Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination

• URL: http://arxiv.org/abs/2510.10665v1
• Date: Sun, 12 Oct 2025 15:42:44 GMT
• Title: Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination
• Authors: Ilias Diakonikolas, Chao Gao, Daniel M. Kane, John Lafferty, Ankit Pensia,
• Abstract summary: We show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $tildeOmega(d1/2/alpha2)$.
• Score: 65.37519531362157
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $y = x^\top \beta + z$, where $z$ is drawn independently of $x$ from an unknown distribution $E$. Moreover, $z$ satisfies $\mathbb{P}_E[z = 0] = \alpha>0$. The goal is to accurately recover the regressor $\beta$ to small $\ell_2$-error. Ignoring computational considerations, this problem is known to be solvable using $O(d/\alpha)$ samples. On the other hand, the best known polynomial-time algorithms require $\Omega(d/\alpha^2)$ samples. Here we provide formal evidence that the quadratic dependence in $1/\alpha$ is inherent for efficient algorithms. Specifically, we show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $\tilde{\Omega}(d^{1/2}/\alpha^2)$.

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