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this article with the regression case. To explain this, we will introduce a suitable definition of a margin that is maximized in both cases

Much recent attention, both experimental and theoretical, has been focussed on classification algorithms which produce voted combinations of classifiers. Recent theoretical work has shown that the impressive generalization performance of algorithms like AdaBoost can be attributed to the classifier having large margins on the training data. We present an abstract algorithm for finding linear combinations of functions that minimize arbitrary cost functionals (i.e functionals that do not necessarily depend on the margin). Many existing voting methods can be shown to be special cases of this abstract algorithm. Then, following previous theoretical results bounding the generalization performance of convex combinations of classifiers in terms of general cost functions of the margin, we present a new algorithm (DOOM II) for performing a gradient descent optimization of such cost functions. Experiments on

Recent theoretical results have shown that the generalization performance of thresholded convex combinations of base classifiers is greatly improved if the underlying convex combination has large margins on the training data (correct examples are classified well away from the decision boundary). Neural network algorithms and AdaBoost have been shown to implicitly maximize margins, thus providing some theoretical justification for their remarkably good generalization performance. In this paper we are concerned with maximizing the margin explicitly. In particular, we prove a theorem bounding the generalization performance of convex combinations in terms of general cost functions of the margin (previous results were stated in terms of the particular cost function sgn(`;margin). We then present an algorithm (DOOM) for directly optimizing a piecewise-linear family of cost functions satisfying the conditions of the theorem. Experiments on several of the datasets in the UC Irvine database are presented in which AdaBoost was used to generate a set of base classifiers and then DOOM was used to find the optimal convex combination of those classifiers. In all but one case the convex combination generated by DOOM had lower test error than AdaBoost's combination. In many cases DOOM achieves these lower test errors by sacrificing training error, in the interests of reducing the new cost function. The margin plots also show that the size of the minimum margin is not relevant to generalization performance.

Sonar Cumulative training margin distributions for AdaBoost versus our "Direct Optimization Of Margins" (DOOM) algorithm.

Abstract not found

Recent theoretical results haveshown that the accuracy of thresholded real-valued functions (suchasvoting classifiers) is greatly improved if the underlying function has large margins on the training data (that is, correct examples are classified well away from the decision boundary). In this paper, wegive bounds on the misclassification probabilityofconvex combinations of classifiers in terms of general cost functions of the margin.

Bounds on the risk play a crucial role in statistical learning theory. They usually involve as capacity measure of the model studied the VC dimension or one of its extensions. In classification, such VC dimensions exist for models taking values in {0, 1}, [ 1, Q], and R. We introduce the generalizations appropriate for the missing case, the one of models with values in RQ. This provides us with a new guaranteed risk for M-SVMs. For those models, a sharper bound is obtained by using the Rademacher complexity.

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