- Abstract: In this paper, we develop a trust region method for training deep neural networks. At each iteration, trust region method computes the search direction by solving a non-convex subproblem. Solving this subproblem is non-trivial---existing methods have only sub-linear convergence rate. In the first part, we show that a simple modification of gradient descent algorithm can converge to a global minimizer of the subproblem with an asymptotic linear convergence rate. Moreover, our method only requires Hessian-vector products, which can be computed efficiently by back-propagation in neural networks. In the second part, we apply our algorithm to train large-scale convolutional neural networks, such as VGG and MobileNets. Although trust region method is about 3 times slower than SGD in terms of running time, we observe it finds a model that has lower generalization (test) error than SGD, and this difference is even more significant in large batch training. We conduct several interesting experiments to support our conjecture that the trust region method can avoid sharp local minimas.