On the Convergence Rates of Federated Q-Learning Across Heterogeneous Environments

Abstract

Large-scale multi-agent systems are often deployed across wide geographic areas, where agents interact with heterogeneous environments. There is an emerging interest in understanding the role of heterogeneity in the performance of the federated versions of classic reinforcement learning algorithms. In this paper, we study synchronous federated Q-learning, which aims to learn an optimal Q-function by having $K$ agents average their local Q-estimates per $E$ iterations. We provide a fine-grained characterization of the error evolution, which decays to zero as the number of iterations $T$ increases. When $K(E-1)$ is below a certain threshold, similar to the homogeneous environment settings, there is a linear speed-up concerning $K$. The slow convergence of having $E>1$ turns out to be fundamental rather than an artifact of our analysis. We prove that, for a wide range of stepsizes, the $\ell_{\infty}$ norm of the error cannot decay faster than $\Theta_R (\frac{E}{(1-\gamma)T})$, where $\Theta_R$ only hides numerical constants and the specific choice of reward values. In addition, our experiments demonstrate that the convergence exhibits an interesting two-phase phenomenon. For any given stepsize, there is a sharp phase transition of the convergence: the error decays rapidly in the beginning yet later bounces up and stabilizes.