Efficient List-Decodable Regression using Batches

We begin the study of list-decodable linear regression using batches. In this setting only an $\alpha \in (0,1]$ fraction of the batches are genuine. Each genuine batch contains $\ge n$ i.i.d. samples from a common unknown distribution and the remaining batches may contain arbitrary or even adversarial samples. We derive a polynomial time algorithm that for any $n\ge \tilde \Omega(1/\alpha)$ returns a list of size $\mathcal O(1/\alpha^2)$ such that one of the items in the list is close to the true regression parameter. The algorithm requires only $\tilde{\mathcal{O}}(d/\alpha^2)$ genuine batches and works under fairly general assumptions on the distribution. The results demonstrate the utility of batch structure, which allows for the first polynomial time algorithm for list-decodable regression, which may be impossible for the non-batch setting, as suggested by a recent SQ lower bound \cite{diakonikolas2021statistical} for the non-batch setting.