Statistical Query Lower Bounds for List-Decodable Linear Regression

We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set T of labeled examples (x, y) \in \mathbb{R} d \times \mathbb{R} and a parameter 0 \alpha 1/2 such that an \alpha -fraction of the points in T are i.i.d. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our main result is a Statistical Query (SQ) lower bound of d {\mathrm{poly}(1/\alpha)} for this problem. Our SQ lower bound qualitatively matches the performance of previously developed algorithms, providing evidence that current upper bounds for this task are nearly best possible.