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.

翻译：我们研究在存在污染任务情况下的多任务线性回归问题。考虑设定：大部分任务的未知参数在ℓ₂范数意义下彼此接近，而部分任务为任意异常值。现有针对该问题的理论框架严重依赖每个任务经验二阶矩的最小特征值远离零（阶数为Ω(1)）这一假设。关键问题在于，该假设在许多高维场景下无法成立，导致先前结论无效。为克服这一限制，我们提出基于矩阵加权范数正则化的估计量。同时引入相对平衡性条件（通过平衡性常数量化），该条件将每个任务的二阶矩与平均内点几何结构进行比较，从而放宽对任务级二阶矩下界的要求。在适度平衡性的有利情境下，我们的预测均方误差界在显著更弱的谱假设下达到Duan与Wang（2023）的速率；由此得到的任务整体均方误差在对数因子意义下达到极小极大最优。进一步证明，该估计量具有安全性保证：当相关平衡性常数较大或无穷大，或任务之间无关联时，其性能不劣于独立任务学习。