  Due to the rapid growth of data and computational resources, distributed
optimization has become an active research area in recent years. While
first-order methods seem to dominate the field, second-order methods are
nevertheless attractive as they potentially require fewer communication rounds
to converge. However, there are significant drawbacks that impede their wide
adoption, such as the computation and the communication of a large Hessian
matrix. In this paper we present a new algorithm for distributed training of
generalized linear models that only requires the computation of diagonal blocks
of the Hessian matrix on the individual workers. To deal with this approximate
information we propose an adaptive approach that - akin to trust-region methods
- dynamically adapts the auxiliary model to compensate for modeling errors. We
provide theoretical rates of convergence for a wide class of problems including
L1-regularized objectives. We also demonstrate that our approach achieves
state-of-the-art results on multiple large benchmark datasets.
