The resulting combinatorial problem - finding the best agreement half-space over an input sample - is NP hard to approximate to within some constant factor. We suggest a way to circumvent this theoretical bound by introducing a new measure of success for such algorithms. An algorithm is ILmargin successful if the agreement ratio of the half-space it outputs is as good as that of any half-space once training points that are inside the ILmargins of its separating hyper-plane are disregarded. We prove crisp computational complexity results with respect to this success measure: On one hand, for every positive IL, there exist efficient (poly-time) ILmargin successful learning algorithms. On the other hand, we prove that unless P NP, there is no algorithm that runs in time polynomial in the sample size and in 1/ IL that is ILmargin successful for all IL O. 1 Introduction We consider the computational complexity of learning linear perceptrons for arbitrary (Le.

Subsequently, SVMs have been modified to handle regression [12] and GPs have been adapted to the problem of classification [8]. Both schemes essentially work in the same function space that is characterised by kernels (SVM) and covariance functions (GP), respectively. While the formal similarity of the two methods is striking the underlying paradigms of inference are very different. The SVM was inspired by results from statistical/PAC learning theory while GPs are usually considered in a Bayesian framework. This ideological clash can be viewed as a continuation in machine learning of the by now classical disagreement between Bayesian and frequentistic statistics.

The resulting combinatorial problem - finding the best agreement half-space over an input sample - is NP hard to approximate to within some constant factor. We suggest a way to circumvent this theoretical bound by introducing a new measure of success for such algorithms. An algorithm is ILmargin successful if the agreement ratio of the half-space it outputs is as good as that of any half-space once training points that are inside the ILmargins of its separating hyper-plane are disregarded. We prove crisp computational complexity results with respect to this success measure: On one hand, for every positive IL, there exist efficient (poly-time) ILmargin successful learning algorithms. On the other hand, we prove that unless P NP, there is no algorithm that runs in time polynomial in the sample size and in 1/ IL that is ILmargin successful for all IL O. 1 Introduction We consider the computational complexity of learning linear perceptrons for arbitrary (Le.

Subsequently, SVMs have been modified to handle regression [12] and GPs have been adapted to the problem of classification [8]. Both schemes essentially work in the same function space that is characterised by kernels (SVM) and covariance functions (GP), respectively. While the formal similarity of the two methods is striking the underlying paradigms of inference are very different. The SVM was inspired by results from statistical/PAC learning theory while GPs are usually considered in a Bayesian framework. This ideological clash can be viewed as a continuation in machine learning of the by now classical disagreement between Bayesian and frequentistic statistics.

A stable learner is one for which the learned solution does not change much with small changes in the training set. The bounds we obtain do not depend on any measure of the complexity of the hypothesis space (e.g. VC dimension) but rather depend on how the learning algorithm searches this space, and can thus be applied even when the VC dimension is infinite. We demonstrate that regularization networks possess the required stability property and apply our method to obtain new bounds on their generalization performance.

The resulting combinatorial problem - finding the best agreement half-space over an input sample - is NP hard to approximate to within some constant factor. We suggest a way to circumvent this theoretical bound by introducing a new measure of success for such algorithms. An algorithm is ILmargin successful if the agreement ratio of the half-space it outputs is as good as that of any half-space once training points that are inside the ILmargins of its separating hyper-plane are disregarded. We prove crisp computational complexity resultswith respect to this success measure: On one hand, for every positive IL, there exist efficient (poly-time) ILmargin successful learningalgorithms. On the other hand, we prove that unless P NP, there is no algorithm that runs in time polynomial in the sample size and in 1/IL that is ILmargin successful for all IL O. 1 Introduction We consider the computational complexity of learning linear perceptrons for arbitrary (Le.non -separable) data sets.

I consider the problem of learning an optimal path graphical model from data and show the problem to be NP-hard for the maximum likelihood and minimum description length approaches and a Bayesian approach. This hardness result holds despite the fact that the problem is a restriction of the polynomially solvable problem of finding the optimal tree graphical model.

The scientific method aims to derive mathematical models that help us to understand and exploit phenomena, whether they be natural or human made. Machine learning, and more particularly learning with neural networks, can be viewed as just such a phenomenon. Frequently remarkable performance is obtained by training networks to perform relatively complex AI tasks. Despite this success, most practitioners would readily admit that they are far from fully understanding why and, more importantly, when the techniques can be expected to be effective. The need for a fuller theoretical analysis and understanding of their performance has been a major research objective for the last decade. Neural Network Learning: Theoretical Foundations reports on important developments that have been made toward this goal within the computational learning theory framework.

Simply changing the potential function allows one to create new algorithms related to AdaBoost. However, these new algorithms are generally not known to have the formal boosting property. This paper examines the question of which potential functions lead to new algorithms that are boosters. The two main results are general sets of conditions on the potential; one set implies that the resulting algorithm is a booster, while the other implies that the algorithm is not. These conditions are applied to previously studied potential functions, such as those used by LogitBoost and Doom II.

Simply changing the potential function allows one to create new algorithms related to AdaBoost. However, these new algorithms are generally not known to have the formal boosting property. This paper examines the question of which potential functions lead to new algorithms that are boosters. The two main results are general sets of conditions on the potential; one set implies that the resulting algorithm is a booster, while the other implies that the algorithm is not. These conditions are applied to previously studied potential functions, such as those used by LogitBoost and Doom II.