Bootstrap-Based Inference for Cube Root Asymptotics

This paper proposes a valid bootstrap-based distributional approximation for M-estimators exhibiting a Chernoff (1964)-type limiting distribution. For estimators of this kind, the standard nonparametric bootstrap is inconsistent. The method proposed herein is based on the nonparametric bootstrap, but restores consistency by altering the shape of the criterion function defining the estimator whose distribution we seek to approximate. This modification leads to a generic and easy-to-implement resampling method for inference that is conceptually distinct from other available distributional approximations. We illustrate the applicability of our results with four examples in econometrics and machine learning

Active Sampling for Class Probability Estimation and Ranking

In many cost-sensitive environments class probability estimates are used by decision makers to evaluate the expected utility from a set of alternatives. Supervised learning can be used to build class probability estimates; however, it often is very costly to obtain training data with class labels. Active sampling acquires data incrementally, at each phase identifying especially useful additional data for labeling, and can be used to economize on examples needed for learning. We outline the critical features for an active sampling approach and present an active sampling method for estimating class probabilities and ranking. BOOTSTRAP-LV identifies particularly informative new data for learning based on the variance in probability estimates, and by accounting for a particular data item's informative value for the rest of the input space. We show empirically that the method reduces the number of data items that must be obtained and labeled, across a wide variety of domains. We investigate the contribution of the components of the algorithm and show that each provides valuable information to help identify informative examples. We also compare BOOTSTRAP-LV with UNCERTAINTY SAMPLING,a n existing active sampling method designed to maximize classification accuracy. The results show that BOOTSTRAP-LV uses fewer examples to exhibit a certain class probability estimation accuracy and provide insights on the behavior of the algorithms. Finally, to further our understanding of the contributions made by the elements of BOOTSTRAP-LV, we experiment with a new active sampling algorithm drawing from both UNCERTAINIY SAMPLING and BOOTSTRAP-LV and show that it is significantly more competitive with BOOTSTRAP-LV compared to UNCERTAINTY SAMPLING. The analysis suggests more general implications for improving existing active sampling algorithms for classification.Information Systems Working Papers Serie

The Theory Behind Overfitting, Cross Validation, Regularization, Bagging, and Boosting: Tutorial

In this tutorial paper, we first define mean squared error, variance, covariance, and bias of both random variables and classification/predictor models. Then, we formulate the true and generalization errors of the model for both training and validation/test instances where we make use of the Stein's Unbiased Risk Estimator (SURE). We define overfitting, underfitting, and generalization using the obtained true and generalization errors. We introduce cross validation and two well-known examples which are $K$-fold and leave-one-out cross validations. We briefly introduce generalized cross validation and then move on to regularization where we use the SURE again. We work on both $\ell_2$ and $\ell_1$ norm regularizations. Then, we show that bootstrap aggregating (bagging) reduces the variance of estimation. Boosting, specifically AdaBoost, is introduced and it is explained as both an additive model and a maximum margin model, i.e., Support Vector Machine (SVM). The upper bound on the generalization error of boosting is also provided to show why boosting prevents from overfitting. As examples of regularization, the theory of ridge and lasso regressions, weight decay, noise injection to input/weights, and early stopping are explained. Random forest, dropout, histogram of oriented gradients, and single shot multi-box detector are explained as examples of bagging in machine learning and computer vision. Finally, boosting tree and SVM models are mentioned as examples of boosting.Comment: 23 pages, 9 figure

A bagging SVM to learn from positive and unlabeled examples

We consider the problem of learning a binary classifier from a training set of positive and unlabeled examples, both in the inductive and in the transductive setting. This problem, often referred to as \emph{PU learning}, differs from the standard supervised classification problem by the lack of negative examples in the training set. It corresponds to an ubiquitous situation in many applications such as information retrieval or gene ranking, when we have identified a set of data of interest sharing a particular property, and we wish to automatically retrieve additional data sharing the same property among a large and easily available pool of unlabeled data. We propose a conceptually simple method, akin to bagging, to approach both inductive and transductive PU learning problems, by converting them into series of supervised binary classification problems discriminating the known positive examples from random subsamples of the unlabeled set. We empirically demonstrate the relevance of the method on simulated and real data, where it performs at least as well as existing methods while being faster