In this work, we study stochastic quasi-Newton methods for solving the non-linear and non-convex optimization problems arising in the training of deep neural networks. We consider the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) update in the framework of a trust-region approach. We provide an almost comprehensive overview of recent improvements in quasi-Newton based training algorithms, such as accurate selection of the initial Hessian approximation, efficient solution of the trust-region subproblem with a direct method in high accuracy and an overlap sampling strategy to assure stable quasi-Newton updating by computing gradient differences based on this overlap. We provide a comparison of the standard L-BFGS method with a variant of this algorithm based on a modified secant condition which is theoretically shown to provide an increased order of accuracy in the approximation of the curvature of the Hessian. In our experiments, both quasi-Newton updates exhibit comparable performances. Our results show that with a fixed computational time budget the proposed quasi-Newton methods provide comparable or better testing accuracy than the state-of-the-art first-order Adam optimizer.
