Title: Learning to Abstain in the Presence of Uninformative Data. Abstract: Learning and decision making in domains with naturally high noise-to-signal ratios – such as Finance or Public Health – can be challenging and yet extremely important. In this paper, we study a problem of learning on datasets in which a significant proportion of samples does not contain useful information. To analyze this setting, we introduce a noisy generative process with a clear distinction between uninformative/not learnable/purely random data and a structured/informative component. This dichotomy is present both during the training and in the inference phase. We propose a novel approach to learn under these conditions via a loss inspired by the selective learning theory. By minimizing the loss, our method is guaranteed to make a near-optimal decision by simultaneously distinguishing structured data from the non-learnable and making predictions, even in a highly imbalanced setting. 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 conditions.