  We propose a stochastic optimization method for minimizing loss functions,
expressed as an expected value, that adaptively controls the batch size used in
the computation of gradient approximations and the step size used to move along
such directions, eliminating the need for the user to tune the learning rate.
The proposed method exploits local curvature information and ensures that
search directions are descent directions with high probability using an
acute-angle test and can be used as a method that has a global linear rate of
convergence on self-concordant functions with high probability. Numerical
experiments show that this method is able to choose the best learning rates and
compares favorably to fine-tuned SGD for training logistic regression and DNNs.
We also propose an adaptive version of ADAM that eliminates the need to tune
the base learning rate and compares favorably to fine-tuned ADAM on training
DNNs. In our DNN experiments, we rarely encountered negative curvature at the
current point along the step direction in DNNs.
