Convex optimization based on global lower second-order models

In this paper, we present new second-order algorithms for composite convex optimization, called Contracting-domain Newton methods. These algorithms are affine-invariant and based on global second-order lower approximation for the smooth component of the objective. Our approach has an interpretation both as a second-order generalization of the conditional gradient method, or as a variant of trust-region scheme. Under the assumption, that the problem domain is bounded, we prove O(1/k2) global rate of convergence in functional residual, where k is the iteration counter, minimizing convex functions with Lipschitz continuous Hessian. This significantly improves the previously known bound O(1/k) for this type of algorithms. Additionally, we propose a stochastic extension of our method, and present computational results for solving empirical risk minimization problem.