A scalable trust-region algorithm with application to mixed-norm regression

We present a new algorithm for minimizing a convex loss-function subject to regularization. Our framework applies to numerous problems in machine learning and statistics; notably, for sparsity-promoting regularizers such as l ₁ or l_1,∞,OQ norms, it enables efficient computation of sparse solutions. Our approach is based on the trust-region framework with nonsmooth objectives, which allows us to build on known results to provide convergence analysis. We avoid the computational overheads associated with the conventional Hessian approximation used by trust-region methods by instead using a simple separable quadratic approximation. This approximation also enables use of proximity operators for tackling nonsmooth regularizers. We illustrate the versatility of our resulting algorithm by specializing it to three mixed-norm regression problems: group lasso [36], group logistic regression [21], and multi-task lasso [19]. We experiment with both synthetic and real-world large-scale data-our method is seen to be competitive, robust, and scalable.