  First-order methods like stochastic gradient descent(SGD) are recently the
popular optimization method to train deep neural networks (DNNs), but
second-order methods are scarcely used because of the overpriced computing cost
in getting the high-order information. In this paper, we propose the Damped
Newton Stochastic Gradient Descent(DN-SGD) method and Stochastic Gradient
Descent Damped Newton(SGD-DN) method to train DNNs for regression problems with
Mean Square Error(MSE) and classification problems with Cross-Entropy
Loss(CEL), which is inspired by a proved fact that the hessian matrix of last
layer of DNNs is always semi-definite. Different from other second-order
methods to estimate the hessian matrix of all parameters, our methods just
accurately compute a small part of the parameters, which greatly reduces the
computational cost and makes convergence of the learning process much faster
and more accurate than SGD. Several numerical experiments on real datesets are
performed to verify the effectiveness of our methods for regression and
classification problems.
