Ensemble methods such as boosting combine multiple learners to obtain better prediction than could be obtained from any individual learner. Here we propose a principled framework for directly constructing ensemble learning methods from kernel methods. Unlike previous studies showing the equivalence between boosting and support vector machines (SVMs), which needs a translation procedure, we show that it is possible to design boosting-like procedure to solve the SVM optimization problems. In other words, it is possible to design ensemble methods directly from SVM without any middle procedure. This finding not only enables us to design new ensemble learning methods directly from kernel methods, but also makes it possible to take advantage of those highly-optimized fast linear SVM solvers for ensemble learning. We exemplify this framework for designing binary ensemble learning as well as a new multi-class ensemble learning methods. Experimental results demonstrate the flexibility and usefulness of the proposed framework.
If one uses an infinitely many weak learners in boosting @cite_15 (or hidden units in neural networks @cite_16 ), the model is equivalent to SVM with a certain kernel. In particular, it shows that when the feature mapping function @math contains infinitely many randomly distributed decision stumps, the kernel function @math is the stump kernel of the form @math . Here @math is a constant, which has no impact on the SVM training. Moreover, when @math , , a perceptron, the corresponding kernel is called the perceptron kernel @math @math .
@cite_15 @cite_16
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Ensemble learning algorithms such as boosting can achieve better performance by averaging over the predictions of some base hypotheses. Nevertheless, most existing algorithms are limited to combining only a finite number of hypotheses, and the generated ensemble is usually sparse. Thus, it is not clear whether we should construct an ensemble classifier with a larger or even an infinite number of hypotheses. In addition, constructing an infinite ensemble itself is a challenging task. In this paper, we formulate an infinite ensemble learning framework based on the support vector machine (SVM). The framework can output an infinite and nonsparse ensemble through embedding infinitely many hypotheses into an SVM kernel. We use the framework to derive two novel kernels, the stump kernel and the perceptron kernel. The stump kernel embodies infinitely many decision stumps, and the perceptron kernel embodies infinitely many perceptrons. We also show that the Laplacian radial basis function kernel embodies infinitely many decision trees, and can thus be explained through infinite ensemble learning. Experimental results show that SVM with these kernels is superior to boosting with the same base hypothesis set. In addition, SVM with the stump kernel or the perceptron kernel performs similarly to SVM with the Gaussian radial basis function kernel, but enjoys the benefit of faster parameter selection. These properties make the novel kernels favorable choices in practice. This article extends neural networks to the case of an uncountable number of hidden units, in several ways. In the first approach proposed, a finite parametrization is possible, allowing gradient-based learning. While having the same number of parameters as an ordinary neural network, its internal structure suggests that it can represent some smooth functions much more compactly. Under mild assumptions, we also find better error bounds than with ordinary neural networks. Furthermore, this parametrization may help reducing the problem of saturation of the neurons. In a second approach, the input-to-hidden weights are fully nonparametric, yielding a kernel machine for which we demonstrate a simple kernel formula. Interestingly, the resulting kernel machine can be made hyperparameter-free and still generalizes in spite of an absence of explicit regularization.