Abstract
We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over n elements, and they wish to estimate within ±ε the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log n factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater Than is Θ (n log n).
- Anurag Anshu, Aleksandrs Belovs, Shalev Ben-David, Mika Göös, Rahul Jain, Robin Kothari, Troy Lee, and Miklos Santha. 2016. Separations in communication complexity using cheat sheets and information complexity. In Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS’16). IEEE, 555--564.Google Scholar
Cross Ref
- Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. 2004. An information statistics approach to data stream and communication complexity. J. Comput. System Sci. 68, 4 (2004), 702--732. Google Scholar
Digital Library
- Tugkan Batu, Lance Fortnow, Ronitt Rubinfeld, Warren Smith, and Patrick White. 2013. Testing closeness of discrete distributions. J. ACM 60, 1 (2013), 4. Google Scholar
Digital Library
- Andrej Bogdanov, Elchanan Mossel, and Salil Vadhan. 2008. The complexity of distinguishing Markov random fields. In Proceedings of the 12th International Workshop on Randomization and Computation (RANDOM’08). Springer, 331--342. Google Scholar
Digital Library
- Mark Braverman. 2015. Interactive information complexity. SIAM J. Comput. 44, 6 (2015), 1698--1739.Google Scholar
Cross Ref
- Mark Braverman and Omri Weinstein. 2015. An interactive information odometer and applications. In Proceedings of the 47th Symposium on Theory of Computing (STOC’15). ACM, 341--350. Google Scholar
Digital Library
- Mark Braverman and Omri Weinstein. 2016. A discrepancy lower bound for information complexity. Algorithmica 76, 3 (2016), 846--864. Google Scholar
Digital Library
- Clément Canonne. 2015. A Survey on Distribution Testing: Your Data is Big. But Is It Blue? Technical Report TR15-063. Electronic Colloquium on Computational Complexity (ECCC). Retrieved from http://eccc.hpi-web.de/report/2015/063.Google Scholar
- Amit Chakrabarti and Oded Regev. 2012. An optimal lower bound on the communication complexity of Gap-Hamming-Distance. SIAM J. Comput. 41, 5 (2012), 1299--1317.Google Scholar
Cross Ref
- Siu On Chan, Ilias Diakonikolas, Paul Valiant, and Gregory Valiant. 2014. Optimal algorithms for testing closeness of discrete distributions. In Proceedings of the 25th Symposium on Discrete Algorithms (SODA’14). ACM-SIAM, 1193--1203. Google Scholar
Digital Library
- Joan Feigenbaum, Sampath Kannan, Martin Strauss, and Mahesh Viswanathan. 2002. An approximate L1-difference algorithm for massive data streams. SIAM J. Comput. 32, 1 (2002), 131--151. Google Scholar
Digital Library
- Jessica Fong and Martin Strauss. 2001. An approximate Lp-difference algorithm for massive data streams. Discrete Math. Theor. Comput. Sci. 4, 2 (2001), 301--322.Google Scholar
- Oded Goldreich, Amit Sahai, and Salil Vadhan. 1999. Can statistical zero knowledge be made non-interactive? or On the relationship of SZK and NISZK. In Proceedings of the 19th International Cryptology Conference (CRYPTO’99). Springer, 467--484. Google Scholar
Digital Library
- Oded Goldreich and Salil Vadhan. 1999. Comparing entropies in statistical zero-knowledge with applications to the structure of SZK. In Proceedings of the 14th Conference on Computational Complexity (CCC’99). IEEE, 54--73. Google Scholar
Digital Library
- Oded Goldreich and Salil Vadhan. 2011. On the complexity of computational problems regarding distributions. Studies in Complexity and Cryptography. (2011), 390--405.Google Scholar
- Mika Göös, T. S. Jayram, Toniann Pitassi, and Thomas Watson. 2017. Randomized communication vs. partition number. In Proceedings of the 44th International Colloquium on Automata, Languages, and Programming (ICALP). Schloss Dagstuhl, 52:1--52:15.Google Scholar
- Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. 2016. Rectangles are nonnegative juntas. SIAM J. Comput. 45, 5 (2016), 1835--1869.Google Scholar
Cross Ref
- Thomas Holenstein. 2009. Parallel repetition: Simplification and the no-signaling case. Theory Comput. 5, 1 (2009), 141--172.Google Scholar
Cross Ref
- Rahul Jain and Hartmut Klauck. 2010. The partition bound for classical communication complexity and query complexity. In Proceedings of the 25th Conference on Computational Complexity (CCC’10). IEEE, 247--258. Google Scholar
Digital Library
- Iordanis Kerenidis, Sophie Laplante, Virginie Lerays, Jérémie Roland, and David Xiao. 2015. Lower bounds on information complexity via zero-communication protocols and applications. SIAM J. Comput. 44, 5 (2015), 1550--1572.Google Scholar
Cross Ref
- Jon Kleinberg and Éva Tardos. 2002. Approximation algorithms for classification problems with pairwise relationships: Metric labeling and Markov random fields. J. ACM 49, 5 (2002), 616--639. Google Scholar
Digital Library
- Eyal Kushilevitz and Noam Nisan. 1997. Communication Complexity. Cambridge University Press. Google Scholar
Digital Library
- Anup Rao. 2011. Parallel repetition in projection games and a concentration bound. SIAM J. Comput. 40, 6 (2011), 1871--1891. Google Scholar
Digital Library
- Ran Raz. 2011. A counterexample to strong parallel repetition. SIAM J. Comput. 40, 3 (2011), 771--777. Google Scholar
Digital Library
- Ronitt Rubinfeld. 2012. Taming big probability distributions. ACM Crossroads 19, 1 (2012), 24--28. Google Scholar
Digital Library
- Amit Sahai and Salil Vadhan. 2003. A complete problem for statistical zero knowledge. J. ACM 50, 2 (2003), 196--249. Google Scholar
Digital Library
- Alexander Sherstov. 2012. The communication complexity of Gap Hamming Distance. Theory Comput. 8, 1 (2012), 197--208.Google Scholar
Cross Ref
- Paul Valiant. 2011. Testing symmetric properties of distributions. SIAM J. Comput. 40, 6 (2011), 1927--1968. Google Scholar
Digital Library
- Thomas Vidick. 2012. A concentration inequality for the overlap of a vector on a large set, with application to the communication complexity of the Gap-Hamming-Distance problem. Chicago J. Theoret. Comput. Sci. 2012, 1 (2012), 1--12.Google Scholar
Cross Ref
- Thomas Watson. 2015. The complexity of deciding statistical properties of samplable distributions. Theory Comput. 11 (2015), 1--34.Google Scholar
Cross Ref
- Thomas Watson. 2016. The complexity of estimating min-entropy. Comput. Complex. 25, 1 (2016), 153--175. Google Scholar
Digital Library
- Thomas Watson. 2017. Communication complexity of statistical distance. In Proceedings of the 21st International Workshop on Randomization and Computation (RANDOM’17). Schloss Dagstuhl, 49:1--49:10.Google Scholar
Index Terms
Communication Complexity of Statistical Distance
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