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

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & & Analytics, Mathematics & Computational Sciences · Depth: Expert, long

Summary

This research addresses the multi-task linear regression problem, particularly when some tasks are contaminated outliers and the unknown parameters of a majority of tasks are close in $\ell_{2}$-norm. Existing theoretical frameworks for this problem typically assume that the empirical second moment of each task has a minimum eigenvalue bounded away from zero (order $\Omega(1)$), an assumption that often fails in high-dimensional scenarios. To overcome this, the authors propose a novel estimator based on matrix-weighted norm regularization. They introduce a "balancedness condition," quantified by a constant $B$, which compares each task's second moment with the average inlier geometry, thereby relaxing the need for task-wise second-moment lower bounds. In favorable regimes with moderate balancedness, their prediction MSE bounds match the rate of Duan and Wang (2023) under significantly weaker spectral assumptions, achieving minimax optimality up to logarithmic factors. The estimator also includes a safety guarantee, ensuring performance no worse than independent task learning even when balancedness is poor or tasks are unrelated.

Key takeaway

Research Scientists working on multi-task learning with potentially ill-conditioned or high-dimensional data should consider adopting matrix-weighted regularization. This approach offers robust and adaptive performance, even with outliers, and crucially, provides a safety net by performing no worse than independent task learning when transfer is unhelpful. You can achieve minimax optimal MSE bounds in favorable conditions without relying on restrictive eigenvalue lower bounds.

Key insights

A new multi-task linear regression estimator offers adaptivity, robustness, and safety without strong eigenvalue assumptions.

Principles

• Matrix-weighted regularization adapts to task geometry.
• Balancedness quantifies transfer favorability.
• Safety ensures performance no worse than independent learning.

Method

The proposed method minimizes a matrix-weighted regularized objective function, penalizing deviations from a shared centroid parameter in prediction space rather than raw parameter space, using a regularization parameter scaled as $\lambda_{j}\asymp\sqrt{d/n_{j}}$.

In practice

• Use matrix-weighted regularization for ill-conditioned data.
• Evaluate transfer geometry with the $B_{\mathrm{emp}}$ diagnostic.
• Consider GLMs with this framework for diverse applications.

Topics

• Multi-task Linear Regression
• Matrix-Weighted Regularization
• Balancedness Condition
• Outlier Robustness
• Adaptive Learning

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.