The Complexity of Gradient Descent: CLS = PPAD ∩ PLS

IF 2.3 2区计算机科学 Q2 COMPUTER SCIENCE, HARDWARE & ARCHITECTURE

Journal of the ACM Pub Date : 2022-12-19 DOI:https://dl.acm.org/doi/10.1145/3568163

John Fearnley, Paul Goldberg, Alexandros Hollender, Rahul Savani

引用次数: 0

Abstract

We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain [0,1]² is PPAD ∩ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) – which was defined by Daskalakis and Papadimitriou as a more “natural” counterpart to PPAD ∩ PLS and contains many interesting problems – is itself equal to PPAD ∩ PLS.

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梯度下降的复杂度:CLS = PPAD∩PLS

我们研究了可以通过在有界凸多边形域上执行梯度下降来解决的搜索问题，并证明该类等于两个已知类:PPAD和PLS的交集。作为我们主要的基础技术贡献，我们证明了在域[0,1]2上计算连续可微函数的Karush-Kuhn-Tucker (KKT)点是PPAD∩PLS完备的。这是本课程第一个完整的非人工问题。我们的结果还意味着CLS(连续本地搜索)类——由Daskalakis和Papadimitriou定义为PPAD∩PLS更“自然”的对应，包含许多有趣的问题——本身等于PPAD∩PLS。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the ACM 工程技术-计算机：理论方法

CiteScore

7.50

自引率

0.00%

发文量

审稿时长

3 months

期刊介绍： The best indicator of the scope of the journal is provided by the areas covered by its Editorial Board. These areas change from time to time, as the field evolves. The following areas are currently covered by a member of the Editorial Board: Algorithms and Combinatorial Optimization; Algorithms and Data Structures; Algorithms, Combinatorial Optimization, and Games; Artificial Intelligence; Complexity Theory; Computational Biology; Computational Geometry; Computer Graphics and Computer Vision; Computer-Aided Verification; Cryptography and Security; Cyber-Physical, Embedded, and Real-Time Systems; Database Systems and Theory; Distributed Computing; Economics and Computation; Information Theory; Logic and Computation; Logic, Algorithms, and Complexity; Machine Learning and Computational Learning Theory; Networking; Parallel Computing and Architecture; Programming Languages; Quantum Computing; Randomized Algorithms and Probabilistic Analysis of Algorithms; Scientific Computing and High Performance Computing; Software Engineering; Web Algorithms and Data Mining