  This paper proposes a framework of L-BFGS based on the (approximate)
second-order information with stochastic batches, as a novel approach to the
finite-sum minimization problems. Different from the classical L-BFGS where
stochastic batches lead to instability, we use a smooth estimate for the
evaluations of the gradient differences while achieving acceleration by
well-scaling the initial Hessians. We provide theoretical analyses for both
convex and nonconvex cases. In addition, we demonstrate that within the popular
applications of least-square and cross-entropy losses, the algorithm admits a
simple implementation in the distributed environment. Numerical experiments
support the efficiency of our algorithms.
