An Online Multiobjective Policy Gradient for Long-run Average-reward Markov Decision Process

We propose a reinforcement learning (RL) framework for multi-objective decision-making, where the agent seeks to optimize a vector of rewards rather than a single scalar value. The objective is to ensure that the time-averaged reward vector converges asymptotically to a predefined target set. Since standard RL algorithms operate on scalar rewards, we introduce a dynamic scalarization mechanism guided by Blackwell's Approachability Theorem. This theorem enables adaptive updates of the scalarization vector to guarantee convergence toward the target set. Assuming ergodicity, the Markov chain induced by the learned policies admits a stationary distribution, ensuring all states recur with finite return times. Our algorithm exploits this property by defining an inner loop that applies a policy gradient method (with baseline) between successive visits to a designated recurrent state, enforcing Blackwell's condition at each iteration. An outer loop then updates the scalarization vector after each recurrence. We establish theoretical convergence of the long-run average reward vector to the target set and validate the approach through a numerical example.