  We initiate the study of federated reinforcement learning under environmental
heterogeneity by considering a policy evaluation problem. Our setup involves
$N$ agents interacting with environments that share the same state and action
space but differ in their reward functions and state transition kernels.
Assuming agents can communicate via a central server, we ask: Does exchanging
information expedite the process of evaluating a common policy? To answer this
question, we provide the first comprehensive finite-time analysis of a
federated temporal difference (TD) learning algorithm with linear function
approximation, while accounting for Markovian sampling, heterogeneity in the
agents' environments, and multiple local updates to save communication. Our
analysis crucially relies on several novel ingredients: (i) deriving
perturbation bounds on TD fixed points as a function of the heterogeneity in
the agents' underlying Markov decision processes (MDPs); (ii) introducing a
virtual MDP to closely approximate the dynamics of the federated TD algorithm;
and (iii) using the virtual MDP to make explicit connections to federated
optimization. Putting these pieces together, we rigorously prove that in a
low-heterogeneity regime, exchanging model estimates leads to linear
convergence speedups in the number of agents.
