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Convex optimization based on global lower second-order models
Nikita Doikov · Yurii Nesterov

Wed Dec 09 06:45 AM -- 07:00 AM (PST) @ Orals & Spotlights: Optimization
In this work, 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 $O(1/k^2)$ 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.

Author Information

Nikita Doikov (Catholic University of Louvain)
Yurii Nesterov (Catholic University of Louvain (UCL))

Yurii Nesterov is a professor at Center for Operations Research and Econometrics (CORE) in Catholic University of Louvain (UCL), Belgium. He received Ph.D. degree (Applied Mathematics) in 1984 at Institute of Control Sciences, Moscow. Starting from 1993 he works at CORE. His research interests are related to complexity issues and efficient methods for solving various optimization problems. The main results are obtained in Convex Optimization (optimal methods for smooth problems, polynomial-time interior-point methods, smoothing technique for structural optimization, cubic regularization of Newton method, optimization methods for huge-scale problems). He is an author of 4 monographs and more than 80 refereed papers in the leading optimization journals. He got the Dantzig Prize from SIAM and Mathematical Programming society in 2000, von Neumann Theory Prize from INFORMS in 2009, and SIAM Outstanding paper award in 2014.

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