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A Strong Direct Sum Theorem for Distributional Query Complexity

Authors Guy Blanc, Caleb Koch, Carmen Strassle, Li-Yang Tan

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Author Details

Guy Blanc
  • Department of Computer Science, Stanford University, CA, USA
Caleb Koch
  • Department of Computer Science, Stanford University, CA, USA
Carmen Strassle
  • Department of Computer Science, Stanford University, CA, USA
Li-Yang Tan
  • Department of Computer Science, Stanford University, CA, USA

Funding

The authors are supported by NSF awards 1942123, 2211237, 2224246 and a Google Research Scholar award. Guy is also supported by a Jane Street Graduate Research Fellowship, Caleb by an NDSEG fellowship, and Carmen by a Stanford Computer Science Distinguished Fellowship.

Cite AsGet BibTex

Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A Strong Direct Sum Theorem for Distributional Query Complexity. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 16:1-16:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.16

Abstract

Consider the expected query complexity of computing the k-fold direct product f^{⊗ k} of a function f to error ε with respect to a distribution μ^k. One strategy is to sequentially compute each of the k copies to error ε/k with respect to μ and apply the union bound. We prove a strong direct sum theorem showing that this naive strategy is essentially optimal. In particular, computing a direct product necessitates a blowup in both query complexity and error. Strong direct sum theorems contrast with results that only show a blowup in query complexity or error but not both. There has been a long line of such results for distributional query complexity, dating back to (Impagliazzo, Raz, Wigderson 1994) and (Nisan, Rudich, Saks 1994), but a strong direct sum theorem that holds for all functions in the standard query model had been elusive. A key idea in our work is the first use of the Hardcore Theorem (Impagliazzo 1995) in the context of query complexity. We prove a new resilience lemma that accompanies it, showing that the hardcore of f^{⊗k} is likely to remain dense under arbitrary partitions of the input space.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Query complexity
  • direct product theorem
  • hardcore theorem

