In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration.

In our proposed methods, we approximate the Hessian by a block-diagonal matrix and use the structure of the gradient and Hessian to further approximate these blocks, each of which corresponds to a layer, as the Kronecker product of two much smaller matrices.

The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration. This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases.

The limited memory BFGS (L-BFGS) method is one of the popular methods for solving large-scale unconstrained optimization. Since the standard L-BFGS method uses a line search to guarantee its global convergence, it sometimes requires a large number of function evaluations. To overcome the difficulty, we propose a new L-BFGS with a certain regularization technique. We show its global convergence under the usual assumptions. In order to make the method more robust and efficient, we also extend it with several techniques such as nonmonotone technique and simultaneous use of the Wolfe line search. Finally, we present some numerical results for test problems in CUTEst, which show that the proposed method is robust in terms of solving number of problems.

Abstract: Motivated by the potential for parallel implementation of batch-based algorithms and the accelerated convergence achievable with approximated second order information a limited memory version of the BFGS algorithm has been receiving increasing attention in recent years for large neural network training problems. As the shape of the cost function is generally not quadratic and only becomes approximately quadratic in the vicinity of a minimum, the use of second order information by L-BFGS can be unreliable during the initial phase of training, i.e. when far from a minimum. Therefore, to control the influence of second order information as training progresses, we propose a multi-batch L-BFGS algorithm, namely MB-AM, that gradually increases its trust in the curvature information by implementing a progressive storage and use of curvature data through a development-based increase (dev-increase) scheme. Using six discriminative modelling benchmark problems we show empirically that MB-AM has slightly faster convergence and, on average, achieves better solutions than the standard multi-batch L-BFGS algorithm when training MLP and CNN models. Keywords: Deep learning, L-BFGS, variable memory, quasi-Newton methods, neural networks 1. INTRODUCTION currently an active area of research due to the accelerated convergence achievable with curvature information and In the last twenty years significant advances have been the ability to exploit parallelism with large batch sizes made towards making artificial neural networks able to to achieve efficient algorithm implementations (Berahas compete with their biological counterparts (Dodge and et al.(2016); Yousefian et al.(2017)).

The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration. This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases.

The question of how to parallelize the stochastic gradient descent (SGD) method has received much attention in the literature. In this paper, we focus instead on batch methods that use a sizeable fraction of the training set at each iteration to facilitate parallelism, and that employ second-order information. In order to improve the learning process, we follow a multi-batch approach in which the batch changes at each iteration. This can cause difficulties because L-BFGS employs gradient differences to update the Hessian approximations, and when these gradients are computed using different data points the process can be unstable. This paper shows how to perform stable quasi-Newton updating in the multi-batch setting, illustrates the behavior of the algorithm in a distributed computing platform, and studies its convergence properties for both the convex and nonconvex cases.