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 ${(x_i,y_i)}_{i leq n}$ of $n$ linear equations where $alpha n$ of the equations satisfy $y_i = langle x_i,ell^*rangle +zeta$ 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 emph{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 standard Gaussian. For discrete product distributions that are anti-concentrated only in emph{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. Our algorithm is based on a new framework for list-decodable learning that strengthens the `identifiability to algorithms paradigm based on the sum-of-squares method. In an independent and concurrent work, Raghavendra and Yau also used the Sum-of-Squares method to give a similar result for list-decodable regression.