Online Gradient Method Research Articles

Convergence analysis of feedforward neural networks using the online gradient method with smoothing L1 regularization

Convergence analysis of feedforward neural networks using the online gradient method with smoothing L1 regularization

Boundedness and Convergence Analysis of a Pi-Sigma Neural Network Based on Online Gradient Method and Sparse Optimization

Boundedness and Convergence Analysis of a Pi-Sigma Neural Network Based on Online Gradient Method and Sparse Optimization

Convergence Analysis of Online Gradient Method for High-Order Neural Networks and Their Sparse Optimization.

In this article, we investigate the boundedness and convergence of the online gradient method with the smoothing group L1/2 regularization for the sigma-pi-sigma neural network (SPSNN). This enhances the sparseness of the network and improves its generalization ability. For the original group L1/2 regularization, the error function is nonconvex and nonsmooth, which can cause oscillation of the error function. To ameliorate this drawback, we propose a simple and effective smoothing technique, which can effectively eliminate the deficiency of the original group L1/2 regularization. The group L1/2 regularization effectively optimizes the network structure from two aspects redundant hidden nodes tending to zero and redundant weights of surviving hidden nodes in the network tending to zero. This article shows the strong and weak convergence results for the proposed method and proves the boundedness of weights. Experiment results clearly demonstrate the capability of the proposed method and the effectiveness of redundancy control. The simulation results are observed to support the theoretical results.

Proximal Online Gradient Is Optimum for Dynamic Regret: A General Lower Bound.

In online learning, the dynamic regret metric chooses the reference oracle that may change over time, while the typical (static) regret metric assumes the reference solution to be constant over the whole time horizon. The dynamic regret metric is particularly interesting for applications, such as online recommendation (since the customers' preference always evolves over time). While the online gradient (OG) method has been shown to be optimal for the static regret metric, the optimal algorithm for the dynamic regret remains unknown. In this article, we show that proximal OG (a general version of OG) is optimum to the dynamic regret by showing that the proved lower bound matches the upper bound. It is highlighted that we provide a new and general lower bound of dynamic regret. It provides new understanding about the difficulty to follow the dynamics in the online setting.

Convergence of Online Gradient Method with Momentum for BP Neural Network

The BP neural network, which uses the steepest descent method (gradient descent method) as the basic idea for learning, is often used to deal with approximation problems because of its strong nonlinear mapping ability. The gradient method with added momentum can improve the learning speed of BP neural network. We study the convergence of the online gradient method with momentum for two-layer BP neural network when the training samples are randomly arranged in each iteration. Choosing appropriate learning rates, and selecting momentum coefficients in an adaptive manner, we prove the weak and strong convergence theorems of the algorithm.

An online self-organizing modular neural network for nonlinear system modeling

Modular neural network (MNN) has distinct advantage in many fields such as pattern recognition and pattern recognition. However it is still a challenge to dynamically adjust the MNN structure for dynamic nonlinear system modeling. This paper proposes a novel online self-organizing MNN (OSOMNN) for nonlinear system modeling. In OSOMNN, an online task decomposition algorithm and a self-organizing algorithm for subnetwork are introduced. Firstly, the task decomposition algorithm is implemented by the online clustering method based on distance and local density, which can online divide the original task into several simpler subtasks. Then subnetworks with single-layer feedforward neural network are built to learn the divided subtasks. Moreover, this paper develops a self-organizing algorithm for subnetwork, which can dynamically adjust its structure and is trained by the improved online gradient method with fixed memory mechanism (FMOGM). To demonstrate the effectiveness of OSOMNN for nonlinear system modeling, experimental investigations using four benchmark nonlinear systems and the monthly sunspots time series show that OSOMNN can automatically add or merge the subnetwork modules and optimize the structure of subnetworks for nonlinear system modeling with a better generalization performance than the established alternatives.

An online self-organizing algorithm for feedforward neural network

Feedforward neural network (FNN) is the most popular network model, and the appropriate structure and learning algorithms are the key of its performance. This paper proposes an online self-organizing algorithm for feedforward neural network (OSNN) with a single hidden layer. The proposed OSNN optimizes the structure of FNN for time-varying system including structure design and parameter learning. In structure design, this paper measures the contribution ratios of hidden nodes by local sensitivity analysis based on differentiation method. OSNN merges hidden nodes with the others that have the highest correlation when their contribution ratios are almost zero and adds new hidden nodes by error reparation. For parameter learning, an improved online gradient method (OGM), called online gradient method with fixed memory (FMOGM), is proposed to improve the convergence speed and accuracy of OGM. In addition, this paper calculates the contribution ratios and the network error and estimates the local minima by using the fixed-sized training set of FMOGM instead of one sample at the current time, which can obtain more effective local information and a compact network structure. Finally, the proposed OSNN is verified using a number of benchmark problems and a practical problem for biochemical oxygen demand prediction in wastewater treatment. The experimental results show that OSNN has better convergence speed and accuracy than other algorithms.

