Bounds for Linear Multi-Task Learning

We give dimension-free and data-dependent bounds for linear multi-task learning where a common linear operator is chosen to preprocess data for a vector of task specific linear-thresholding classifiers. The complexity penalty of multi-task learning is bounded by a simple expression involving the margins of the task-specific classifiers, the Hilbert-Schmidt norm of the selected preprocessor and the Hilbert-Schmidt norm of the covariance operator for the total mixture of all task distributions, or, alternatively, the Frobenius norm of the total Gramian matrix for the data-dependent version. The results can be compared to state-of-the-art results on linear single-task learning.