Oracle complexity of second-order methods for smooth convex optimization

Oracle complexity of second-order methods for smooth convex optimization Math. Program., Ser. A https://doi.org/10.1007/s10107-018-1293-1 FULL LENGTH PAPER Oracle complexity of second-order methods for smooth convex optimization 1 1 1 Yossi Arjevani · Ohad Shamir · Ron Shiff Received: 12 September 2017 / Accepted: 7 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract Second-order methods, which utilize gradients as well as Hessians to opti- mize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth convex functions, or equivalently, the worst-case number of iterations required to optimize such functions to a given accuracy. In particular, these bounds indicate when such methods can or cannot improve on gradient-based methods, whose oracle com- plexity is much better understood. We also provide generalizations of our results to higher-order methods. Keywords Smooth convex optimization · Oracle complexity Mathematics Subject Classification 90C25 · 65K05 · 49M37 1 Introduction We consider an unconstrained optimization problem of the form min f (w), (1) w∈R Ron Shiff ron.shiff1@gmail.com Yossi Arjevani yossi.arjevani@weizmann.ac.il Ohad Shamir ohad.shamir@weizmann.ac.il Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel 123 Y. Arjevani et al. where f is a http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

Oracle complexity of second-order methods for smooth convex optimization

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-018-1293-1
Publisher site
See Article on Publisher Site

Abstract

Math. Program., Ser. A https://doi.org/10.1007/s10107-018-1293-1 FULL LENGTH PAPER Oracle complexity of second-order methods for smooth convex optimization 1 1 1 Yossi Arjevani · Ohad Shamir · Ron Shiff Received: 12 September 2017 / Accepted: 7 May 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract Second-order methods, which utilize gradients as well as Hessians to opti- mize a given function, are of major importance in mathematical optimization. In this work, we prove tight bounds on the oracle complexity of such methods for smooth convex functions, or equivalently, the worst-case number of iterations required to optimize such functions to a given accuracy. In particular, these bounds indicate when such methods can or cannot improve on gradient-based methods, whose oracle com- plexity is much better understood. We also provide generalizations of our results to higher-order methods. Keywords Smooth convex optimization · Oracle complexity Mathematics Subject Classification 90C25 · 65K05 · 49M37 1 Introduction We consider an unconstrained optimization problem of the form min f (w), (1) w∈R Ron Shiff ron.shiff1@gmail.com Yossi Arjevani yossi.arjevani@weizmann.ac.il Ohad Shamir ohad.shamir@weizmann.ac.il Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel 123 Y. Arjevani et al. where f is a

Journal

Mathematical ProgrammingSpringer Journals

Published: May 28, 2018

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