This paper investigates a new learning model in which the input data is corrupted with noise. We present a general statistical framework to tackle this problem. Based on the statistical reasoning, we propose a novel formulation of support vector classification, which allows uncertainty ininput data. We derive an intuitive geometric interpretation of the proposed formulation, and develop algorithms to efficiently solve it. Empirical results are included to show that the newly formed method is superior to the standard SVM for problems with noisy input.

The purpose of this paper is to investigate infinity-sample properties of risk minimization based multi-category classification methods. These methods can be considered as natural extensions to binary large margin classification. We establish conditions that guarantee the infinity-sample consistency of classifiers obtained in the risk minimization framework. Examples are provided for two specific forms of the general formulation, which extend a number of known methods. Using these examples, we show that some risk minimization formulations can also be used to obtain conditional probability estimates for the underlying problem. Such conditional probability information will be useful for statistical inferencing tasks beyond classification.

In this paper we obtain convergence bounds for the concentration of Bayesian posterior distributions (around the true distribution) using a novel method that simplifies and enhances previous results. Based on the analysis, we also introduce a generalized family of Bayesian posteriors, and show that the convergence behavior of these generalized posteriors is completely determined by the local prior structure around the true distribution. This important and surprising robustness property does not hold for the standard Bayesian posterior in that it may not concentrate when there exist "bad" prior structures even at places far away from the true distribution.

In this paper we obtain convergence bounds for the concentration of Bayesian posterior distributions (around the true distribution) using a novel method that simplifies and enhances previous results. Based on the analysis, we also introduce a generalized family of Bayesian posteriors, and show that the convergence behavior of these generalized posteriors is completely determined by the local prior structure around the true distribution. Thisimportant and surprising robustness property does not hold for the standard Bayesian posterior in that it may not concentrate when there exist "bad" prior structures even at places far away from the true distribution.

The purpose of this paper is to investigate infinity-sample properties of risk minimization based multi-category classification methods. These methods can be considered as natural extensions to binary large margin classification. We establish conditions that guarantee the infinity-sample consistency of classifiers obtained in the risk minimization framework. Examples are provided for two specific forms of the general formulation, which extend a number of known methods. Using these examples, we show that some risk minimization formulations can also be used to obtain conditionalprobability estimates for the underlying problem. Such conditional probability information will be useful for statistical inferencing tasksbeyond classification.

We investigate the generalization performance of some learning problems inHilbert function Spaces. We introduce a concept of scalesensitive effectivedata dimension, and show that it characterizes the convergence rateof the underlying learning problem. Using this concept, we can naturally extend results for parametric estimation problems in finite dimensional spaces to nonparametric kernel learning methods. We derive upperbounds on the generalization performance and show that the resulting convergent rates are optimal under various circumstances.

We consider Bayesian mixture approaches, where a predictor is constructed by forming a weighted average of hypotheses from some space of functions. While such procedures are known to lead to optimal predictors in several cases, where sufficiently accurate prior information is available, it has not been clear how they perform when some of the prior assumptions are violated. In this paper we establish data-dependent bounds for such procedures, extending previous randomized approaches such as the Gibbs algorithm to a fully Bayesian setting. The finite-sample guarantees established in this work enable the utilization of Bayesian mixture approaches in agnostic settings, where the usual assumptions of the Bayesian paradigm fail to hold. Moreover, the bounds derived can be directly applied to non-Bayesian mixture approaches such as Bagging and Boosting.

We investigate the generalization performance of some learning problems in Hilbert function Spaces. We introduce a concept of scalesensitive effective data dimension, and show that it characterizes the convergence rate of the underlying learning problem. Using this concept, we can naturally extend results for parametric estimation problems in finite dimensional spaces to nonparametric kernel learning methods. We derive upper bounds on the generalization performance and show that the resulting convergent rates are optimal under various circumstances.

We consider Bayesian mixture approaches, where a predictor is constructed by forming a weighted average of hypotheses from some space of functions. While such procedures are known to lead to optimal predictors in several cases, where sufficiently accurate prior information is available, it has not been clear how they perform when some of the prior assumptions are violated. In this paper we establish data-dependent bounds for such procedures, extending previous randomized approaches such as the Gibbs algorithm to a fully Bayesian setting. The finite-sample guarantees established in this work enable the utilization of Bayesian mixture approaches in agnostic settings, where the usual assumptions of the Bayesian paradigm fail to hold. Moreover, the bounds derived can be directly applied to non-Bayesian mixture approaches such as Bagging and Boosting.

Large margin linear classification methods have been successfully applied to many applications. For a linearly separable problem, it is known that under appropriate assumptions, the expected misclassification error of the computed "optimal hyperplane" approaches zero at a rate proportional to the inverse training sample size. This rate is usually characterized by the margin and the maximum norm of the input data. In this paper, we argue that another quantity, namely the robustness of the input data distribution, also plays an important role in characterizing the convergence behavior of expected misclassification error. Based on this concept of robustness, we show that for a large margin separable linear classification problem, the expected misclassification error may converge exponentially in the number of training sample size.