Weighted QMIX: Expanding Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning

QMIX is a popular Q-learning algorithm for cooperative MARL in the training and decentralised execution paradigm. In order to enable decentralisation, QMIX restricts the joint action Q-values it can to be a monotonic mixing of each agent's utilities. However, this prevents it from representing value functions in which an agent's over its actions can depend on other agents' actions. To analyse this limitation, we first formalise the objective QMIX optimises, allows us to view QMIX as an operator that first computes theQ-learning targets and then projects them into the space representable by. This projection returns a representable Q-value that minimises the squared error across all joint actions. We show in particular that projection can fail to recover the optimal policy even with access toQ^*, which primarily stems from the equal weighting placed on each joint. We rectify this by introducing a weighting into the projection, in to place more importance on the better joint actions. We propose two schemes and prove that they recover the correct maximal action for joint action Q-values, and therefore for Q^* as well. Based on our and results in the tabular setting, we introduce two scalable versions our algorithm, Centrally-Weighted (CW) QMIX and Optimistically-Weighted (OW) and demonstrate improved performance on both predator-prey and challenging-agent StarCraft benchmark tasks.