Multi-task regression that works without strong spectral assumptions

Multi-task learning assumes most tasks share similar parameters while some are outliers. Prior work required each task's second moment matrix to have eigenvalues bounded away from zero—an assumption that breaks down in high dimensions. This work proposes a matrix-weighted regularization estimator that instead uses a relative balancedness condition comparing each task's geometry to the average inlier geometry. On well-balanced problems, the method achieves minimax-optimal prediction error rates. Critically, when balancedness is poor or tasks are unrelated, the estimator safely reverts to independent learning, ensuring no performance loss from attempted transfer.