Review for NeurIPS paper: Personalized Federated Learning with Moreau Envelopes

Weaknesses: I am not completely convinced by the theoretical results. Specifically, I am not sure that Theorems 1 and 2 prove the right notion of convergence. Conceptually, I think you want to show that sum_i f_i(theta_i) is small and/or sum_i f_i(w) is small. The theorem statements, I suppose, upper bound sum_i f_i(theta_i), but \ w - theta_i \ is involved, and I don't see why that quantity should be relevant. I don't really see any reason to care about \ w - theta_i \; so what if you need to move far away from w to optimally personalize for one particular objective? In short, I think that the proposed objective pFedMe makes sense as a *training/surrogate objective*, but does not make as much sense as a criterion for evaluating a model, if that makes sense.