  Inspired by recent work of Islamov et al (2021), we propose a family of
Federated Newton Learn (FedNL) methods, which we believe is a marked step in
the direction of making second-order methods applicable to FL. In contrast to
the aforementioned work, FedNL employs a different Hessian learning technique
which i) enhances privacy as it does not rely on the training data to be
revealed to the coordinating server, ii) makes it applicable beyond generalized
linear models, and iii) provably works with general contractive compression
operators for compressing the local Hessians, such as Top-$K$ or Rank-$R$,
which are vastly superior in practice. Notably, we do not need to rely on error
feedback for our methods to work with contractive compressors. Moreover, we
develop FedNL-PP, FedNL-CR and FedNL-LS, which are variants of FedNL that
support partial participation, and globalization via cubic regularization and
line search, respectively, and FedNL-BC, which is a variant that can further
benefit from bidirectional compression of gradients and models, i.e., smart
uplink gradient and smart downlink model compression. We prove local
convergence rates that are independent of the condition number, the number of
training data points, and compression variance. Our communication efficient
Hessian learning technique provably learns the Hessian at the optimum. Finally,
we perform a variety of numerical experiments that show that our FedNL methods
have state-of-the-art communication complexity when compared to key baselines.
