  Multi-task learning aims to boost the generalization performance of multiple
related tasks simultaneously by leveraging information contained in those
tasks. In this paper, we propose a multi-task learning framework, where we
utilize prior knowledge about the relations between features. We also impose a
penalty on the coefficients changing for each specific feature to ensure
related tasks have similar coefficients on common features shared among them.
In addition, we capture a common set of features via group sparsity. The
objective is formulated as a non-smooth convex optimization problem, which can
be solved with various methods, including gradient descent method with fixed
stepsize, iterative shrinkage-thresholding algorithm (ISTA) with back-tracking,
and its variation -- fast iterative shrinkage-thresholding algorithm (FISTA).
In light of the sub-linear convergence rate of the methods aforementioned, we
propose an asymptotically linear convergent algorithm with theoretical
guarantee. Empirical experiments on both regression and classification tasks
with real-world datasets demonstrate that our proposed algorithms are capable
of improving the generalization performance of multiple related tasks.
