Learning from Similar Linear Representations: Adaptivity, Minimaxity, and Robustness

Representation multi-task learning (MTL) has achieved tremendous success in practice. However, the theoretical understanding of these methods is still lacking. Most existing theoretical works focus on cases where all tasks share the same representation, and claim that MTL almost always improves performance. Nevertheless, as the number of tasks grows, assuming all tasks share the same representation is unrealistic. Furthermore, empirical findings often indicate that a shared representation does not necessarily improve single-task learning performance. In this paper, we aim to understand how to learn from tasks with textitsimilar but not exactly the same linear representations, while dealing with outlier tasks. Assuming a known intrinsic dimension, we propose a penalized empirical risk minimization method and a spectral method that are textitadaptive to the similarity structure and textitrobust to outlier tasks. Both algorithms outperform single-task learning when representations across tasks are sufficiently similar and the proportion of outlier tasks is small. Moreover, they always perform at least as well as single-task learning, even when the representations are dissimilar. We provide information-theoretic lower bounds to demonstrate that both methods are nearly textitminimax optimal in a large regime, with the spectral method being optimal in the absence of outlier tasks. Additionally, we introduce a thresholding algorithm to adapt to an unknown intrinsic dimension. We conduct extensive numerical experiments to validate our theoretical findings.