effort by presenting the first exploration into regret bound computation for the general parameterized policy gradient algorithm in the context of average reward scenarios.

This paper explores the realm of infinite horizon average reward Constrained Markov Decision Processes (CMDPs). 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, our proposed algorithm achieves $\tilde{\mathcal{O}}({T}^{4/5})$ objective regret and $\tilde{\mathcal{O}}({T}^{4/5})$ constraint violation bounds.

We consider the problem of constrained Markov decision process (CMDP) in continuous state actions spaces where the goal is to maximize the expected cumulative reward subject to some constraints. We propose a novel Conservative Natural Policy Gradient Primal Dual Algorithm (CNPGPD) to achieve zero constraint violation while achieving state of the art convergence results for the objective value function. For general policy parametrization, we prove convergence of value function to global optimal upto an approximation error due to restricted policy class. We improve the sample complexity of existing constrained NPGPD algorithm. To the best of our knowledge, this is the first work to establish zero constraint violation with Natural policy gradient style algorithms for infinite horizon discounted CMDPs. We demonstrate the merits of proposed algorithm via experimental evaluations.

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.

Reinforcement learning is widely used in applications where one needs to perform sequential decisions while interacting with the environment. The problem becomes more challenging when the decision requirement includes satisfying some safety constraints. The problem is mathematically formulated as constrained Markov decision process (CMDP). In the literature, various algorithms are available to solve CMDP problems in a model-free manner to achieve epsilon-optimal cumulative reward with epsilon feasible policies. An epsilon-feasible policy implies that it suffers from constraint violation. An important question here is whether we can achieve epsilon-optimal cumulative reward with zero constraint violations or not. To achieve that, we advocate the use of a randomized primal-dual approach to solve the CMDP problems and propose a conservative stochastic primal-dual algorithm (CSPDA) which is shown to exhibit O(1/epsilon^2) sample complexity to achieve epsilon-optimal cumulative reward with zero constraint violations. In the prior works, the best available sample complexity for the epsilon-optimal policy with zero constraint violation is O(1/epsilon^5). Hence, the proposed algorithm provides a significant improvement compared to the state of the art.

We consider the problem of constrained Markov Decision Process (CMDP) where an agent interacts with an ergodic Markov Decision Process. At every interaction, the agent obtains a reward and incurs $K$ costs. The agent aims to maximize the long-term average reward while simultaneously keeping the $K$ long-term average costs lower than a certain threshold. In this paper, we propose \NAM, a posterior sampling based algorithm using which the agent can learn optimal policies to interact with the CMDP. We show that with the assumption of slackness, characterized by $\kappa$, the optimization problem is feasible for the sampled MDPs. Further, for MDP with $S$ states, $A$ actions, and mixing time $T_M$, we prove that following \NAM{} algorithm, the agent can bound the regret of not accumulating rewards from an optimal policy by $\Tilde{O}(T_MS\sqrt{AT})$. Further, we show that the violations for any of the $K$ constraints is also bounded by $\Tilde{O}(T_MS\sqrt{AT})$. To the best of our knowledge, this is the first work that obtains a $\Tilde{O}(\sqrt{T})$ regret bounds for ergodic MDPs with long-term average constraints using a posterior sampling method.

Many engineering problems have multiple objectives, and the overall aim is to optimize a non-linear function of these objectives. In this paper, we formulate the problem of maximizing a non-linear concave function of multiple long-term objectives. A policy-gradient based model-free algorithm is proposed for the problem. To compute an estimate of the gradient, an asymptotically biased estimator is proposed. The proposed algorithm is shown to achieve convergence to within an ε of the global optima after sampling O(M4 σ2/(1-γ)8ε4) trajectories where γ is the discount factor and M is the number of the agents, thus achieving the same dependence on ε as the policy gradient algorithm for the standard reinforcement learning.

In this paper, we consider the problem of optimization and learning for constrained and multi-objective Markov decision processes, for both discounted rewards and expected average rewards. We formulate the problems as zero-sum games where one player (the agent) solves a Markov decision problem and its opponent solves a bandit optimization problem, which we here call Markov-Bandit games. We extend $Q$-learning to solve Markov-Bandit games and show that our new $Q$-learning algorithms converge to the optimal solutions of the zero-sum Markov-Bandit games, and hence converge to the optimal solutions of the constrained and multi-objective Markov decision problems. We provide numerical examples where we calculate the optimal policies and show by simulations that the algorithm converges to the calculated optimal policies. To the best of our knowledge, this is the first time Q-learning algorithms guarantee convergence to optimal stationary policies for the multi-objective Reinforcement Learning problem with discounted and expected average rewards, respectively.