  While first-order methods are popular for solving optimization problems that
arise in large-scale deep learning problems, they come with some acute
deficiencies. To diminish such shortcomings, there has been recent interest in
applying second-order methods such as quasi-Newton based methods which
construct Hessians approximations using only gradient information. The main
focus of our work is to study the behaviour of stochastic quasi-Newton
algorithms for training deep neural networks. We have analyzed the performance
of two well-known quasi-Newton updates, the limited memory
Broyden-Fletcher-Goldfarb-Shanno (BFGS) and the Symmetric Rank One (SR1). This
study fills a gap concerning the real performance of both updates and analyzes
whether more efficient training is obtained when using the more robust BFGS
update or the cheaper SR1 formula which allows for indefinite Hessian
approximations and thus can potentially help to better navigate the
pathological saddle points present in the non-convex loss functions found in
deep learning. We present and discuss the results of an extensive experimental
study which includes the effect of batch normalization and network's
architecture, the limited memory parameter, the batch size and the type of
sampling strategy. we show that stochastic quasi-Newton optimizers are
efficient and able to outperform in some instances the well-known first-order
Adam optimizer run with the optimal combination of its numerous
hyperparameters.