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References

  1. Andris Ambainis, Loïck Magnin, Martin Roetteler, and Jérémie Roland. Symmetry-assisted adversaries for quantum state generation. In 2011 IEEE 26th Annual Conference on Computational Complexity (CCC), pages 167-177. IEEE, 2011. Google Scholar
  2. Andris Ambainis, Robert Špalek, and Ronald de Wolf. A new quantum lower bound method, with applications to direct product theorems and time-space tradeoffs. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC), pages 618-633, 2006. Google Scholar
  3. Sepehr Assadi and Vishvajeet N. Graph streaming lower bounds for parameter estimation and property testing via a streaming XOR lemma. In Samir Khuller and Virginia Vassilevska Williams, editors, Proceedings of the 53rd Annual ACM Symposium on Theory of Computing (STOC), pages 612-625, 2021. Google Scholar
  4. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 67-76, 2010. Google Scholar
  5. Shalev Ben-David and Robin Kothari. Randomized query complexity of sabotaged and composed functions. Theory of Computing, 14(5):1-27, 2018. URL: https://doi.org/10.4086/toc.2018.v014a005.
  6. Eric Blais and Joshua Brody. Optimal Separation and Strong Direct Sum for Randomized Query Complexity. In 34th Computational Complexity Conference (CCC), volume 137, pages 29:1-29:17, 2019. URL: https://doi.org/10.4230/LIPIcs.CCC.2019.29.
  7. Guy Blanc, Caleb Koch, Carmen Strassle, and Li-Yang Tan. A strong composition theorem for junta complexity and the boosting of property testers. In Proceedings of the 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1757-1777, 2023. Google Scholar
  8. Mark Braverman, Anup Rao, Omri Weinstein, and Amir Yehudayoff. Direct products in communication complexity. In Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 746-755, 2013. Google Scholar
  9. Joshua Brody, Jae Tak Kim, Peem Lerdputtipongporn, and Hariharan Srinivasulu. A strong xor lemma for randomized query complexity. Theory of Computing, 19(11):1-14, 2023. URL: https://doi.org/10.4086/toc.2023.v019a011.
  10. Andrew Drucker. Improved direct product theorems for randomized query complexity. computational complexity, 21(2):197-244, 2012. Google Scholar
  11. Andrew Drucker. Nondeterministic direct product reductions and the success probability of sat solvers. In Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS), pages 736-745, 2013. Google Scholar
  12. Oded Goldreich, Noam Nisan, and Avi Wigderson. On yao’s xor-lemma. Studies in Complexity and Cryptography, 6650:273-301, 2011. Google Scholar
  13. William Hoza. A technique for hardness amplification against AC0. ECCC preprint TR23-176, 2023. Google Scholar
  14. Russell Impagliazzo. Hard-core distributions for somewhat hard problems. In Proceedings of IEEE 36th Annual Foundations of Computer Science (FOCS), pages 538-545, 1995. Google Scholar
  15. Russell Impagliazzo, Ragesh Jaiswal, Valentine Kabanets, and Avi Wigderson. Uniform direct product theorems: simplified, optimized, and derandomized. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pages 579-588, 2008. Google Scholar
  16. Russell Impagliazzo, Ran Raz, and Avi Wigderson. A direct product theorem. In Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory, pages 88-96, 1994. Google Scholar
  17. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC), pages 220-229, 1997. Google Scholar
  18. Rahul Jain. New strong direct product results in communication complexity. Journal of the ACM (JACM), 62(3):1-27, 2015. Google Scholar
  19. Rahul Jain, Hartmut Klauck, and Miklos Santha. Optimal direct sum results for deterministic and randomized decision tree complexity. Information Processing Letters, 110(20):893-897, 2010. Google Scholar
  20. Rahul Jain, Attila Pereszlényi, and Penghui Yao. A direct product theorem for the two-party bounded-round public-coin communication complexity. In Proceedings of the 53rd Annual Symposium on Foundations of Computer Science (FOCS), pages 167-176, 2012. Google Scholar
  21. Hartmut Klauck. A strong direct product theorem for disjointness. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pages 77-86, 2010. Google Scholar
  22. Hartmut Klauck, Robert Špalek, and Ronald de Wolf. Quantum and classical strong direct product theorems and optimal time-space tradeoffs. SIAM Journal on Computing, 36(5):1472-1493, 2007. Preliminary version in FOCS 2004. Google Scholar
  23. Adam R Klivans and Rocco A Servedio. Boosting and hard-core set construction. Machine Learning, 51:217-238, 2003. Google Scholar
  24. Troy Lee and Jérémie Roland. A strong direct product theorem for quantum query complexity. computational complexity, 22:429-462, 2013. Google Scholar
  25. Troy Lee, Adi Shraibman, and Robert Špalek. A direct product theorem for discrepancy. In Proceedings of the 23rd Annual IEEE Conference on Computational Complexity (CCC), pages 71-80, 2008. Google Scholar
  26. Leonid A Levin. One-way functions and pseudorandom generators. In Proceedings of the 17th Annual ACM Symposium on Theory of Computing (STOC), pages 363-365, 1985. Google Scholar
  27. Noam Nisan, Steven Rudich, and Michael Saks. Products and help bits in decision trees. In Proceedings 35th Annual Symposium on Foundations of Computer Science (FOCS), pages 318-329, 1994. Google Scholar
  28. Noam Nisan and Avi Wigderson. Hardness vs randomness. Journal of computer and System Sciences, 49(2):149-167, 1994. Google Scholar
  29. Ryan O'Donnell. Hardness amplification within np. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pages 751-760, 2002. Google Scholar
  30. Ronen Shaltiel. Towards proving strong direct product theorems. Computational Complexity, 12(1/2):1-22, 2004. Google Scholar
  31. Alexander A Sherstov. Strong direct product theorems for quantum communication and query complexity. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 41-50, 2011. Google Scholar
  32. Robert Špalek. The multiplicative quantum adversary. In 23rd Annual IEEE Conference on Computational Complexity (CCC), pages 237-248, 2008. Google Scholar
  33. Volker Strassen. Vermeidung von divisionen. Journal für die reine und angewandte Mathematik, 264:184-202, 1973. Google Scholar
  34. Luca Trevisan. The Impagliazzo Hard-Core-Set Theorem. https://lucatrevisan.wordpress.com/2007/11/06/the-impagliazzo-hard-core-set-theorem/, 2007.
  35. Emanuele Viola and Avi Wigderson. Norms, xor lemmas, and lower bounds for polynomials and protocols. Theory of Computing, 4(1):137-168, 2008. Google Scholar
  36. Andrew Yao. Theory and application of trapdoor functions. In Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (FOCS), pages 80-91, 1982. Google Scholar
  37. Huacheng Yu. Strong xor lemma for communication with bounded rounds. In Proceedings of the 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 1186-1192, 2022. Google Scholar
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