research-article

Hardness of Continuous Local Search: : Query Complexity and Cryptographic Lower Bounds

Authors: Pavel Hubáček,

Eylon YogevAuthors Info & Claims

SIAM Journal on Computing, Volume 49, Issue 6

Pages 1128 - 1172

https://doi.org/10.1137/17M1118014

Published: 01 January 2020 Publication History

Abstract

Local search proved to be an extremely useful tool when facing hard optimization problems (e.g., via the simplex algorithm, simulated annealing, or genetic algorithms). Although powerful, it has its limitations: there are functions for which exponentially many queries are needed to find a local optimum. In many contexts, the optimization problem is defined by a continuous function which might offer an advantage when performing the local search. This leads us to study the following natural question: How hard is continuous local search? The computational complexity of such search problems is captured by the complexity class ${CLS}$ [C. Daskalakis and C. H. Papadimitriou, Proceedings of SODA'11, 2011], which is contained in the intersection of ${PLS}$ and ${PPAD}$, two important subclasses of ${TFNP}$ (the class of ${NP}$ search problems with a guaranteed solution). In this work, we show the first hardness results for ${CLS}$ (the smallest nontrivial class among the currently defined subclasses of $\mathbf{TFNP}$). Our hardness results are in terms of black-box (where only oracle access to the function is given) and white-box (where the function is represented succinctly by a circuit). In the black-box case, we show instances for which any (computationally unbounded) randomized algorithm must perform exponentially many queries in order to find a local optimum. In the white-box case, we show hardness for computationally bounded algorithms under cryptographic assumptions. Our results demonstrate a strong conceptual barrier precluding design of efficient algorithms for solving local search problems even over continuous domains. As our main technical contribution we introduce a new total search problem which we call End-of-Metered-Line. The special structure of End-of-Metered-Line enables us to (1) show that it is contained in ${CLS}$, (2) prove hardness for it in both the black-box and the white-box setting, and (3) extend to ${CLS}$ a variety of results previously known only for ${PPAD}$.

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 49, Issue 6

ISSN:0097-5397

DOI:10.1137/smjcat.49.6

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© 2020, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

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Published: 01 January 2020

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Göös MHollender AJain SMaystre GPires WRobere RTao R(2024)Separations in Proof Complexity and TFNPJournal of the ACM10.1145/366375871:4(1-45)Online publication date: 9-May-2024
https://dl.acm.org/doi/10.1145/3663758
Fearnley JGoldberg PHollender ASavani RMohar BShinkar IO'Donnell R(2024)The Complexity of Computing KKT Solutions of Quadratic ProgramsProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649647(892-903)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649647
Göös MHollender AJain SMaystre GPires WRobere RTao R(2022)Further collapses in TFNPProceedings of the 37th Computational Complexity Conference10.4230/LIPIcs.CCC.2022.33(1-15)Online publication date: 20-Jul-2022
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2022.33
Fearnley JGoldberg PHollender ASavani R(2022)The Complexity of Gradient Descent: CLS = PPAD ∩ PLSJournal of the ACM10.1145/356816370:1(1-74)Online publication date: 19-Dec-2022
https://dl.acm.org/doi/10.1145/3568163

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