Deep learning algorithms often require solving a highly non-linear and nonconvex unconstrained optimization problem. Methods for solving optimization problems in large-scale machine learning, such as deep learning and deep reinforcement learning (RL), are generally restricted to the class of first-order algorithms, like stochastic gradient descent (SGD). While SGD iterates are inexpensive to compute, they have slow theoretical convergence rates. Furthermore, they require exhaustive

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... or to fine-tune many learning parameters. Using second-order curvature information to find search directions can help with more robust convergence for non-convex optimization problems. However, computing Hessian matrices for large-scale problems is not computationally practical. Alternatively, quasi-Newton methods construct an approximate of the Hessian matrix to build a quadratic model of the objective function. Quasi-Newton methods, like SGD, require only first-order gradient information, but they can result in superlinear convergence, which makes them attractive alternatives to SGD. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) approach is one of the most popular quasi-Newton methods that construct positive definite Hessian approximations. In this chapter, we propose efficient optimization methods based on L-BFGS quasi-Newton methods using line search and trust-region strategies. Our methods bridge the disparity between first- and second-order methods by using gradient information to calculate low-rank updates to Hessian approximations. We provide formal convergence analysis of these methods as well as empirical results on deep learning applications, such as image classification tasks and deep reinforcement learning on a set of ATARI 2600 video games. Our results show a robust convergence with preferred generalization characteristics as well as fast training time.