By Alexander J. Smola, Peter Bartlett, Bernhard Schölkopf, Dale Schuurmans
The concept that of huge margins is a unifying precept for the research of many various methods to the category of knowledge from examples, together with boosting, mathematical programming, neural networks, and help vector machines. the truth that it's the margin, or self belief point, of a classification--that is, a scale parameter--rather than a uncooked education mistakes that issues has turn into a key device for facing classifiers. This e-book exhibits how this concept applies to either the theoretical research and the layout of algorithms.The ebook presents an summary of modern advancements in huge margin classifiers, examines connections with different tools (e.g., Bayesian inference), and identifies strengths and weaknesses of the strategy, in addition to instructions for destiny examine. one of the individuals are Manfred Opper, Vladimir Vapnik, and style Wahba.
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The similarity is most obvious in regression, where the Support Vector solution is the maximum a posteriori estimate of the corre sponding Bayesian inference scheme [Williams, 1998]. ik(Xi, x) is given by P(f) ex exp (_! jk(Xi ' Xj) ) . 84) t ,3 Bayesian methods, however, require averaging over the posterior distribution P(f I X, Y) in order to obtain the final estimate and to derive error bounds. In classification the situation is more complicated, since we have Bernoulli distributed random variables for the labels of the classifier.
For such a problem, the dual method has no advantage. The potential advantage of the dual method for regression is that it can be applied to very large feature vectors. The coefficient matrix XXT contains the scalar products of pairs of feature vectors: the ijth element of XXT is Vi . Vj' In the dual calculation, it is only scalar products of feature vectors that are used feature vectors never appear on their own. The matrix of scalar products of the feature vectors encodes the lengths and relative orientations of the features, and this geometric information is enough for most linear computations.
The main practical contribution of this chapter is the introduction of a new (sigmoidal) margin cost functional that can be optimized by a heuristic search procedure (DOOM II). The resulting procedure achieves good theoretical bounds on its generalization perfor mance but also demonstrates systematic improvements over AdaBoost in empirical tests-especially in domains with significant classification noise. In their chapter entitled Towards a Strategy for Boosting Regressors, Karakoulas and Shawe-Taylor describe a new strategy for combining regressors (as opposed to classifiers) in a boosting framework.