Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination
Year: 2025
Type: preprint
Abstract: We study the task of noiseless linear regression under Gaussian covariates in the presence of additive oblivious contamination. Specifically, we are given i.i.d.\ samples from a distribution $(x, y)$ on $\mathbb{R}^d \times \mathbb{R}$ with $x \sim \mathcal{N}(0,\mathbf{I}_d)$ and $y = x^\top β+ z$, where $z$ is drawn independently of $x$ from an unknown distribution $E$. Moreover, $z$ satisfies $\mathbb{P}_E[z = 0] = α>0$. The goal is to accurately recover the regressor $β$ to small $\ell_2$-error. Ignoring computational considerations, this problem is known to be solvable using $O(d/α)$ samples. On the other hand, the best known polynomial-time algorithms require $Ω(d/α^2)$ samples. Here we provide formal evidence that the quadratic dependence in $1/α$ is inherent for efficient algorithms. Specifically, we show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $\tildeΩ(d^{1/2}/α^2)$.
Source: arXiv (Cornell University)
Authors: Diakonikolas, Ilias, Gao, Chao, Kane, Daniel M., Lafferty, John, Pensia, Ankit
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7091188154
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