新方法让多任务线性回归在高维下更稳健，无需强假设
Multi-task Linear Regression without Eigenvalue Lower Bounds: Adaptivity, Robustness, and Safety
- 用矩阵加权正则化处理任务间参数相似性，放宽了对协方差矩阵的谱条件要求
- 在中等平衡性条件下，预测均方误差达到最优率，且对异常任务有良好鲁棒性
- 即使任务无关或平衡性差，也不会比独立建模更差，适合高维、含噪声场景
研究存在污染任务的多任务线性回归问题。当多数任务参数在ℓ₂范数下相近，而部分任务为任意异常值时，现有理论依赖于每个任务的经验二阶矩最小特征值远离零（Ω(1)），但在高维场景常不成立，导致已有结果失效。为此，提出基于矩阵加权范数正则化的估计器，并引入由平衡常数量化的关系平衡性条件，通过比较各任务二阶矩与整体内点几何结构，弱化了任务级二阶矩下界的要求。在中等平衡性条件下，预测均方误差（MSE）界与Duan and Wang (2023) 的最优率相当，且在满足该条件时总体任务均方误差达到最小极大最优（对数因子内）。此外，证明了该估计器具备安全保证：当平衡常数较大或无穷，或任务无关时，其性能不低于独立任务学习。
原文摘要 · Abstract (English)
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.
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