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Recently, an idling mechanism has been introduced in the context ofdistributed \emph{first order} methods for minimization of a sum of nodes'local convex costs over a generic, connected network. With the idlingmechanism, each node $i$, at each iteration $k$, is active -- updates itssolution estimate and exchanges messages with its network neighborhood -- withprobability $p_k$, and it stays idle with probability $1-p_k$, while theactivations are independent both across nodes and across iterations. In thispaper, we demonstrate that the idling mechanism can be successfullyincorporated in \emph{distributed second order methods} also. Specifically, weapply the idling mechanism to the recently proposed Distributed Quasi Newtonmethod (DQN). We first show theoretically that, when $p_k$ grows to one acrossiterations in a controlled manner, DQN with idling exhibits very similartheoretical convergence and convergence rates properties as the standard DQNmethod, thus achieving the same order of convergence rate (R-linear) as thestandard DQN, but with significantly cheaper updates. Simulation examplesconfirm the benefits of incorporating the idling mechanism, demonstrate themethod's flexibility with respect to the choice of the $p_k$'s, and compare theproposed idling method with related algorithms from the literature.

Distributed second order methods with increasing number of working nodes