We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widely held belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of performance as the training set size is increased, one in which each algorithm does better. This stems from the observation -- which is borne out in repeated experiments -- that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster.

The problem of statistical learning can be considered as a problem of nonparametric estimation of sets, where the risk is de ned by means of a speci c distance function between sets associated to the misclassi cation error. The rates of convergence of classi ers depend on two parameters: the complexity of the class of candidate sets and the "margin" parameter. The dependence is explicitly given, in particular the optimal rates up to O(n ) can be attained, where n is the sample size, and the proposed classi ers have the property of robustness to the margin. The main result of the paper concerns optimal aggregation of classi ers: we suggest a classi er that automatically adapts both to the complexity and to the margin, and attains the optimal fast rates, up to a logarithmic factor.

A major problem in machine learning is that of inductive bias: how to choose a learner's hypothesis space so that it is large enough to contain a solution to the problem being learnt, yet small enough to ensure reliable generalization from reasonably-sized training sets. Typically such bias is supplied by hand through the skill and insights of experts. In this paper a model for automatically learning bias is investigated. The central assumption of the model is that the learner is embedded within an environment of related learning tasks. Within such an environment the learner can sample from multiple tasks, and hence it can search for a hypothesis space that contains good solutions to many of the problems in the environment. Under certain restrictions on the set of all hypothesis spaces available to the learner, we show that a hypothesis space that performs well on a sufficiently large number of training tasks will also perform well when learning novel tasks in the same environment. Exp...

Dedicated to A.V. Skorohod on his seventieth birthday We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such as boosting and bagging. The bounds are in terms of the empirical distribution of the margin of the combined classifier. They are based on the methods of the theory of Gaussian and empirical processes (comparison inequalities, symmetrization method, concentration inequalities) and they improve previous results of Bartlett (1998) on bounding the generalization error of neural networks in terms of ℓ1-norms of the weights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding the generalization error of boosting. We also obtain rates of convergence in Lévy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates.

We state and analyze the first active learning algorithm which works in the presence of arbitrary forms of noise. The algorithm, A2 (for Agnostic Active), relies only upon the assumption that the samples are drawn i.i.d. from a fixed distribution. We show that A2 achieves an exponential improvement (i.e., requires only O � ln 1 ɛ samples to find an ɛ-optimal classifier) over the usual sample complexity of supervised learning, for several settings considered before in the realizable case. These include learning threshold classifiers and learning homogeneous linear separators with respect to an input distribution which is uniform over the unit sphere. 1.

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We derive new bounds for the generalization error of kernel machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs make use of a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinite-dimensional unit ball in feature space into a finite-dimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence, we are able to theoretically explain the effect of the choice of kernel function on the generalization performance of support vector machines.

We introduce constraint classification, a framework capturing many flavors of multiclass classification including multilabel classification and ranking, and present a meta-algorithm for learning in this framework. We provide generalization bounds when using a collection of k linear functions to represent each hypothesis. We also present empirical and theoretical evidence that constraint classification is more powerful than existing methods of multiclass classification. 1

Abstract. We consider batch reinforcement learning problems in continuous space, expected total discounted-reward Markovian Decision Problems. As opposed to previous theoretical work, we consider the case when the training data consists of a single sample path (trajectory) of some behaviour policy. In particular, we do not assume access to a generative model of the environment. The algorithm studied is policy iteration where in successive iterations the Q-functions of the intermediate policies are obtained by means of minimizing a novel Bellman-residual type error. PACstyle polynomial bounds are derived on the number of samples needed to guarantee near-optimal performance where the bound depends on the mixing rate of the trajectory, the smoothness properties of the underlying Markovian Decision Problem, the approximation power and capacity of the function set used. 1

A fundamental problem in statistical parsing is the choice of criteria and algorithms used to estimate the parameters in a model. The predominant approach in computational linguistics has been to use a parametric model with some variant of maximum-likelihood estimation. The assumptions under which maximum-likelihood estimation is justified are arguably quite strong. This paper discusses the statistical theory underlying various parameter-estimation methods, and gives algorithms which depend on alternatives to (smoothed) maximumlikelihood estimation. We first give an overview of results from statistical learning theory. We then show how important concepts from the classification literature -- specifically, generalization results based on margins on training data -- can be derived for parsing models. Finally, we describe parameter estimation algorithms which are motivated by these generalization bounds.