  Despite their high computation and communication costs, Newton-type methods
remain an appealing option for distributed training due to their robustness
against ill-conditioned convex problems. In this work, we study ommunication
compression and aggregation mechanisms for curvature information in order to
reduce these costs while preserving theoretically superior local convergence
guarantees. We prove that the recently developed class of three point
compressors (3PC) of Richtarik et al. [2022] for gradient communication can be
generalized to Hessian communication as well. This result opens up a wide
variety of communication strategies, such as contractive compression} and lazy
aggregation, available to our disposal to compress prohibitively costly
curvature information. Moreover, we discovered several new 3PC mechanisms, such
as adaptive thresholding and Bernoulli aggregation, which require reduced
communication and occasional Hessian computations. Furthermore, we extend and
analyze our approach to bidirectional communication compression and partial
device participation setups to cater to the practical considerations of
applications in federated learning. For all our methods, we derive fast
condition-number-independent local linear and/or superlinear convergence rates.
Finally, with extensive numerical evaluations on convex optimization problems,
we illustrate that our designed schemes achieve state-of-the-art communication
complexity compared to several key baselines using second-order information.
