List-decodable Linear Regression

We give the first polynomial-time algorithm for robust regression in the list-decodable setting where an adversary can corrupt a greater than 1/2 fraction of examples. For any \alpha < 1, our algorithm takes as input a sample {(xi,yi)}{i \leq n} of n linear equations where \alpha n of the equations satisfy yi = \langle x_i,\ell^\rangle +\zeta for some small noise \zeta and (1-\alpha) n of the equations are {\em arbitrarily} chosen. It outputs a list L of size O(1/\alpha) - a fixed constant - that contains an \ell that is close to \ell^. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. In particular, this gives a (d/\alpha)^{O(1/\alpha^8)} time algorithm to find a O(1/\alpha) size list when the inlier distribution is a standard Gaussian. For discrete product distributions that are anti-concentrated only in regular directions, we give an algorithm that achieves similar guarantee under the promise that \ell^* has all coordinates of the same magnitude. To complement our result, we prove that the anti-concentration assumption on the inliers is information-theoretically necessary. To solve the problem we introduce a new framework for list-decodable learning that strengthens the ``identifiability to algorithms'' paradigm based on the sum-of-squares method.