Convergence Rates of Distributed TD(0) with Linear Function Approximation for Multi-Agent Reinforcement Learning

We study the policy evaluation problem in multi-agent reinforcement learning. In this problem, a group of agents work cooperatively to evaluate the value function for the global discounted accumulative reward problem, which is composed of local rewards observed by the agents. Over a series of time steps, the agents act, get rewarded, update their local estimate of the value function, then communicate with their neighbors. The local update at each agent can be interpreted as a distributed consensus-based variant of the popular temporal difference learning algorithm TD(0). While distributed reinforcement learning algorithms have been presented in the literature, almost nothing is known about their convergence rate. Our main contribution is providing a finite-time analysis for the convergence of the distributed TD(0) algorithm. We do this when the communication network between the agents is time-varying in general. We obtain an explicit upper bound on the rate of convergence of this algorithm as a function of the network topology and the discount factor. Our results mirror what we would expect from using distributed stochastic gradient descent for solving convex optimization problems.