Risk-Averse Classification
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We develop a new approach to solving classification problems, in which the labeled training data is viewed as random samples from populations with unknown distributions and we base our analysis on the theory of coherent measures of risk and risk sharing. The proposed approach aims at designing a risk-averse classifier. We stipulate that misclassification in different classes is associated with different risk. Therefore, we employ non-linear (in probability) risk functionals specific to each class. We analyze the structure of the new classifier design problem and establish its theoretical relation to the risk-neutral design problem. In particular, we show that the risk-sharing classification problem is equivalent to an implicitly defined optimization problem with unequal, implicitly defined but unknown weights for each data point. We implement our methodology in a binary classification scenario on several different data sets and carry out numerical comparison with classifiers which are obtained using the Huber loss function and other popular loss functions. In these applications, we use linear support vector machines in order to demonstrate the viability of our method.
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