An Inexact Semismooth Newton-Based Augmented Lagrangian Algorithm for Multi-Task Lasso Problems

This paper is concerned with the ℓ1,∞-norm ball constrained multi-task learning problem, which has received extensive attention in many research areas such as machine learning, cognitive neuroscience, and signal processing. To address the challenges of solving large-scale multi-task Lasso problems, this paper develops an inexact semismooth Newton-based augmented Lagrangian (Ssnal) algorithm. When solving the inner problems in the Ssnal algorithm, the semismooth Newton (Ssn) algorithm with superlinear or even quadratic convergence is applied. Theoretically, this paper presents the global and asymptotically superlinear local convergence of the Ssnal algorithm under standard conditions. Computationally, we derive an efficient procedure to construct the generalized Jacobian of the projector onto ℓ1,∞-norm ball, which is an important component of the Ssnal algorithm, making the computational cost in the Ssn algorithm very cheap. Comprehensive numerical experiments on the multi-task Lasso problems demonstrate that the Ssnal algorithm is more efficient and robust than several existing state-of-the-art first-order algorithms.