An Online Gradient Method with Smoothing L_0 Regularization for Pi-Sigma Network

The description of this study is to make possibility analysis solution of online gradient method with smoothing regularization for pi-sigma network training. Due to the effectiveness computational and theoretical analysis are a very important issues to improve the generalization performance of networks and the gradient descent algorithm with regularization is widely used method. However, regularization is reefed to NP-hard nature problems, which has not differentiable objective functional-penalty term. In this paper to avoid this trick, we use a smoothing function to recover the origin regularization into smoothing regularization. Under this condition, the resulting obtained as a good decreases solution when compared with others. The monotonically of the error function, weak and strong convergence theorems are proved.

Online gradient method with smoothing ℓ0 regularization for feedforward neural networks

ℓp regularization has been a popular pruning method for neural networks. The parameter p was usually set as 0<p≤2 in the literature, and practical training algorithms with ℓ0 regularization are lacking due to the NP-hard nature of the ℓ0 regularization problem; however, the ℓ0 regularization tends to produce the sparsest solution, corresponding to the most parsimonious network structure which is desirable in view of the generalization ability. To this end, this paper considers an online gradient training algorithm with smoothing ℓ0 regularization (OGTSL0) for feedforward neural networks, where the ℓ0 regularizer is approximated by a series of smoothing functions. The underlying principle for the sparsity of OGTSL0 is provided, and the convergence of the algorithm is also theoretically analyzed. Simulation examples support the theoretical analysis and illustrate the superiority of the proposed algorithm.

Convergence of Online Gradient Method for Pi-sigma Neural Networks with Inner-penalty Terms

This paper investigates an online gradient method with inner- penalty for a novel feed forward network it is called pi-sigma network. This network utilizes product cells as the output units to indirectly incorporate the capabilities of higher-order networks while using a fewer number of weights and processing units. Penalty term methods have been widely used to improve the generalization performance of feed forward neural networks and to control the magnitude of the network weights. The monotonicity of the error function and weight boundedness with inner- penalty term and both weak and strong convergence theorems in the training iteration are proved.

Fast convergence of regularised Region-based Mixture of Gaussians for dynamic background modelling

The momentum term has long been used in machine learning algorithms, especially back-propagation, to improve their speed of convergence. In this paper, we derive an expression to prove the O(1/k2) convergence rate of the online gradient method, with momentum type updates, when the individual gradients are constrained by a growth condition. We then apply these type of updates to video background modelling by using it in the update equations of the Region-based Mixture of Gaussians algorithm. Extensive evaluations are performed on both simulated data, as well as challenging real world scenarios with dynamic backgrounds, to show that these regularised updates help the mixtures converge faster than the conventional approach and consequently improve the algorithm’s performance.

Convergence of online gradient method for feedforward neural networks with smoothing L1/2 regularization penalty

Minimization of the training regularization term has been recognized as an important objective for sparse modeling and generalization in feedforward neural networks. Most of the studies so far have been focused on the popular L2 regularization penalty. In this paper, we consider the convergence of online gradient method with smoothing L1/2 regularization term. For normal L1/2 regularization, the objective function is the sum of a non-convex, non-smooth, and non-Lipschitz function, which causes oscillation of the error function and the norm of gradient. However, using the smoothing approximation techniques, the deficiency of the normal L1/2 regularization term can be addressed. This paper shows the strong convergence results for the smoothing L1/2 regularization. Furthermore, we prove the boundedness of the weights during the network training. The assumption that weights are bounded is no longer needed for the proof of convergence. Simulation results support the theoretical findings and demonstrate that our algorithm has better performance than two other algorithms with L2 and normal L1/2 regularizations respectively.

Analysis of boundedness and convergence of online gradient method for two-layer feedforward neural networks.

