LIPIcs.CCC.2022.27.pdf
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Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs
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37th Computational Complexity Conference (CCC 2022)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Computational Complexity Conference (CCC) - License:
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- Publication Date: 2022-07-11
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Acknowledgements
S.V. thanks Dean Doron, Raghu Meka, Omer Reingold, and Avishay Tal, conversations with whom inspired some of this research. We also thank the anonymous CCC reviewers for helpful comments that have improved the presentation.
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Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 27:1-27:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.27
https://doi.org/10.4230/LIPIcs.CCC.2022.27
Abstract
We study the pseudorandomness of random walks on expander graphs against tests computed by symmetric functions and permutation branching programs. These questions are motivated by applications of expander walks in the coding theory and derandomization literatures. A line of prior work has shown that random walks on expanders with second largest eigenvalue λ fool symmetric functions up to a O(λ) error in total variation distance, but only for the case where the vertices are labeled with symbols from a binary alphabet, and with a suboptimal dependence on the bias of the labeling. We generalize these results to labelings with an arbitrary alphabet, and for the case of binary labelings we achieve an optimal dependence on the labeling bias. We extend our analysis to unify it with and strengthen the expander-walk Chernoff bound. We then show that expander walks fool permutation branching programs up to a O(λ) error in 𝓁₂-distance, and we prove that much stronger bounds hold for programs with a certain structure. We also prove lower bounds to show that our results are tight. To prove our results for symmetric functions, we analyze the Fourier coefficients of the relevant distributions using linear-algebraic techniques. Our analysis for permutation branching programs is likewise linear-algebraic in nature, but also makes use of the recently introduced singular-value approximation notion for matrices (Ahmadinejad et al. 2021).
Subject Classification
ACM Subject Classification
- Theory of computation → Pseudorandomness and derandomization
Keywords
- Expander graph
- Random walk
- Pseudorandomness
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References
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AmirMahdi Ahmadinejad. Computing stationary distributions: perron vectors, random walks, and ride-sharing competition. PhD thesis, Stanford University, Stanford, California, 2020.
- AmirMahdi Ahmadinejad, Jonathan Kelner, Jack Murtagh, John Peebles, Aaron Sidford, and Salil Vadhan. High-precision Estimation of Random Walks in Small Space. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 1295-1306, November 2020. ISSN: 2575-8454. URL: https://doi.org/10.1109/FOCS46700.2020.00123.
-
AmirMahdi Ahmadinejad, John Peebles, Aaron Sidford, and Salil Vadhan. Personal Communication, 2021.
- Avraham Ben-Aroya and Amnon Ta-Shma. A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag Product. SIAM Journal on Computing, 40(2):267-290, January 2011. URL: https://doi.org/10.1137/080732651.
- Gil Cohen, Dor Minzer, Shir Peleg, Aaron Potechin, and Amnon Ta-Shma. Expander Random Walks: The General Case and Limitations. Electronic Colloquium on Computational Complexity, 2021. URL: https://eccc.weizmann.ac.il/report/2021/091/.
- Gil Cohen, Noam Peri, and Amnon Ta-Shma. Expander random walks: a Fourier-analytic approach. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 1643-1655, New York, NY, USA, June 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3406325.3451049.
- David Gillman. A Chernoff Bound for Random Walks on Expander Graphs. SIAM Journal on Computing, 27(4):1203-1220, August 1998. URL: https://doi.org/10.1137/S0097539794268765.
-
Louis Golowich. A Berry-Esseen Theorem for Expander Walks. Forthcoming, 2022.
- Louis Golowich and Salil Vadhan. Pseudorandomness of Expander Random Walks for Symmetric Functions and Permutation Branching Programs. Electronic Colloquium on Computational Complexity, 2022. URL: https://eccc.weizmann.ac.il/report/2022/024/.
- Venkatesan Guruswami. Guest column: error-correcting codes and expander graphs. ACM SIGACT News, 35(3):25-41, September 2004. URL: https://doi.org/10.1145/1027914.1027924.
- Venkatesan Guruswami and Vinayak M. Kumar. Pseudobinomiality of the Sticky Random Walk. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 48:1-48:19, Dagstuhl, Germany, 2021. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.48.
- Alexander D. Healy. Randomness-Efficient Sampling within NC1. computational complexity, 17(1):3-37, April 2008. URL: https://doi.org/10.1007/s00037-007-0238-5.
- Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006. URL: https://doi.org/10.1090/S0273-0979-06-01126-8.
- William M. Hoza, Edward Pyne, and Salil Vadhan. Pseudorandom Generators for Unbounded-Width Permutation Branching Programs. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021), volume 185 of Leibniz International Proceedings in Informatics (LIPIcs), pages 7:1-7:20, Dagstuhl, Germany, 2021. Schloss Dagstuhl–Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ITCS.2021.7.
- Russell Impagliazzo, Noam Nisan, and Avi Wigderson. Pseudorandomness for network algorithms. In Proceedings of the twenty-sixth annual ACM symposium on Theory of Computing, STOC '94, pages 356-364, New York, NY, USA, May 1994. Association for Computing Machinery. URL: https://doi.org/10.1145/195058.195190.
- Akhil Jalan and Dana Moshkovitz. Near-Optimal Cayley Expanders for Abelian Groups. arXiv:2105.01149 [cs], May 2021. URL: http://arxiv.org/abs/2105.01149v1.
- A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, September 1988. URL: https://doi.org/10.1007/BF02126799.
-
Grigorii Aleksandrovich Margulis. Explicit constructions of expanders. Problemy Peredachi Informatsii, 9(4):71-80, 1973. Publisher: Russian Academy of Sciences, Branch of Informatics, Computer Equipment and Automatization.
- Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders. The Annals of Mathematics, 155(1):157, January 2002. URL: https://doi.org/10.2307/3062153.
- Amnon Ta-Shma. Explicit, almost optimal, epsilon-balanced codes. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 238-251, Montreal Canada, June 2017. ACM. URL: https://doi.org/10.1145/3055399.3055408.
- Salil P. Vadhan. Pseudorandomness. Foundations and Trends® in Theoretical Computer Science, 7(1–3):1-336, December 2012. URL: https://doi.org/10.1561/0400000010.