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Growth transformations constitute a class of fixed-point multiplicative update algorithms that were originally proposed for optimizing polynomial and rational functions over a domain of probability measures. In this paper, we extend this framework to the domain of bounded real variables which can be applied towards optimizing the dual cost function of a generic support vector machine (SVM). The approach can, therefore, not only be used to train traditional soft-margin binary SVMs, one-class SVMs, and probabilistic SVMs but can also be used to design novel variants of SVMs with different types of convex and quasi-convex loss functions. In this paper, we propose an efficient training algorithm based on polynomial growth transforms, and compare and contrast the properties of different SVM variants using several synthetic and benchmark data sets. The preliminary experiments show that the proposed multiplicative update algorithm is more scalable and yields better convergence compared to standard quadratic and nonlinear programming solvers. While the formulation and the underlying algorithms have been validated in this paper only for SVM-based learning, the proposed approach is general and can be applied to a wide variety of optimization problems and statistical learning models. [ABSTRACT FROM AUTHOR]

Memory is one of the most important mechanisms in recurrent neural networks (RNNs) learning. It plays a crucial role in practical applications, such as sequence learning. With a good memory mechanism, long term history can be fused with current information, and can thus improve RNNs learning. Developing a suitable memory mechanism is always desirable in the field of RNNs. This paper proposes a novel memory mechanism for RNNs. The main contributions of this paper are: 1) an auxiliary memory unit (AMU) is proposed, which results in a new special RNN model (AMU-RNN), separating the memory and output explicitly and 2) an efficient learning algorithm is developed by employing the technique of error flow truncation. The proposed AMU-RNN model, together with the developed learning algorithm, can learn and maintain stable memory over a long time range. This method overcomes both the learning conflict problem and gradient vanishing problem. Unlike the traditional method, which mixes the memory and output with a single neuron in a recurrent unit, the AMU provides an auxiliary memory neuron to maintain memory in particular. By separating the memory and output in a recurrent unit, the problem of learning conflicts can be eliminated easily. Moreover, by using the technique of error flow truncation, each auxiliary memory neuron ensures constant error flow during the learning process. The experiments demonstrate good performance of the proposed AMU-RNNs and the developed learning algorithm. The method exhibits quite efficient learning performance with stable convergence in the AMU-RNN learning and outperforms the state-of-the-art RNN models in sequence generation and sequence classification tasks. [ABSTRACT FROM AUTHOR]

The limited memory version of the Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm is the most popular quasi-Newton algorithm in machine learning and optimization. Recently, it was shown that the stochastic L-BFGS (sL-BFGS) algorithm with the variance-reduced stochastic gradient converges linearly. In this paper, we propose a new sL-BFGS algorithm by importing a proper momentum. We prove an accelerated linear convergence rate under mild conditions. The experimental results on different data sets also verify this acceleration advantage. [ABSTRACT FROM AUTHOR]

Conjugate gradient (CG) methods are a class of important methods for solving linear equations and nonlinear optimization problems. In this paper, we propose a new stochastic CG algorithm with variance reduction and we prove its linear convergence with the Fletcher and Reeves method for strongly convex and smooth functions. We experimentally demonstrate that the CG with variance reduction algorithm converges faster than its counterparts for four learning models, which may be convex, nonconvex or nonsmooth. In addition, its area under the curve performance on six large-scale data sets is comparable to that of the LIBLINEAR solver for the $L2$ -regularized $L2$ -loss but with a significant improvement in computational efficiency. 1 CGVR algorithm is available on github: https://github.com/xbjin/cgvr [ABSTRACT FROM AUTHOR]

In this paper, we consider the composition optimization with two expected-value functions in the form of $({1}/{n})\sum _{i = 1}^{n} F_{i}\left({({1}/{m})\sum _{j = 1}^{m} G_{j}(x)}\right)+R(x)$ , which formulates many important problems in statistical learning and machine learning such as solving Bellman equations in reinforcement learning and nonlinear embedding. Full gradient- or classical stochastic gradient descent-based optimization algorithms are unsuitable or computationally expensive to solve this problem due to the inner expectation $({1}/{m})\sum _{j = 1}^{m} G_{j}(x)$. We propose a dualityfree-based stochastic composition method that combines the variance reduction methods to address the stochastic composition problem. We apply the stochastic variance reduction gradient- and stochastic average gradient algorithm-based methods to estimate the inner function and the dualityfree method to estimate the outer function. We prove the linear convergence rate not only for the convex composition problem but also for the case that the individual outer functions are nonconvex, while the objective function is strongly convex. We also provide the results of experiments that show the effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

