Guang-Bin Huang

Due to the simplicity of their implementations, least square support vector machine (LS-SVM) and proximal support vector machine (PSVM) have been widely used in binary classification applications. The conventional LS-SVM and PSVM cannot be used in regression and multiclass classification applications directly, although variants of LS-SVM and PSVM have been proposed to handle such cases. This paper shows that both LS-SVM and PSVM can be simplified further and a unified learning framework of LS-SVM, PSVM, and other regularization algorithms referred to extreme learning machine (ELM) can be built. ELM works for the "generalized" single-hidden-layer feedforward networks (SLFNs), but the hidden layer (or called feature mapping) in ELM need not be tuned. Such SLFNs include but are not limited to SVM, polynomial network, and the conventional feedforward neural networks. This paper shows the following: 1) ELM provides a unified learning platform with a widespread type of feature mappings and can be applied in regression and multiclass classification applications directly; 2) from the optimization method point of view, ELM has milder optimization constraints compared to LS-SVM and PSVM; 3) in theory, compared to ELM, LS-SVM and PSVM achieve suboptimal solutions and require higher computational complexity; and 4) in theory, ELM can approximate any target continuous function and classify any disjoint regions. As verified by the simulation results, ELM tends to have better scalability and achieve similar (for regression and binary class cases) or much better (for multiclass cases) generalization performance at much faster learning speed (up to thousands times) than traditional SVM and LS-SVM.

It is clear that the learning speed of feedforward neural networks is in general far slower than required and it has been a major bottleneck in their applications for past decades. Two key reasons behind may be: 1) the slow gradient-based learning algorithms are extensively used to train neural networks, and 2) all the parameters of the networks are tuned iteratively by using such learning algorithms. Unlike these traditional implementations, this paper proposes a new learning algorithm called extreme learning machine (ELM) for single-hidden layer feedforward neural networks (SLFNs) which randomly chooses the input weights and analytically determines the output weights of SLFNs. In theory, this algorithm tends to provide the best generalization performance at extremely fast learning speed. The experimental results based on real-world benchmarking function approximation and classification problems including large complex applications show that the new algorithm can produce best generalization performance in some cases and can learn much faster than traditional popular learning algorithms for feedforward neural networks.

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 integral of R g(x)dx not equal to 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.

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 et al. 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.