Oracle complexity of second-order methods for smooth convex optimization

• 成果类型: Article
• 署名作者: Arjevani, Yossi; Shamir, Ohad; Shiff, Ron
• 署名单位: Weizmann Institute of Science
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1293-1
• 发表日期: 2019
• 页码: 327-360
• 关键词: cubic regularization; newtons

摘要

Second-order methods, which utilize gradients as well as Hessians to optimize 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 complexity is much better understood. We also provide generalizations of our results to higher-order methods.