We study the problem of fair binary classification using the notion of Equal Opportunity. It requires the true positive rate to distribute equally across the sensitive groups. Within this setting we show that the fair optimal classifier is obtained by recalibrating the Bayes classifier by a group-dependent threshold. We provide a constructive expression for the threshold. This result motivates us to devise a plug-in classification procedure based on both unlabeled and labeled datasets. While the latter is used to learn the output conditional probability, the former is used for calibration. The overall procedure can be computed in polynomial time and it is shown to be statistically consistent both in terms of classification error and fairness measure. Finally, we present numerical experiments which indicate that our method is often superior or competitive with the state-of-the-art methods on benchmark datasets.
More recently, classification problems with a distribution dependent threshold have emerged. Prominent examples include classification with non-decomposable measures @cite_24 @cite_36 @cite_0 , classification with reject option @cite_17 @cite_18 , and confidence set setup of multi-class classification @cite_34 @cite_8 @cite_26 , among others. A typical estimation algorithm in these scenarios is based on the plug-in strategy, which uses extra data to estimate the unknown threshold. Interestingly, in some setups a practitioner does not need to have access to two labeled samples and optimal estimation can be efficiently performed in semi-supervised manner @cite_34 @cite_26 .
@cite_18 @cite_26 @cite_8 @cite_36 @cite_24 @cite_0 @cite_34 @cite_17
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Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the output set while minimizing the classification risk. We show the statistical optimality of the procedure and establish rates of convergence under the Tsybakov margin condition. It turns out that these rates are linear on the number of labels. We apply our methodology to convex aggregation of confidence sets based on the V-fold cross validation principle also known as the superlearning principle. We illustrate the numerical performance of the procedure on real data and demonstrate in particular that with moderate expected size, w.r.t. the number of labels, the procedure provides significant improvement of the classification risk. ABSTRACTIn most classification tasks, there are observations that are ambiguous and therefore difficult to correctly label. Set-valued classifiers output sets of plausible labels rather than a single label, thereby giving a more appropriate and informative treatment to the labeling of ambiguous instances. We introduce a framework for multiclass set-valued classification, where the classifiers guarantee user-defined levels of coverage or confidence (the probability that the true label is contained in the set) while minimizing the ambiguity (the expected size of the output). We first derive oracle classifiers assuming the true distribution to be known. We show that the oracle classifiers are obtained from level sets of the functions that define the conditional probability of each class. Then we develop estimators with good asymptotic and finite sample properties. The proposed estimators build on existing single-label classifiers. The optimal classifier can sometimes output the empty set, but we provide two ... Multilabel classification is rapidly developing as an important aspect of modern predictive modeling, motivating study of its theoretical aspects. To this end, we propose a framework for constructing and analyzing multilabel classification metrics which reveals novel results on a parametric form for population optimal classifiers, and additional insight into the role of label correlations. In particular, we show that for multilabel metrics constructed as instance-, micro- and macro-averages, the population optimal classifier can be decomposed into binary classifiers based on the marginal instance-conditional distribution of each label, with a weak association between labels via the threshold. Thus, our analysis extends the state of the art from a few known multilabel classification metrics such as Hamming loss, to a general framework applicable to many of the classification metrics in common use. Based on the population-optimal classifier, we propose a computationally efficient and general-purpose plug-in classification algorithm, and prove its consistency with respect to the metric of interest. Empirical results on synthetic and benchmark datasets are supportive of our theoretical findings. Fano's inequality lower bounds the probability of transmission error through a communication channel. Applied to classification problems, it provides a lower bound on the Bayes error rate and motivates the widely used Infomax principle. In modern machine learning, we are often interested in more than just the error rate. In medical diagnosis, different errors incur different cost; hence, the overall risk is cost-sensitive. Two other popular criteria are balanced error rate (BER) and F-score. In this work, we focus on the two-class problem and use a general definition of conditional entropy (including Shannon's as a special case) to derive upper lower bounds on the optimal F-score, BER and cost-sensitive risk, extending Fano's result. As a consequence, we show that Infomax is not suitable for optimizing F-score or cost-sensitive risk, in that it can potentially lead to low F-score and high risk. For cost-sensitive risk, we propose a new conditional entropy formulation which avoids this inconsistency. In addition, we consider the common practice of using a threshold on the posterior probability to tune performance of a classifier. As is widely known, a threshold of 0.5, where the posteriors cross, minimizes error rate--we derive similar optimal thresholds for F-score and BER. In this work we study the semi-supervised framework of confidence set classification with controlled expected size in minimax settings. We obtain semi-supervised minimax rates of convergence under the margin assumption and a H o lder condition on the regression function. Besides, we show that if no further assumptions are made, there is no supervised method that outperforms the semi-supervised estimator proposed in this work. We establish that the best achievable rate for any supervised method is n^ --1 2 , even if the margin assumption is extremely favorable. On the contrary, semi-supervised estimators can achieve faster rates of convergence provided that sufficiently many unlabeled samples are available. We additionally perform numerical evaluation of the proposed algorithms empirically confirming our theoretical findings. Confident prediction is highly relevant in machine learning; for example, in applications such as medical diagnoses, wrong prediction can be fatal. For classification, there already exist procedures that allow to not classify data when the confidence in their prediction is weak. This approach is known as classification with reject option. In the present paper, we provide new methodology for this approach. Predicting a new instance via a confidence set, we ensure an exact control of the probability of classification. Moreover, we show that this methodology is easily implementable and entails attractive theoretical and numerical properties.