In this work, we consider a novel stochastic optimization algorithm to solve the unconstrained, nonlinear, and non-convex optimization problems arising in the training of deep neural networks. The new algorithm is based on the combination of first- and second-order information, namely, at each step the computed search direction linearly combines a variance-reduced gradient and a stochastic limited memory quasi-Newton direction. We report computational experiments showing the performance of the proposed optimizer in the training of a modern deep residual neural network for image classification tasks. The numerical results show that the proposed algorithm exhibits comparable or superior performance than the state-of-the-art Adam optimizer, without the agonizing pain of tuning its many hyperparameters.