Calibrated multi-task learning

This paper proposes a novel algorithm, named Non-Convex Calibrated Multi-Task Learning (NC-CMTL), for learning multiple related regression tasks jointly. Instead of utilizing the nuclear norm, NC-CMTL adopts a non-convex low rank regularizer to explore the shared information among different tasks. In addition, considering that the regularization parameter for each regression task desponds on its noise level, we replace the least squares loss function by square-root loss function. Computationally, as proposed model has a non-smooth loss function and a non-convex regularization term, we construct an efficient re-weighted method to optimize it. Theoretically, we first present the convergence analysis of constructed method, and then prove that the derived solution is a stationary point of original problem. Particularly, the regularizer and optimization method used in this paper are also suitable for other rank minimization problems. Numerical experiments on both synthetic and real data illustrate the advantages of NC-CMTL over several state-of-the-art methods.