  In this work, we investigate stochastic quasi-Newton methods for minimizing a
finite sum of cost functions over a decentralized network. In Part I, we
develop a general algorithmic framework that incorporates stochastic
quasi-Newton approximations with variance reduction so as to achieve fast
convergence. At each time each node constructs a local, inexact quasi-Newton
direction that asymptotically approaches the global, exact one. To be specific,
(i) A local gradient approximation is constructed by using dynamic average
consensus to track the average of variance-reduced local stochastic gradients
over the entire network; (ii) A local Hessian inverse approximation is assumed
to be positive definite with bounded eigenvalues, and how to construct it to
satisfy these assumptions will be given in Part II. Compared to the existing
decentralized stochastic first-order methods, the proposed general framework
introduces the second-order curvature information without incurring extra
sampling or communication. With a fixed step size, we establish the conditions
under which the proposed general framework linearly converges to an exact
optimal solution.
