Complexities of convex combinations and bounding the generalization error in classification

Many popular learning algorithms (E.g. Regression, Fourier-Transform based algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the problem to a convex optimization problem over a vector space of functions. These methods offer the currently best approach to several central problems such as learning half spaces and learning DNF's. In addition they are widely used in numerous application domains. Despite their importance, there are still very few proof techni...

Ensemble classifiers have been investigated by many in the artificial intelligence and machine learning community. Majority voting and weighted majority voting are two commonly used combination schemes in ensemble learning. However, understanding of them is incomplete at best, with some properties even misunderstood. In this paper, we present a group of properties of these two schemes formally under a dataset-level geometric framework. Two key factors, every component base cl...

Ensembles, as a widely used and effective technique in the machine learning community, succeed within a key element -- "diversity." The relationship between diversity and generalization, unfortunately, is not entirely understood and remains an open research issue. To reveal the effect of diversity on the generalization of classification ensembles, we investigate three issues on diversity, i.e., the measurement of diversity, the relationship between the proposed diversity and ...

The number of component classifiers chosen for an ensemble greatly impacts the prediction ability. In this paper, we use a geometric framework for a priori determining the ensemble size, which is applicable to most of existing batch and online ensemble classifiers. There are only a limited number of studies on the ensemble size examining Majority Voting (MV) and Weighted Majority Voting (WMV). Almost all of them are designed for batch-mode, hardly addressing online environmen...

The question of whether to use one classifier or a combination of classifiers is a central topic in Machine Learning. We propose here a method for finding an optimal linear combination of classifiers derived from a bias-variance framework for the classification task.

When randomized ensembles such as bagging or random forests are used for binary classification, the prediction error of the ensemble tends to decrease and stabilize as the number of classifiers increases. However, the precise relationship between prediction error and ensemble size is unknown in practice. In the standard case when classifiers are aggregated by majority vote, the present work offers a way to quantify this convergence in terms of "algorithmic variance," i.e. the...

We study the behavior of error bounds for multiclass classification under suitable margin conditions. For a wide variety of methods we prove that the classification error under a hard-margin condition decreases exponentially fast without any bias-variance trade-off. Different convergence rates can be obtained in correspondence of different margin assumptions. With a self-contained and instructive analysis we are able to generalize known results from the binary to the multicla...

We present here a PAC-Bayesian point of view on adaptive supervised classification. Using convex analysis, we show how to get local measures of the complexity of the classification model involving the relative entropy of posterior distributions with respect to Gibbs posterior measures. We discuss relative bounds, comparing two classification rules, to show how the margin assumption of Mammen and Tsybakov can be replaced with some empirical measure of the covariance structure ...

In statistical learning theory, a generalization bound usually involves a complexity measure imposed by the considered theoretical framework. This limits the scope of such bounds, as other forms of capacity measures or regularizations are used in algorithms. In this paper, we leverage the framework of disintegrated PAC-Bayes bounds to derive a general generalization bound instantiable with arbitrary complexity measures. One trick to prove such a result involves considering a ...

Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the u...