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 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.
Bibliographical noteFunding Information:
\ast Received by the editors February 22, 2017; accepted for publication (in revised form) July 29, 2020; published electronically November 19, 2020. A preliminary version of this work appeared in Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, 2017. https://doi.org/10.1137/17M1118014 Funding: This work was performed at the Weizmann Institute of Science, Israel, and was supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation, BSF, and the Israeli Ministry of Science and Technology. \dagger Charles University, Prague, Czech Republic (email@example.com). \ddagger Boston University, Boston, MA 02215 USA, and Tel Aviv University, Tel Aviv 6997801, Israel (firstname.lastname@example.org).
© 2020 Society for Industrial and Applied Mathematics.
- Continuous local search
- Cryptographic hardness
- Query complexity