Efficient first-order methods for convex minimization: a constructive approach

• 成果类型: Article
• 署名作者: Drori, Yoel; Taylor, Adrien B.
• 署名单位: Alphabet Inc.; Google Incorporated; Centre National de la Recherche Scientifique (CNRS); Universite PSL; Ecole Normale Superieure (ENS); Inria
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01410-2
• 发表日期: 2020
• 页码: 183-220
• 关键词: worst-case performance; optimization algorithms; gradient-method

摘要

We describe a novel constructive technique for devising efficient first-order methods for a wide range of large-scale convex minimization settings, including smooth, non-smooth, and strongly convex minimization. The technique builds upon a certain variant of the conjugate gradient method to construct a family of methods such that (a) all methods in the family share the same worst-case guarantee as the base conjugate gradient method, and (b) the family includes a fixed-step first-order method. We demonstrate the effectiveness of the approach by deriving optimal methods for the smooth and non-smooth cases, including new methods that forego knowledge of the problem parameters at the cost of a one-dimensional line search per iteration, and a universal method for the union of these classes that requires a three-dimensional search per iteration. In the strongly convex case, we show how numerical tools can be used to perform the construction, and show that the resulting method offers an improved worst-case bound compared to Nesterov's celebrated fast gradient method.