Statistical Query Lower Bounds for List-Decodable Linear Regression

Ilias Diakonikolas, Daniel Kane, Ankit Pensia, Thanasis Pittas, Alistair Stewart

Keywords: learning theory, high-dimensional robust statistics, statistical query model, list-decodable learning, linear regression

TL;DR: We prove a superpolynomial statistical query lower bound for the problem of learning the regression vector of a Gaussian linear model when outliers constitute the majority of the dataset.

Abstract: 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. samples from a linear regression model with Gaussian covariates, and the remaining $(1-\alpha)$-fraction of the points are drawn from an arbitrary noise distribution. 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.

Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.