Robust Average-Reward Markov Decision Processes
This work addresses a gap in robust MDP theory for applications requiring long-term average performance under uncertainty, but it is incremental as it extends existing robust MDP frameworks to the average-reward case.
The paper tackles the problem of robust average-reward Markov decision processes (MDPs), which were previously unexplored, by developing theoretical convergence results and algorithms, proving that robust discounted value functions converge to robust average-reward as the discount factor approaches 1 and designing robust dynamic programming and relative value iteration methods that provably find optimal policies.
In robust Markov decision processes (MDPs), the uncertainty in the transition kernel is addressed by finding a policy that optimizes the worst-case performance over an uncertainty set of MDPs. While much of the literature has focused on discounted MDPs, robust average-reward MDPs remain largely unexplored. In this paper, we focus on robust average-reward MDPs, where the goal is to find a policy that optimizes the worst-case average reward over an uncertainty set. We first take an approach that approximates average-reward MDPs using discounted MDPs. We prove that the robust discounted value function converges to the robust average-reward as the discount factor $γ$ goes to $1$, and moreover, when $γ$ is large, any optimal policy of the robust discounted MDP is also an optimal policy of the robust average-reward. We further design a robust dynamic programming approach, and theoretically characterize its convergence to the optimum. Then, we investigate robust average-reward MDPs directly without using discounted MDPs as an intermediate step. We derive the robust Bellman equation for robust average-reward MDPs, prove that the optimal policy can be derived from its solution, and further design a robust relative value iteration algorithm that provably finds its solution, or equivalently, the optimal robust policy.