Learning from Untrusted Data

The vast majority of theoretical results in machine learning and statistics assume that the training data is a reliable reflection of the phenomena to be learned. Similarly, most learning techniques used in practice are brittle to the presence of large amounts of biased or malicious data. Motivated by this, we consider two frameworks for studying estimation, learning, and optimization in the presence of significant fractions of arbitrary data. The first framework, list-decodable learning, asks whether it is possible to return a list of answers such that at least one is accurate. For example, given a dataset of n points for which an unknown subset of $\alpha$n points are drawn from a distribution of interest, and no assumptions are made about the remaining (1 - $\alpha$)n points, is it possible to return a list of poly(1/$\alpha$) answers? The second framework, which we term the semi-verified model, asks whether a small dataset of trusted data (drawn from the distribution in question) can be used to extract accurate information from a much larger but untrusted dataset (of which only an $\alpha$-fraction is drawn from the distribution). We show strong positive results in both settings, and provide an algorithm for robust learning in a very general stochastic optimization setting. This result has immediate implications for robustly estimating the mean of distributions with bounded second moments, robustly learning mixtures of such distributions, and robustly finding planted partitions in random graphs in which significant portions of the graph have been perturbed by an adversary.