List-decodeable 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 a < 1, our algorithm takes as input a sample {(x_i, y_i)}_i=_n of n linear equations where an of the equations satisfy yi = hx_i, l^*i+ ? for some small noise ? and (1 - a)n of the equations are arbitrarily chosen. It outputs a list L of size O(1/a) - a fixed constant - that contains an l that is close to l^*. Our algorithm succeeds whenever the inliers are chosen from a certifiably anti-concentrated distribution D. As a corollary of our algorithmic result, we obtain a (d/a)^O(1^/a8) time algorithm to find a O(1/a) size list when the inlier distribution is 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 l^* 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.