Complexities of convex combinations and bounding the generalization error in classification

This paper generalizes an important result from the PAC-Bayesian literature for binary classification to the case of ensemble methods for structured outputs. We prove a generic version of the \Cbound, an upper bound over the risk of models expressed as a weighted majority vote that is based on the first and second statistical moments of the vote's margin. This bound may advantageously $(i)$ be applied on more complex outputs such as multiclass labels and multilabel, and $(ii)...

This paper presents an algorithm, Voted Kernel Regularization , that provides the flexibility of using potentially very complex kernel functions such as predictors based on much higher-degree polynomial kernels, while benefitting from strong learning guarantees. The success of our algorithm arises from derived bounds that suggest a new regularization penalty in terms of the Rademacher complexities of the corresponding families of kernel maps. In a series of experiments we dem...

Boosting and other ensemble methods combine a large number of weak classifiers through weighted voting to produce stronger predictive models. To explain the successful performance of boosting algorithms, Schapire et al. (1998) showed that AdaBoost is especially effective at increasing the margins of the training data. Schapire et al. (1998) also developed an upper bound on the generalization error of any ensemble based on the margins of the training data, from which it was co...

We address the problem of aggregating an ensemble of predictors with known loss bounds in a semi-supervised binary classification setting, to minimize prediction loss incurred on the unlabeled data. We find the minimax optimal predictions for a very general class of loss functions including all convex and many non-convex losses, extending a recent analysis of the problem for misclassification error. The result is a family of semi-supervised ensemble aggregation algorithms whi...

We study the generalisation properties of majority voting on finite ensembles of classifiers, proving margin-based generalisation bounds via the PAC-Bayes theory. These provide state-of-the-art guarantees on a number of classification tasks. Our central results leverage the Dirichlet posteriors studied recently by Zantedeschi et al. [2021] for training voting classifiers; in contrast to that work our bounds apply to non-randomised votes via the use of margins. Our contributio...

We consider a problem of risk estimation for large-margin multi-class classifiers. We propose a novel risk bound for the multi-class classification problem. The bound involves the marginal distribution of the classifier and the Rademacher complexity of the hypothesis class. We prove that our bound is tight in the number of classes. Finally, we compare our bound with the related ones and provide a simplified version of the bound for the multi-class classification with kernel b...

We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that such notions are not only sufficient for learning using linear predictors or a kernel, but unlike the exact variants, are also necessary. Thus they are better suited for discussing limitations of linear or kernel methods.

Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers. This chapter provides an analytical framework to quantify the improvements in classification results due to combining. The results apply to both linear combiners and order statistics combiners. We first show that to a first order approximation, the error rate obtained over and abo...

We study prediction and estimation problems using empirical risk minimization, relative to a general convex loss function. We obtain sharp error rates even when concentration is false or is very restricted, for example, in heavy-tailed scenarios. Our results show that the error rate depends on two parameters: one captures the intrinsic complexity of the class, and essentially leads to the error rate in a noise-free (or realizable) problem; the other measures interactions betw...

In boosting, we aim to leverage multiple weak learners to produce a strong learner. At the center of this paradigm lies the concept of building the strong learner as a voting classifier, which outputs a weighted majority vote of the weak learners. While many successful boosting algorithms, such as the iconic AdaBoost, produce voting classifiers, their theoretical performance has long remained sub-optimal: the best known bounds on the number of training examples necessary for ...