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  <title>arXiv Query: search_query=&amp;id_list=2605.17126&amp;start=0&amp;max_results=10</title>
  <updated>2026-07-21T22:53:23Z</updated>
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    <id>http://arxiv.org/abs/2605.17126v2</id>
    <title>Multi-task Linear Regression without Eigenvalue Lower Bounds: Adaptivity, Robustness, and Safety</title>
    <updated>2026-05-29T08:34:47Z</updated>
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    <summary>We study the multi-task linear regression problem in the presence of contaminated tasks. We address the setting where the unknown parameters of a majority of tasks are close in the $\ell_2$-norm, while a fraction of tasks are arbitrary outliers. Existing theoretical frameworks for this problem rely heavily on the assumption that the empirical second moment of each task has a minimum eigenvalue bounded away from zero (order $Ω(1)$). Crucially, this assumption fails in many high-dimensional scenarios, rendering prior guarantees vacuous. To overcome this limitation, we propose an estimator based on matrix-weighted norm regularization. We also introduce a relative balancedness condition, quantified by a balancedness constant, that compares each task's second moment with the average inlier geometry and relaxes the need for taskwise second-moment lower bounds. In favorable regimes with moderate balancedness, our prediction MSE bounds match the rate of Duan and Wang (2023) under substantially weaker spectral assumptions; the resulting task-overall MSE is minimax optimal up to logarithmic factors. Furthermore, we demonstrate that our estimator enjoys a safety guarantee: when the relevant balancedness constant is large or infinite, or when tasks are unrelated, the method performs no worse than independent task learning.</summary>
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    <published>2026-05-16T19:06:54Z</published>
    <arxiv:comment>Accepted at ICML 2026</arxiv:comment>
    <arxiv:primary_category term="stat.ML"/>
    <author>
      <name>Seok-Jin Kim</name>
    </author>
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