On efficient robust regression with subquadratic samples

• URL: http://arxiv.org/abs/2605.18042v1
• Date: Mon, 18 May 2026 08:34:07 GMT
• Title: On efficient robust regression with subquadratic samples
• Abstract summary: We show that a near-linear-time algorithm that uses $widetildeO(d/4)$ samples achieves prediction error $O(sqrt)$ under the condition $lesssim 1$.<n>We also prove a low-degree lower bound that gives fine-grained evidence that, without assumptions such as $lesssim 1$, efficient algorithms may require $tildeleft(mind22, 2d2right)$ samples.
• Score: 2.017649213266693
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We revisit the problem of robust linear regression under Gaussian covariates with an unknown covariance matrix of condition number $κ$. For this fundamental problem, significant gaps remain in our understanding of the trade-offs among sample complexity, condition number, runtime, and prediction error for efficient algorithms. Our first result is a near-linear-time algorithm that uses $\widetilde{O}(d/ε^4)$ samples, where $d$ is the dimension and $ε$ is the corruption rate, and achieves prediction error $O(\sqrt{εκ})$ under the condition $εκ\lesssim 1$, improving over all prior works. We complement this result with a Statistical Query (SQ) lower bound showing that efficient SQ algorithms achieving error $o(\sqrt{εκ})$ when $εκ\lesssim 1$ require queries that take $Ω(d^2)$ samples to simulate. Finally, we prove a low-degree polynomial lower bound that gives fine-grained evidence that, without assumptions such as $εκ\lesssim 1$, efficient algorithms may require $\tildeΩ\left(\min\{dε^{2}κ^{2},\ ε^{2}d^{2}\}\right)$ samples to significantly outperform the trivial estimator that always guesses $0$.

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