  In this paper, we investigate the problem of robust Reconfigurable
Intelligent Surface (RIS) phase-shifts configuration over heterogeneous
communication environments. The problem is formulated as a distributed learning
problem over different environments in a Federated Learning (FL) setting.
Equivalently, this corresponds to a game played between multiple RISs, as
learning agents, in heterogeneous environments. Using Invariant Risk
Minimization (IRM) and its FL equivalent, dubbed FL Games, we solve the RIS
configuration problem by learning invariant causal representations across
multiple environments and then predicting the phases. The solution corresponds
to playing according to Best Response Dynamics (BRD) which yields the Nash
Equilibrium of the FL game. The representation learner and the phase predictor
are modeled by two neural networks, and their performance is validated via
simulations against other benchmarks from the literature. Our results show that
causality-based learning yields a predictor that is 15% more accurate in unseen
Out-of-Distribution (OoD) environments.
