Robust Polynomial Regression up to the Information Theoretic Limit

We consider the problem of robust polynomial regression, where one receives samples that are usually within a small additive error of a target polynomial, but have a chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate the target polynomial only when the outlier probability was subconstant in the degree of the target polynomial. We give an algorithm that works for the entire feasible range of outlier probabilities, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a 1.09 approximation is impossible even with infinitely many samples.