For distributed computing environment, we consider the empirical risk minimization problem and propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, which is sent to the main driver. The main driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT is highly communication efficient and naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. Theoretically, we show that GIANT enjoys an improved convergence rate as compared with first-order methods and existing distributed Newton-type methods. Further, and in sharp contrast with many existing distributed Newton-type methods, as well as popular first-order methods, a highly advantageous practical feature of GIANT is that it only involves one tuning parameter. We conduct large-scale experiments on a computer cluster and, empirically, demonstrate the superior performance of GIANT.

The paper introduces GIANT, a distributed variant of Newton algorithm. The considered problem is important and the paper gives a nice contribution to the field of distributed optimisation. The paper is very clear and nice to read, and propose nice theoretical contributions and experiments, with a detailed bibliography and positioning with respect to priori work. Here is my main criticism: * Authors acknowledge that their approach is close to previous works, namely DANE, for which GIANT seem to coincide to DANE in the least-squares loss case. However, the rate obtained in the paper is much better, certainly thanks to the introduction of the incoherence assumption, which is well known in the field of compressed sensing and randomized linear algebra.

For distributed computing environment, we consider the empirical risk minimization problem and propose a distributed and communication-efficient Newton-type optimization method. At every iteration, each worker locally finds an Approximate NewTon (ANT) direction, which is sent to the main driver. The main driver, then, averages all the ANT directions received from workers to form a Globally Improved ANT (GIANT) direction. GIANT is highly communication efficient and naturally exploits the trade-offs between local computations and global communications in that more local computations result in fewer overall rounds of communications. Theoretically, we show that GIANT enjoys an improved convergence rate as compared with first-order methods and existing distributed Newton-type methods.