Multi-task Linear Regression without Eigenvalue Lower Bounds: Adaptivity, Robustness and Safety
Authors: Seok-Jin Kim
Summary
arXiv:2605. 17126v1 Announce Type: new Abstract: We study the multi-task linear regression problem in the presence of contaminated tasks.
Relevance
Read next because Multi-task Linear Regression without Eigenvalue Lower Bounds: Adaptivity, Robustness and Safety overlaps with clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)", clean result "The marker is a representational handle, not a behavioural one — sharing it between a villain persona and the assistant transfers no misalignment (HIGH confidence)", experiment "Implement Chen et al. persona-vector extraction recipe and compare to project's centroid-difference recipe". Matching terms: under, line, rate, compare, without, factor. Source: arxiv stat.ML (Machine Learning).
Threat model
Potential threat/caveat for clean result "Language-mismatch LoRA SFT on Qwen2.5-7B leaks the trained completion language into bystander directives the model was never trained on, absent under same-language SFT (LOW confidence)": this item discusses limitation, robustness.
Abstract
arXiv:2605.17126v1 Announce Type: new Abstract: 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 $\Omega(1)$). 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. Consequently, our methodology achieves simultaneous adaptivity to task similarity, robustness to outliers, and safety outside favorable transfer regimes.