Effects of sampling skewness of the importance-weighted risk estimator on model selection

Importance-weighting is a popular and well-researched technique for dealing with sample selection bias and covariate shift. It has desirable characteristics such as unbiasedness, consistency and low computational complexity. However, weighting can have a detrimental effect on an estimator as well. In this work, we empirically show that the sampling distribution of an importance-weighted estimator can be skewed. For sample selection bias settings, and for small sample sizes, the importance-weighted risk estimator produces overestimates for datasets in the body of the sampling distribution, i.e. the majority of cases, and large underestimates for data sets in the tail of the sampling distribution. These over- and underestimates of the risk lead to suboptimal regularization parameters when used for importance-weighted validation.Comment: Conference paper, 6 pages, 5 figure

On Regularization Parameter Estimation under Covariate Shift

This paper identifies a problem with the usual procedure for L2-regularization parameter estimation in a domain adaptation setting. In such a setting, there are differences between the distributions generating the training data (source domain) and the test data (target domain). The usual cross-validation procedure requires validation data, which can not be obtained from the unlabeled target data. The problem is that if one decides to use source validation data, the regularization parameter is underestimated. One possible solution is to scale the source validation data through importance weighting, but we show that this correction is not sufficient. We conclude the paper with an empirical analysis of the effect of several importance weight estimators on the estimation of the regularization parameter.Comment: 6 pages, 2 figures, 2 tables. Accepted to ICPR 201

The Law of Total Odds

The law of total probability may be deployed in binary classification exercises to estimate the unconditional class probabilities if the class proportions in the training set are not representative of the population class proportions. We argue that this is not a conceptually sound approach and suggest an alternative based on the new law of total odds. We quantify the bias of the total probability estimator of the unconditional class probabilities and show that the total odds estimator is unbiased. The sample version of the total odds estimator is shown to coincide with a maximum-likelihood estimator known from the literature. The law of total odds can also be used for transforming the conditional class probabilities if independent estimates of the unconditional class probabilities of the population are available. Keywords: Total probability, likelihood ratio, Bayes' formula, binary classification, relative odds, unbiased estimator, supervised learning, dataset shift.Comment: 12 pages, 1 figure, new reference

Discriminative Density-ratio Estimation

The covariate shift is a challenging problem in supervised learning that results from the discrepancy between the training and test distributions. An effective approach which recently drew a considerable attention in the research community is to reweight the training samples to minimize that discrepancy. In specific, many methods are based on developing Density-ratio (DR) estimation techniques that apply to both regression and classification problems. Although these methods work well for regression problems, their performance on classification problems is not satisfactory. This is due to a key observation that these methods focus on matching the sample marginal distributions without paying attention to preserving the separation between classes in the reweighted space. In this paper, we propose a novel method for Discriminative Density-ratio (DDR) estimation that addresses the aforementioned problem and aims at estimating the density-ratio of joint distributions in a class-wise manner. The proposed algorithm is an iterative procedure that alternates between estimating the class information for the test data and estimating new density ratio for each class. To incorporate the estimated class information of the test data, a soft matching technique is proposed. In addition, we employ an effective criterion which adopts mutual information as an indicator to stop the iterative procedure while resulting in a decision boundary that lies in a sparse region. Experiments on synthetic and benchmark datasets demonstrate the superiority of the proposed method in terms of both accuracy and robustness