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The lack of fine-grained annotations hinders the deployment of automated diagnosis systems, which require human-interpretable justification for their decision process. In this paper, we address the problem of weakly supervised identification and localization of abnormalities in chest radiographs. To that end, we introduce a novel loss function for training convolutional neural networks increasing the \emph{localization confidence} and assisting the overall \emph{disease identification}. The loss leverages both image- and patch-level predictions to generate auxiliary supervision. Rather than forming strictly binary from the predictions as done in previous loss formulations, we create targets in a more customized manner, which allows the loss to account for possible misclassification. We show that the supervision provided within the proposed learning scheme leads to better performance and more precise predictions on prevalent datasets for multiple-instance learning as well as on the NIH~ChestX-Ray14 benchmark for disease recognition than previously used losses.