# Multi-task Linear Regression without Eigenvalue Lower Bounds: Adaptivity, Robustness, and Safety

arXiv: https://arxiv.org/abs/2605.17126 · HTML: https://mlsanity.com/paper/2605.17126/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 34 references that are in the map.
- First posted on arXiv 16 May 2026.

## Builds on (most cited first, 1 of 1)

- [Late Fusion Multi-task Learning for Semiparametric Inference with Nuisance Parameters](https://mlsanity.com/paper/2507.07941/) (2025)

## Built on by (most cited first, 1 of 1)

- [Contaminated Multi-task Learning with Heterogeneity: Fundamental Limits and Optimal Algorithms](https://mlsanity.com/paper/2607.02681/) (2026)

## Abstract (arXiv)

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 ℓ2-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 Omega(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.