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Junta Distance Approximation with Sub-Exponential Queries

Authors Vishnu Iyer , Avishay Tal , Michael Whitmeyer

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Author Details

Vishnu Iyer
  • University of California at Berkeley, CA, USA
Avishay Tal
  • University of California at Berkeley, CA, USA
Michael Whitmeyer
  • University of California at Berkeley, CA, USA


We thank Anindya De, Shafi Goldwasser, Amit Levi, and Orr Paradise for very helpful discussions.

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Vishnu Iyer, Avishay Tal, and Michael Whitmeyer. Junta Distance Approximation with Sub-Exponential Queries. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 24:1-24:38, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:{±1}ⁿ → {±1}: 1) We give a poly(k, 1/(ε)) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k'-juntas, where k' = O(k/(ε²)). 2) In the non-relaxed setting, we extend our ideas to give a 2^{Õ(√{k/ε})} (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [Michel Talagrand, 1994].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Algorithms
  • Complexity Theory
  • Fourier Analysis
  • Juntas
  • Normalized Influence
  • Property Testing
  • Tolerant Property Testing


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  1. Nir Ailon, Bernard Chazelle, Seshadhri Comandur, and Ding Liu. Estimating the distance to a monotone function. Random Structures & Algorithms, 31(3):371-383, 2007. URL:
  2. Andris Ambainis, Aleksandrs Belovs, Oded Regev, and Ronald de Wolf. Efficient quantum algorithms for (gapped) group testing and junta testing. In SODA, pages 903-922. SIAM, 2016. Google Scholar
  3. Mihir Bellare, Oded Goldreich, and Madhu Sudan. Free bits, pcps, and nonapproximability-towards tight results. SIAM J. Comput., 27(3):804-915, 1998. Google Scholar
  4. Eric Blais. Improved bounds for testing juntas. In APPROX-RANDOM, volume 5171 of Lecture Notes in Computer Science, pages 317-330. Springer, 2008. Google Scholar
  5. Eric Blais. Testing juntas nearly optimally. In STOC, pages 151-158. ACM, 2009. Google Scholar
  6. Eric Blais, Clément L. Canonne, Talya Eden, Amit Levi, and Dana Ron. Tolerant junta testing and the connection to submodular optimization and function isomorphism. ACM Trans. Comput. Theory, 11(4):24:1-24:33, 2019. Google Scholar
  7. Guy Blanc, Jane Lange, and Li-Yang Tan. Testing and reconstruction via decision trees. CoRR, abs/2012.08735, 2020. Google Scholar
  8. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. In STOC, pages 73-83. ACM, 1990. Google Scholar
  9. Nader H. Bshouty. Almost optimal distribution-free junta testing. In Computational Complexity Conference, volume 137 of LIPIcs, pages 2:1-2:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  10. Sourav Chakraborty, Eldar Fischer, David García-Soriano, and Arie Matsliah. Junto-symmetric functions, hypergraph isomorphism and crunching. In Computational Complexity Conference, pages 148-158. IEEE Computer Society, 2012. Google Scholar
  11. Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, and Jinyu Xie. Settling the query complexity of non-adaptive junta testing. J. ACM, 65(6):40:1-40:18, 2018. Google Scholar
  12. Hana Chockler and Dan Gutfreund. A lower bound for testing juntas. Inf. Process. Lett., 90(6):301-305, 2004. Google Scholar
  13. Anindya De, Elchanan Mossel, and Joe Neeman. Junta correlation is testable. In FOCS, pages 1549-1563. IEEE Computer Society, 2019. Google Scholar
  14. Ilias Diakonikolas, Homin K. Lee, Kevin Matulef, Krzysztof Onak, Ronitt Rubinfeld, Rocco A. Servedio, and Andrew Wan. Testing for concise representations. In FOCS, pages 549-558. IEEE Computer Society, 2007. Google Scholar
  15. Ilias Diakonikolas, Homin K. Lee, Kevin Matulef, Rocco A. Servedio, and Andrew Wan. Efficiently testing sparse GF(2) polynomials. In ICALP (1), volume 5125 of Lecture Notes in Computer Science, pages 502-514. Springer, 2008. Google Scholar
  16. Eldar Fischer, Guy Kindler, Dana Ron, Shmuel Safra, and Alex Samorodnitsky. Testing juntas. J. Comput. Syst. Sci., 68(4):753-787, 2004. Google Scholar
  17. Oded Goldreich, Shafi Goldwasser, and Dana Ron. Property testing and its connection to learning and approximation. In 37th Annual Symposium on Foundations of Computer Science, FOCS '96, Burlington, Vermont, USA, 14-16 October, 1996, pages 339-348. IEEE Computer Society, 1996. URL:
  18. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. Google Scholar
  19. Jeff Kahn, Gil Kalai, and Nathan Linial. The influence of variables on boolean functions (extended abstract). In FOCS, pages 68-80. IEEE Computer Society, 1988. Google Scholar
  20. Michael J. Kearns and Dana Ron. Testing problems with sublearning sample complexity. J. Comput. Syst. Sci., 61(3):428-456, 2000. Google Scholar
  21. Esty Kelman, Subhash Khot, Guy Kindler, Dor Minzer, and Muli Safra. Theorems of kkl, friedgut, and talagrand via random restrictions and log-sobolev inequality. Electron. Colloquium Comput. Complex., 27:9, 2020. Google Scholar
  22. Amit Levi and Erik Waingarten. Lower bounds for tolerant junta and unateness testing via rejection sampling of graphs. In ITCS, volume 124 of LIPIcs, pages 52:1-52:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  23. Zhengyang Liu, Xi Chen, Rocco A. Servedio, Ying Sheng, and Jinyu Xie. Distribution-free junta testing. ACM Trans. Algorithms, 15(1):1:1-1:23, 2019. Google Scholar
  24. Elchanan Mossel, Ryan O'Donnell, and Rocco A. Servedio. Learning juntas. In STOC, pages 206-212. ACM, 2003. Google Scholar
  25. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. URL:
  26. Ryan O'Donnell and Karl Wimmer. Sharpness of KKL on Schreier graphs. Electronic Communications in Probability, 18:1-12, 2013. URL:
  27. Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and Erik Waingarten. Approximating the distance to monotonicity of boolean functions. In SODA, pages 1995-2009. SIAM, 2020. Google Scholar
  28. Michal Parnas, Dana Ron, and Ronitt Rubinfeld. Tolerant property testing and distance approximation. Electron. Colloquium Comput. Complex., 010, 2004. Google Scholar
  29. Michal Parnas, Dana Ron, and Alex Samorodnitsky. Proclaiming dictators and juntas or testing boolean formulae. In RANDOM-APPROX, volume 2129 of Lecture Notes in Computer Science, pages 273-284. Springer, 2001. Google Scholar
  30. Mert Saglam. Near log-convexity of measured heat in (discrete) time and consequences. In FOCS, pages 967-978. IEEE Computer Society, 2018. Google Scholar
  31. Rocco A. Servedio. Testing by implicit learning: A brief survey. In Property Testing, volume 6390 of Lecture Notes in Computer Science, pages 197-210. Springer, 2010. Google Scholar
  32. Michel Talagrand. On Russo’s Approximate Zero-One Law. The Annals of Probability, 22(3):1576-1587, 1994. URL:
  33. Gregory Valiant. Finding correlations in subquadratic time, with applications to learning parities and juntas. In FOCS, pages 11-20. IEEE Computer Society, 2012. Google Scholar
  34. Xiaojin Zhang. Near-optimal algorithm for distribution-free junta testing. CoRR, abs/1911.10833, 2019. Google Scholar
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