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This brief considers constrained nonconvex stochastic finite-sum and online optimization in deep neural networks. Adaptive-learning-rate optimization algorithms (ALROAs), such as Adam, AMSGrad, and their variants, have widely been used for these optimizations because they are powerful and useful in theory and practice. Here, it is shown that the ALROAs are ϵ -approximations for these optimizations. We provide the learning rates, mini-batch sizes, number of iterations, and stochastic gradient complexity with which to achieve ϵ -approximations of the algorithms.

This article deals with nonconvex stochastic optimization problems in deep learning. Appropriate learning rates, based on theory, for adaptive-learning-rate optimization algorithms (e.g., Adam and AMSGrad) to approximate the stationary points of such problems are provided. These rates are shown to allow faster convergence than previously reported for these algorithms. Specifically, the algorithms are examined in numerical experiments on text and image classification and are shown in experiments to perform better with constant learning rates than algorithms using diminishing learning rates.

This article proposes a Riemannian adaptive optimization algorithm to optimize the parameters of deep neural networks. The algorithm is an extension of both AMSGrad in Euclidean space and RAMSGrad on a Riemannian manifold. The algorithm helps to resolve two issues affecting RAMSGrad. The first is that it can solve the Riemannian stochastic optimization problem directly, in contrast to RAMSGrad which only achieves a low regret. The other is that it can use constant learning rates, which makes it implementable in practice. Additionally, we apply the proposed algorithm to Poincaré embeddings that embed the transitive closure of the WordNet nouns into the Poincaré ball model of hyperbolic space. Numerical experiments show that regardless of the initial value of the learning rate, our algorithm stably converges to the optimal solution and converges faster than the existing algorithms.

This paper considers a classifier ensemble problem with sparsity and diversity learning, which arises in the field of machine learning, and shows that the classifier ensemble problem can be formulated as a convex stochastic optimization problem over the fixed point set of a quasi-nonexpansive mapping. Specifically, for such a problem, this paper proposes an algorithm referred to as the stochastic fixed point optimization algorithm and performs a convergence analysis for three types of step size: 1) constant step size; 2) decreasing step size; and 3) a step size computed by line searches. In the case of a constant step size, the results indicate that a sufficiently small constant step size allows a solution to the problem to be approximated. In the case of a decreasing step size, conditions are shown under which the algorithm converges in probability to a solution. For the third case, a variation of the basic proposed algorithm also achieves convergence in probability to a solution. The high classification accuracies of the proposed algorithms are demonstrated through numerical comparisons with the conventional algorithm.

The existing machine learning algorithms for minimizing the convex function over a closed convex set suffer from slow convergence because their learning rates must be determined before running them. This paper proposes two machine learning algorithms incorporating the line search method, which automatically and algorithmically finds appropriate learning rates at run-time. One algorithm is based on the incremental subgradient algorithm, which sequentially and cyclically uses each of the parts of the objective function; the other is based on the parallel subgradient algorithm, which uses parts independently in parallel. These algorithms can be applied to constrained nonsmooth convex optimization problems appearing in tasks of learning support vector machines without adjusting the learning rates precisely. The proposed line search method can determine learning rates to satisfy weaker conditions than the ones used in the existing machine learning algorithms. This implies that the two algorithms are generalizations of the existing incremental and parallel subgradient algorithms for solving constrained nonsmooth convex optimization problems. We show that they generate sequences that converge to a solution of the constrained nonsmooth convex optimization problem under certain conditions. The main contribution of this paper is the provision of three kinds of experiment showing that the two algorithms can solve concrete experimental problems faster than the existing algorithms. First, we show that the proposed algorithms have performance advantages over the existing ones in solving a test problem. Second, we compare the proposed algorithms with a different algorithm Pegasos, which is designed to learn with a support vector machine efficiently, in terms of prediction accuracy, value of the objective function, and computational time. Finally, we use one of our algorithms to train a multilayer neural network and discuss its applicability to deep learning., (Copyright © 2019 Hishinuma and Iiduka.)