Restarts Subject to Approximate Sharpness: A Parameter-Free and Optimal Scheme For First-Order Methods

IF 2.5 1区 数学 Q2 COMPUTER SCIENCE, THEORY & METHODS
Ben Adcock, Matthew J. Colbrook, Maksym Neyra-Nesterenko
{"title":"Restarts Subject to Approximate Sharpness: A Parameter-Free and Optimal Scheme For First-Order Methods","authors":"Ben Adcock, Matthew J. Colbrook, Maksym Neyra-Nesterenko","doi":"10.1007/s10208-024-09673-8","DOIUrl":null,"url":null,"abstract":"<p>Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through <i>restarts</i>. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of <i>approximate sharpness</i>, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"13 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10208-024-09673-8","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0

Abstract

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.

近似锐度下的重启:一阶方法的无参数最优方案
在连续优化中,锐度几乎是一个通用的假设,它通过目标函数的次优性来限定到最小值的距离。它通过重新启动促进了一阶方法的加速。然而,锐度涉及特定于问题的常量,这些常量通常是未知的,并且重启方案通常会降低收敛速度。此外,这些方案在存在噪声或近似模型类的情况下(例如,在压缩成像或学习问题中)具有挑战性,并且它们通常假设所使用的一阶方法产生可行的迭代。我们考虑近似锐度的假设,这是一种包含未知常数扰动的目标函数误差的锐度概化。这个常数为寻找近似最小值提供了更大的鲁棒性(例如,关于噪声或模型类的松弛)。通过对未知常数的一种新型搜索,我们设计了一种适用于一般一阶方法的重启方案,并且不需要一阶方法产生可行的迭代。我们的方案与常数已知时保持相同的收敛速度。我们为各种一阶方法获得的收敛率与最优速率相匹配,或者在先前建立的速率基础上改进,适用于广泛的问题。我们在几个例子中展示了我们的重启方案,并强调了我们的框架和理论的潜在未来应用和发展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Foundations of Computational Mathematics
Foundations of Computational Mathematics 数学-计算机:理论方法
CiteScore
6.90
自引率
3.30%
发文量
46
审稿时长
>12 weeks
期刊介绍: Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer. With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles. The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信