Generalized eigendecomposition problem has been widely employed in many signal processing applications. In this paper, we propose a unified and self-stabilizing algorithm, which is able to extract the first principal and minor generalized eigenvectors of a matrix pencil of two vector sequences adaptively. Furthermore, we extend the proposed algorithm to extract multiple generalized eigenvectors. The performance analysis shows that only the desired equilibrium point of the proposed algorithm is stable and all others are (unstable) repellers or saddle points. Convergence analysis based on the deterministic discrete-time approach shows that, for a step size within a certain range, the norm of the principal/minor state vector converges to a fixed value that relates to the corresponding principal/minor generalized eigenvalue. Thus, the proposed algorithm is a generalized eigenpairs (eigenvectors and eigenvalues) extraction algorithm. Finally, the simulation experiments are carried to further demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Focal underdetermined system solver (FOCUSS) is a powerful method for basis selection and sparse representation, where it employs the \ell \vphantom {Rjlp} -norm with $p\in (0,2)$ to measure the sparsity of solutions. In this paper, we give a systematical analysis on the rate of convergence of the FOCUSS algorithm with respect to $p\in (0,2)$ . We prove that the FOCUSS algorithm converges superlinearly for $0

This paper proposes a control algorithm based on adaptive dynamic programming to solve the infinite-horizon optimal control problem for known deterministic nonlinear systems with saturating actuators and nonquadratic cost functionals. The algorithm is based on an actor/critic framework, where a critic neural network (NN) is used to learn the optimal cost, and an actor NN is used to learn the optimal control policy. The adaptive control nature of the algorithm requires a persistence of excitation condition to be a priori validated, but this can be relaxed using previously stored data concurrently with current data in the update of the critic NN. A robustifying control term is added to the controller to eliminate the effect of residual errors, leading to the asymptotically stability of the closed-loop system. Simulation results show the effectiveness of the proposed approach for a controlled Van der Pol oscillator and also for a power system plant. [ABSTRACT FROM AUTHOR]

In this paper, we consider the learning problem of multilayer perceptrons (MLPs) formulated as the problem of minimizing a smooth error function. As well known, the learning problem of MLPs can be a difficult nonlinear nonconvex optimization problem. Typical difficulties can be the presence of extensive flat regions and steep sided valleys in the error surface, and the possible large number of training data and of free network parameters. We define a wide class of batch learning algorithms for MLP, based on the use of block decomposition techniques in the minimization of the error function. The learning problem is decomposed into a sequence of smaller and structured minimization problems in order to advantageously exploit the structure of the objective function. Theoretical convergence results are established, and a specific algorithm is constructed and evaluated through an extensive numerical experimentation. The comparisons with the state-of-the-art learning algorithms show the effectiveness of the proposed techniques. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization (NMF) and symmetric NMF (SymNMF) have been shown to be effective for clustering linearly separable data and nonlinearly separable data, respectively. Nevertheless, many practical applications demand constrained algorithms in which a small number of constraints in the form of must-link and cannot-link are available. In this paper, we propose an NMF-based constrained clustering framework in which the similarity between two points on a must-link is enforced to approximate 1 and the similarity between two points on a cannot-link is enforced to approximate 0. We then formulate the framework using NMF and SymNMF to deal with clustering of linearly separable data and nonlinearly separable data, respectively. Furthermore, we present multiplicative update rules to solve them and show the correctness and convergence. Experimental results on various text data sets, University of California, Irvine (UCI) data sets, and gene expression data sets demonstrate the superiority of our algorithms over existing constrained clustering algorithms. [ABSTRACT FROM AUTHOR]

In many real-world applications, data are represented by matrices or high-order tensors. Despite the promising performance, the existing 2-D discriminant analysis algorithms employ a single projection model to exploit the discriminant information for projection, making the model less flexible. In this paper, we propose a novel compound rank- $k$ projection (CRP) algorithm for bilinear analysis. The CRP deals with matrices directly without transforming them into vectors, and it, therefore, preserves the correlations within the matrix and decreases the computation complexity. Different from the existing 2-D discriminant analysis algorithms, objective function values of CRP increase monotonically. In addition, the CRP utilizes multiple rank- $k$ projection models to enable a larger search space in which the optimal solution can be found. In this way, the discriminant ability is enhanced. We have tested our approach on five data sets, including UUIm, CVL, Pointing’04, USPS, and Coil20. Experimental results show that the performance of our proposed CRP performs better than other algorithms in terms of classification accuracy. [ABSTRACT FROM PUBLISHER]

