  RES, a regularized stochastic version of the Broyden-Fletcher-Goldfarb-Shanno
(BFGS) quasi-Newton method is proposed to solve convex optimization problems
with stochastic objectives. The use of stochastic gradient descent algorithms
is widespread, but the number of iterations required to approximate optimal
arguments can be prohibitive in high dimensional problems. Application of
second order methods, on the other hand, is impracticable because computation
of objective function Hessian inverses incurs excessive computational cost.
BFGS modifies gradient descent by introducing a Hessian approximation matrix
computed from finite gradient differences. RES utilizes stochastic gradients in
lieu of deterministic gradients for both, the determination of descent
directions and the approximation of the objective function's curvature. Since
stochastic gradients can be computed at manageable computational cost RES is
realizable and retains the convergence rate advantages of its deterministic
counterparts. Convergence results show that lower and upper bounds on the
Hessian egeinvalues of the sample functions are sufficient to guarantee
convergence to optimal arguments. Numerical experiments showcase reductions in
convergence time relative to stochastic gradient descent algorithms and
non-regularized stochastic versions of BFGS. An application of RES to the
implementation of support vector machines is developed.
