  In this paper we study stochastic quasi-Newton methods for nonconvex
stochastic optimization, where we assume that only stochastic information of
the gradients of the objective function is available via a stochastic
first-order oracle (SFO). Firstly, we propose a general framework of stochastic
quasi-Newton methods for solving nonconvex stochastic optimization. The
proposed framework extends the classic quasi-Newton methods working in
deterministic settings to stochastic settings, and we prove its almost sure
convergence to stationary points. Secondly, we propose a general framework for
a class of randomized stochastic quasi-Newton methods, in which the number of
iterations conducted by the algorithm is a random variable. The worst-case
SFO-calls complexities of this class of methods are analyzed. Thirdly, we
present two specific methods that fall into this framework, namely stochastic
damped-BFGS method and stochastic cyclic Barzilai-Borwein method. Finally, we
report numerical results to demonstrate the efficiency of the proposed methods.

Optimization and Control