Title: Learning to Abstain from Uninformative Data. Abstract: Learning and decision making in domains with naturally high noise-to-signal ratios – such as Finance or Healthcare – can be challenging yet extremely important. In this paper, we study a problem of learning and decision making under a general noisy generative process. The  distribution has a significant proportion of uninformative data with high noise in label, while part of the data contains useful information represented by low label noise. This dichotomy is present during both training and inference, which requires the proper handling of uninformative data at testing time. We propose a novel approach to learn under these conditions via a loss inspired by the selective learning theory. By minimizing the loss, the model is guaranteed to make a near-optimal decision by distinguishing informative data from the uninformative data and making predictions.  We build upon the strength of our theoretical guarantees by describing an iterative algorithm, which jointly optimizes both a predictor and a selector, and evaluate its empirical performance under a variety of settings.