How to propagate the label information from labeled examples to unlabeled examples is a critical problem for graph-based semisupervised learning. Many label propagation algorithms have been developed in recent years and have obtained promising performance on various applications. However, the eigenvalues of iteration matrices in these algorithms are usually distributed irregularly, which slow down the convergence rate and impair the learning performance. This paper proposes a novel label propagation method called Fick’s law assisted propagation (FLAP). Unlike the existing algorithms that are directly derived from statistical learning, FLAP is deduced on the basis of the theory of Fick’s First Law of Diffusion, which is widely known as the fundamental theory in fluid-spreading. We prove that FLAP will converge with linear rate and show that FLAP makes eigenvalues of the iteration matrix distributed regularly. Comprehensive experimental evaluations on synthetic and practical datasets reveal that FLAP obtains encouraging results in terms of both accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Support vector ordinal regression (SVOR) is a popular method to tackle ordinal regression problems. However, until now there were no effective algorithms proposed to address incremental SVOR learning due to the complicated formulations of SVOR. Recently, an interesting accurate on-line algorithm was proposed for training $\nu $ -support vector classification ( $\nu $ -SVC), which can handle a quadratic formulation with a pair of equality constraints. In this paper, we first present a modified SVOR formulation based on a sum-of-margins strategy. The formulation has multiple constraints, and each constraint includes a mixture of an equality and an inequality. Then, we extend the accurate on-line $\nu $ -SVC algorithm to the modified formulation, and propose an effective incremental SVOR algorithm. The algorithm can handle a quadratic formulation with multiple constraints, where each constraint is constituted of an equality and an inequality. More importantly, it tackles the conflicts between the equality and inequality constraints. We also provide the finite convergence analysis for the algorithm. Numerical experiments on the several benchmark and real-world data sets show that the incremental algorithm can converge to the optimal solution in a finite number of steps, and is faster than the existing batch and incremental SVOR algorithms. Meanwhile, the modified formulation has better accuracy than the existing incremental SVOR algorithm, and is as accurate as the sum-of-margins based formulation of Shashua and Levin. [ABSTRACT FROM PUBLISHER]

Nonnegative matrix factorization (NMF) has recently become a very popular unsupervised learning method because of its representational properties of factors and simple multiplicative update algorithms for solving the NMF. However, for the common NMF approach of minimizing the Euclidean distance between approximate and true values, the convergence of multiplicative update algorithms has not been well resolved. This paper first discusses the convergence of existing multiplicative update algorithms. We then propose a new multiplicative update algorithm for minimizing the Euclidean distance between approximate and true values. Based on the optimization principle and the auxiliary function method, we prove that our new algorithm not only converges to a stationary point, but also does faster than existing ones. To verify our theoretical results, the experiments on three data sets have been conducted by comparing our proposed algorithm with other existing methods. [ABSTRACT FROM AUTHOR]

As a special class of recurrent neural network, Zhang neural network (ZNN) has been recently proposed since 2001 for solving various time-varying problems, and has shown high efficiency and excellent performance for solving the problems in the real domain. In this paper, to solve online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem in the complex domain, a new type of complex-valued ZNN is further proposed and investigated. The design of such a complex ZNN is based on a complex Zhang function (ZF) which is indefinite and quite different from the usual error function (specially, the scalar-valued energy function) in the studies of conventional algorithms. By introducing five different complex ZFs, five different complex ZNN models (termed complex ZNN-I, ZNN-II, ZNN-III, ZNN-IV, and ZNN-V models) are proposed, developed, and investigated for the online solution of the time-varying complex generalized inverse matrices. Theoretical results of convergence analysis are presented to show the desirable properties of complex ZNN models. In addition, we discover the link between the proposed complex ZNN models and the Getz–Marsden dynamic system in the complex domain. Computer-simulation results further demonstrate the effectiveness of complex ZNN models based on different complex ZFs for the time-varying complex generalized inverse matrices. [ABSTRACT FROM AUTHOR]