According to conventional neural network theories, single-hidden-layer feedforward networks (SLFNs) with additive or radial basis function (RBF) hidden nodes are universal approximators when all the parameters of the networks are allowed adjustable. However, as observed in most neural network implementations, tuning all the parameters of the networks may cause learning complicated and inefficient, and it may be difficult to train networks with nondifferential activation functions such as threshold networks. Unlike conventional neural network theories, this paper proves in an incremental constructive method that in order to let SLFNs work as universal approximators, one may simply randomly choose hidden nodes and then only need to adjust the output weights linking the hidden layer and the output layer. In such SLFNs implementations, the activation functions for additive nodes can be any bounded nonconstant piecewise continuous functions g:R→R and the activation functions for RBF nodes can be any integrable piecewise continuous functions g:R→R and ∫Rg(x)dx≠0. The proposed incremental method is efficient not only for SFLNs with continuous (including nondifferentiable) activation functions but also for SLFNs with piecewise continuous (such as threshold) activation functions. Compared to other popular methods such a new network is fully automatic and users need not intervene the learning process by manually tuning control parameters.

Universal approximation using incremental constructive feedforward networks with random hidden nodes

In this paper, we develop an online sequential learning algorithm for single hidden layer feedforward networks (SLFNs) with additive or radial basis function (RBF) hidden nodes in a unified framework. The algorithm is referred to as online sequential extreme learning machine (OS-ELM) and can learn data one-by-one or chunk-by-chunk (a block of data) with fixed or varying chunk size. The activation functions for additive nodes in OS-ELM can be any bounded nonconstant piecewise continuous functions and the activation functions for RBF nodes can be any integrable piecewise continuous functions. In OS-ELM, the parameters of hidden nodes (the input weights and biases of additive nodes or the centers and impact factors of RBF nodes) are randomly selected and the output weights are analytically determined based on the sequentially arriving data. The algorithm uses the ideas of ELM of Huang developed for batch learning which has been shown to be extremely fast with generalization performance better than other batch training methods. Apart from selecting the number of hidden nodes, no other control parameters have to be manually chosen. Detailed performance comparison of OS-ELM is done with other popular sequential learning algorithms on benchmark problems drawn from the regression, classification and time series prediction areas. The results show that the OS-ELM is faster than the other sequential algorithms and produces better generalization performance

A Fast and Accurate Online Sequential Learning Algorithm for Feedforward Networks

Computational intelligence techniques have been used in wide applications. Out of numerous computational intelligence techniques, neural networks and support vector machines (SVMs) have been playing the dominant roles. However, it is known that both neural networks and SVMs face some challenging issues such as: (1) slow learning speed, (2) trivial human intervene, and/or (3) poor computational scalability. Extreme learning machine (ELM) as emergent technology which overcomes some challenges faced by other techniques has recently attracted the attention from more and more researchers. ELM works for generalized single-hidden layer feedforward networks (SLFNs). The essence of ELM is that the hidden layer of SLFNs need not be tuned. Compared with those traditional computational intelligence techniques, ELM provides better generalization performance at a much faster learning speed and with least human intervene. This paper gives a survey on ELM and its variants, especially on (1) batch learning mode of ELM, (2) fully complex ELM, (3) online sequential ELM, (4) incremental ELM, and (5) ensemble of ELM.

Extreme learning machines: a survey

Extreme learning machines (ELMs) have been proposed for generalized single-hidden-layer feedforward networks which need not be neuron-like and perform well in both regression and classification applications. In this brief, we propose an ELM with adaptive growth of hidden nodes (AG-ELM), which provides a new approach for the automated design of networks. Different from other incremental ELMs (I-ELMs) whose existing hidden nodes are frozen when the new hidden nodes are added one by one, in AG-ELM the number of hidden nodes is determined in an adaptive way in the sense that the existing networks may be replaced by newly generated networks which have fewer hidden nodes and better generalization performance. We then prove that such an AG-ELM using Lebesgue p-integrable hidden activation functions can approximate any Lebesgue p-integrable function on a compact input set. Simulation results demonstrate and verify that this new approach can achieve a more compact network architecture than the I-ELM.

Universal Approximation of Extreme Learning Machine With Adaptive Growth of Hidden Nodes

We propose an approach for neuro-fuzzy system modeling. A neuro-fuzzy system for a given set of input-output data is obtained in two steps. First, the data set is partitioned automatically into a set of clusters based on input-similarity and output-similarity tests. Membership functions associated with each cluster are defined according to statistical means and variances of the data points included in the cluster. Then, a fuzzy IF-THEN rule is extracted from each cluster to form a fuzzy rule-base. Second, a fuzzy neural network is constructed accordingly and parameters are refined to increase the precision of the fuzzy rule-base. To decrease the size of the search space and to speed up the convergence, we develop a hybrid learning algorithm which combines a recursive singular value decomposition-based least squares estimator and the gradient descent method. The proposed approach has advantages of determining the number of rules automatically and matching membership functions closely with the real distribution of the training data points. Besides, it learns faster, consumes less memory, and produces lower approximation errors than other methods.

A neuro-fuzzy system modeling with self-constructing rule generationand hybrid SVD-based learning

The problem of approximating functions by neural networks using incremental algorithms is studied. For functions belonging to a rather general class, characterized by certain smoothness properties with respect to the L/sub 2/ norm, we compute upper bounds on the approximation error where error is measured by the L/sub q/ norm, 1/spl les/q/spl les//spl infin/. These results extend previous work, applicable in the case q=2, and provide an explicit algorithm to achieve the derived approximation error rate. In the range q/spl les/2 near-optimal rates of convergence are demonstrated. A gap remains, however, with respect to a recently established lower bound in the case q>2, although the rates achieved are provably better than those obtained by optimal linear approximation. Extensions of the results from the L/sub 2/ norm to L/sub p/ are also discussed. A further interesting conclusion from our results is that no loss of generality is suffered using networks with positive hidden-to-output weights. Moreover, explicit bounds on the size of the hidden-to-output weights are established, which are sufficient to guarantee the established convergence rates.

R. Meir

Papers

On the optimality of neural-network approximation using incremental algorithms