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The Frank--Wolfe method is a popular algorithm for solving large-scale convex optimization problems appearing in structured statistical learning. However, the traditional Frank--Wolfe method can only be applied when the feasible region is bounded, which limits its applicability in practice. Motivated by two applications in statistical learning, the l1 trend filtering problem and matrix optimization problems with generalized nuclear norm constraints, we study a family of convex optimization problems where the unbounded feasible region is the direct sum of an unbounded linear subspace and a bounded constraint set. We propose two new Frank--Wolfe methods, the unbounded Frank--Wolfe method and the unbounded away-step Frank--Wolfe method, for solving a family of convex optimization problems with this class of unbounded feasible regions. We show that under proper regularity conditions, the unbounded Frank--Wolfe method has an O(1/k) sublinear convergence rate, and the unbounded away-step Frank--Wolfe method has a linear convergence rate, matching the best-known results for the Frank--Wolfe method when the feasible region is bounded. Furthermore, computational experiments indicate that our proposed methods appear to outperform alternative solvers. [ABSTRACT FROM AUTHOR]

The Gradient Boosting Machine (GBM) introduced by Friedman [J. H. Friedman, Ann. Statist., 29 (2001), pp. 1189-1232] is a powerful supervised learning algorithm that is very widely used in practice--it routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In spite of the usefulness of GBM in practice, our current theoretical understanding of this method is rather limited. In this work, we propose the Randomized Gradient Boosting Machine (RGBM), which leads to substantial computational gains compared to GBM by using a randomization scheme to reduce search in the space of weak learners. We derive novel computational guarantees for RGBM. We also provide a principled guideline towards better step-size selection in RGBM that does not require a line search. Our proposed framework is inspired by a special variant of coordinate descent that combines the benefits of randomized coordinate descent and greedy coordinate descent, and may be of independent interest as an optimization algorithm. As a special case, our results for RGBM lead to superior computational guarantees for GBM. Our computational guarantees depend upon a curious geometric quantity that we call the Minimal Cosine Angle, which relates to the density of weak learners in the prediction space. On a series of numerical experiments on real datasets, we demonstrate the effectiveness of RGBM over GBM in terms of obtaining a model with good training and/or testing data fidelity with a fraction of the computational cost. [ABSTRACT FROM AUTHOR]