margin classifiers

Recently ensemble methods like ADABOOST have been applied successfully in many problems, while seemingly defying the problems of overfitting. ADABOOST rarely overfits in the low noise regime, however, we show that it clearly does so for higher noise levels. Central to the understanding of this fact is the margin distribution. ADABOOST can be viewed as a constraint gradient descent in an error function with respect to the margin. We find that ADABOOST asymptotically achieves a hard margin distribution, i.e. the algorithm concentrates its resources on a few hard-to-learn patterns that are interestingly very similar to Support Vectors. A hard margin is clearly a sub-optimal strategy in the noisy case, and regularization, in our case a “mistrust” in the data, must be introduced in the algorithm to alleviate the distortions that single difficult patterns (e.g. outliers) can cause to the margin distribution. We propose several regularization methods and generalizations of the original ADABOOST algorithm to achieve a soft margin. In particular we suggest (1) regularized ADABOOSTREG where the gradient decent is done directly with respect to the soft margin and (2) regularized linear and quadratic programming (LP/QP-) ADABOOST, where the soft margin is attained by introducing slack variables. Extensive simulations demonstrate that the proposed regularized ADABOOST-type algorithms are useful and yield competitive results for noisy data.

This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized "spectral complexity": their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the mnist and cifar10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.

We present a unifying framework for studying the solution of multiclass categorization problems by reducing them to multiple binary problems that are then solved using a margin-based binary learning algorithm. The proposed framework unifies some of the most popular approaches in which each class is compared against all others, or in which all pairs of classes are compared to each other, or in which output codes with error-correcting properties are used. We propose a general method for combining the classifiers generated on the binary problems, and we prove a general empirical multiclass loss bound given the empirical loss of the individual binary learning algorithms. The scheme and the corresponding bounds apply to many popular classification learning algorithms including support-vector machines, AdaBoost, regression, logistic regression and decision-tree algorithms. We also give a multiclass generalization error analysis for general output codes with AdaBoost as the binary learner. Experimental results with SVM and AdaBoost show that our scheme provides a viable alternative to the most commonly used multiclass algorithms.

Class membership probability estimates are important for many applications of data mining in which classification outputs are combined with other sources of information for decision-making, such as example-dependent misclassification costs, the outputs of other classifiers, or domain knowledge. Previous calibration methods apply only to two-class problems. Here, we show how to obtain accurate probability estimates for multiclass problems by combining calibrated binary probability estimates. We also propose a new method for obtaining calibrated two-class probability estimates that can be applied to any classifier that produces a ranking of examples. Using naive Bayes and support vector machine classifiers, we give experimental results from a variety of two-class and multiclass domains, including direct marketing, text categorization and digit recognition.

Pairwise coupling is a popular multi-class classification method that combines all comparisons for each pair of classes. This paper presents two approaches for obtaining class probabilities. Both methods can be reduced to linear systems and are easy to implement. We show conceptually and experimentally that the proposed approaches are more stable than the two existing popular methods: voting and the method by Hastie and Tibshirani (1998).

We consider the problem of multiclass classification. Our main thesis is that a simple “one-vs-all” scheme is as accurate as any other approach, assuming that the underlying binary classifiers are well-tuned regularized classifiers such as support vector machines. This thesis is interesting in that it disagrees with a large body of recent published work on multiclass classification. We support our position by means of a critical review of the existing literature, a substantial collection of carefully controlled experimental work, and theoretical arguments.

In this paper we describe the algorithmic implementation of multiclass kernel-based vector machines. Our starting point is a generalized notion of the margin to multiclass problems. Using this notion we cast multiclass categorization problems as a constrained optimization problem with a quadratic objective function. Unlike most of previous approaches which typically decompose a multiclass problem into multiple independent binary classification tasks, our notion of margin yields a direct method for training multiclass predictors. By using the dual of the optimization problem we are able to incorporate kernels with a compact set of constraints and decompose the dual problem into multiple optimization problems of reduced size. We describe an efficient fixed-point algorithm for solving the reduced optimization problems and prove its convergence. We then discuss technical details that yield significant running time improvements for large datasets. Finally, we describe various experiments with our approach comparing it to previously studied kernel-based methods. Our experiments indicate that for multiclass problems we attain state-of-the-art accuracy.