This paper presents a theoretical boundedness and convergence analysis of online gradient method for the training of two-layer feedforward neural networks. The well-known linear difference equation is extended to apply to the general case of linear or nonlinear activation functions. Based on this extended difference equation, we investigate the boundedness and convergence of the parameter sequence of concern, which is trained by finite training samples with a constant learning rate. We show that the uniform upper bound of the parameter sequence, which is very important in the training procedure, is the solution of an inequality regarding the bound. It is further verified that, for the case of linear activation function, a solution always exists and, moreover, the parameter sequence can be uniformly upper bounded, while for the case of nonlinear activation function, some simple adjustment methods on the training set or the activation function can be derived to improve the boundedness property. Then, for the convergence analysis, it is shown that the parameter sequence can converge into a zone around an optimal solution at which the error function attains its global minimum, where the size of the zone is associated with the learning rate. Particularly, for the case of perfect modeling, a strong global convergence result, where the parameter sequence can always converge to an optimal solution, is proved.

Convergence of an online gradient method with inner-product penalty and adaptive momentum

In this paper, we study the convergence of an online gradient method with inner-product penalty and adaptive momentum for feedforward neural networks, assuming that the training samples are permuted stochastically in each cycle of iteration. Both two-layer and three-layer neural network models are considered, and two convergence theorems are established. Sufficient conditions are proposed to prove weak and strong convergence results. The algorithm is applied to the classical two-spiral problem and identification of Gabor function problem to support these theoretical findings.

Boundedness and convergence of online gradient method with penalty and momentum

In this paper, the deterministic convergence of an online gradient method with penalty and momentum is investigated for training two-layer feedforward neural networks. The monotonicity of the new error function with the penalty term in the training iteration is firstly proved. Under this conclusion, we show that the weights are uniformly bounded during the training process and the algorithm is deterministically convergent. Sufficient conditions are also provided for both weak and strong convergence results.

Convergence analysis of online gradient method for BP neural networks

This paper considers a class of online gradient learning methods for backpropagation (BP) neural networks with a single hidden layer. We assume that in each training cycle, each sample in the training set is supplied in a stochastic order to the network exactly once. It is interesting that these stochastic learning methods can be shown to be deterministically convergent. This paper presents some weak and strong convergence results for the learning methods, indicating that the gradient of the error function goes to zero and the weight sequence goes to a fixed point, respectively. The conditions on the activation function and the learning rate to guarantee the convergence are relaxed compared with the existing results. Our convergence results are valid for not only S-S type neural networks (both the output and hidden neurons are Sigmoid functions), but also for P-P, P-S and S-P type neural networks, where S and P represent Sigmoid and polynomial functions, respectively.

Convergence of an Online Split‐Complex Gradient Algorithm for Complex‐Valued Neural Networks

The online gradient method has been widely used in training neural networks. We consider in this paper an online split‐complex gradient algorithm for complex‐valued neural networks. We choose an adaptive learning rate during the training procedure. Under certain conditions, by firstly showing the monotonicity of the error function, it is proved that the gradient of the error function tends to zero and the weight sequence tends to a fixed point. A numerical example is given to support the theoretical findings.

Boundedness and Convergence of Online Gradient Method With Penalty for Feedforward Neural Networks

In this brief, we consider an online gradient method with penalty for training feedforward neural networks. Specifically, the penalty is a term proportional to the norm of the weights. Its roles in the method are to control the magnitude of the weights and to improve the generalization performance of the network. By proving that the weights are automatically bounded in the network training with penalty, we simplify the conditions that are required for convergence of online gradient method in literature. A numerical example is given to support the theoretical analysis.

Boundedness and Convergence of Online Gradient Method with Penalty for Linear Output Feedforward Neural Networks

This paper investigates an online gradient method with penalty for training feedforward neural networks with linear output. A usual penalty is considered, which is a term proportional to the norm of the weights. The main contribution of this paper is to theoretically prove the boundedness of the weights in the network training process. This boundedness is then used to prove an almost sure convergence of the algorithm to the zero set of the gradient of the error function.

An online gradient method with momentum for two-layer feedforward neural networks

An online gradient method with momentum for two-layer feedforward neural networks is considered. The momentum coefficient is chosen in an adaptive manner to accelerate and stabilize the learning procedure of the network weights. Corresponding convergence results are proved, that is, the weak convergence result is proved under the uniformly boundedness assumption of the activation function and its derivatives, moreover, if the number of elements of the stationary point set for the error function is finite, then strong convergence result holds.