Robust and Adaptive Spectral Method for Representation Multi-Task Learning with Contamination

Abstract: Representation-based multi-task learning (MTL) improves efficiency by learning a shared structure across tasks, but its practical application is often hindered by contamination, outliers, or adversarial tasks. Most existing methods and theories assume a clean or near-clean setting, failing when contamination is significant. This paper tackles representation MTL with an unknown and potentially large contamination proportion, while also allowing for heterogeneity among inlier tasks. We introduce a Robust and Adaptive Spectral method (RAS) that can distill the shared inlier representation effectively and efficiently, while requiring no prior knowledge of the contamination level or the true representation dimension. Theoretically, we provide non-asymptotic error bounds for both the learned representation and the per-task parameters. These bounds adapt to inlier task similarity and outlier structure, and guarantee that RAS performs at least as well as single-task learning, thus preventing negative transfer. We also extend our framework to transfer learning with corresponding theoretical guarantees for the target task. Extensive experiments confirm our theory, showcasing the robustness and adaptivity of RAS, and its superior performance in regimes with up to 80\% task contamination.