Abstract Classifiers used in the wild, in particular for safetycritical systems, should not only have good generalization properties but also should know when they don’t know, in particular make low confidence predictions far away from the training data. We show that ReLU type neural networks which yield a piecewise linear classi- fier function fail in this regard as they produce almost always high confidence predictions far away from the training data. For bounded domains like images we propose a new robust optimization technique similar to adversarial training which enforces low confidence predictions far away from the training data. We show that this technique is surprisingly effective in reducing the confidence of predictions far away from the training data while maintaining high confidence predictions and test error on the original classification task compared to standard training.