Advertisement

Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 981–1044 | Cite as

A new central limit theorem and decomposition for Gaussian polynomials, with an application to deterministic approximate counting

  • Anindya De
  • Rocco A. Servedio
Article
  • 216 Downloads

Abstract

One of the main results of this paper is a new multidimensional central limit theorem (CLT) for multivariate polynomials under Gaussian inputs. Roughly speaking, the new CLT shows that any collection of Gaussian polynomials with small eigenvalues (suitably defined) must have a joint distribution which is close to a multidimensional Gaussian distribution. The new CLT is proved using tools from Malliavin calculus and Stein’s method. A second main result of the paper, which complements the new CLT, is a new decomposition theorem for low-degree multilinear polynomials over Gaussian inputs. Roughly speaking, this result shows that (up to some small error) any such polynomial is very close to a polynomial which can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. An important feature of this decomposition theorem is the delicate control obtained between the number of polynomials in the decomposition versus their eigenvalues. As the main application of these results, we give a deterministic algorithm for approximately counting satisfying assignments of a degree-d polynomial threshold function (PTF) over the domain \(\{-1,1\}^n\); this is a well-studied problem from theoretical computer science. More precisely, given as input a degree-d polynomial \(p(x_1,\dots ,x_n)\) over \({{\mathbb {R}}}^n\) and a parameter \(\epsilon > 0\), the algorithm approximates
$$\begin{aligned} \mathop {\mathbf{Pr}}_{x \sim \{-1,1\}^n}[p(x) \ge 0] \end{aligned}$$
to within an additive \(\pm \epsilon \) in time \(O_{d,\epsilon }(1)\cdot \mathrm {poly}(n^d)\). (Since it is NP-hard to determine whether the above probability is nonzero, any sort of efficient multiplicative approximation is almost certainly impossible even for randomized algorithms.) Note that the running time of the algorithm (as a function of \(n^d\), the number of coefficients of a degree-d PTF) is a fixed polynomial. The fastest previous algorithm for this problem (Kane, CoRR, arXiv:1210.1280, 2012), based on constructions of unconditional pseudorandom generators for degree-d PTFs, runs in time \(n^{O_{d,c}(1) \cdot \epsilon ^{-c}}\) for all \(c > 0\).

Keywords

Gaussian chaos Central limit theorem Polynomials Malliavin calculus Stein’s method Regularity lemma 

Mathematics Subject Classification

60F05 68Q87 

Notes

Acknowledgements

We thank Ilias Diakonikolas for his contributions in the early stages of this project. We also thank Rafal Latala, Michel Ledoux, Elchanan Mossel, Ivan Nourdin and Krzysztof Oleszkiewicz for answering questions about the CLT. Part of this work was done when A.D. was hosted by Oded Regev and the Simons Institute. A.D. would like to thank them for their kind hospitality and support.

References

  1. 1.
    Aaronson, S.: P \(\stackrel{?}{=}\) NP. http://www.scottaaronson.com/papers/pnp.pdf. Earlier version in “Open Problems in Mathematics”, Nash, J.F., Jr., Rassias, M.Th. (eds.) (2017)
  2. 2.
    Alon, N., Babai, L., Itai, A.: A fast and simple randomized algorithm for the maximal independent set problem. J. Algorithms 7, 567–583 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aziz, H., Paterson, M., Leech, D.: Efficient algorithm for designing weighted voting games. In: IEEE International Multitopic Conference, pp. 1–6 (2007)Google Scholar
  4. 4.
    Ajtai, M., Wigderson, A.: Deterministic simulation of probabilistic constant depth circuits. In: Proceedings of 26th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 11–19 (1985)Google Scholar
  5. 5.
    Adamczak, R., Wolff, P.: Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Relat. Fields 162(3), 531–586 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bhatia, R.: Matrix Analysis. Springer, Basel (2000)zbMATHGoogle Scholar
  7. 7.
    Breuer, P., Major, P.: Central limit theorems for non-linear functionals of Gaussian fields. J. Multivar. Anal. 13(3), 425–441 (1983)CrossRefzbMATHGoogle Scholar
  8. 8.
    Chatterjee, S.: A new method of normal approximation. Ann. Probab. 36(4), 1584–1610, 07 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chatterjee, S.: Fluctuations of eigenvalues and second-order Poincaré inequalities. Probab. Theory Relat. Fields 143, 1–40 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Cartwright, D., Sturmfels, B.: The number of eigenvalues of a tensor. Linear Algebra Appl. 432(2), 942–952 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Carbery, A., Wright, J.: Distributional and \(L^q\) norm inequalities for polynomials over convex bodies in \(R^n\). Math. Res. Lett. 8(3), 233–248 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    De, A., Diakonikolas, I., Feldman, V., Servedio, R.: Near-optimal solutions for the Chow parameters problem and low-weight approximation of halfspaces. In: Proceedings of 44th ACM Symposium on Theory of Computing (STOC), pp. 729–746 (2012)Google Scholar
  13. 13.
    De, A., Diakonikolas, I., Servedio, R.A.: The inverse Shapley value problem. ICALP 1, 266–277 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    De, A., Diakonikolas, I., Servedio, R.: Deterministic approximate counting for degree-2 polynomial threshold functions. Manuscript (2013)Google Scholar
  15. 15.
    De, A., Diakonikolas, I., Servedio, R.: Deterministic approximate counting for juntas of degree-2 polynomial threshold functions. Manuscript (2013)Google Scholar
  16. 16.
    Diakonikolas, I., Harsha, P., Klivans, A., Meka, R., Raghavendra, P., Servedio, R.A., Tan, L.-Y.: Bounding the average sensitivity and noise sensitivity of polynomial threshold functions. In: STOC, pp. 533–542 (2010)Google Scholar
  17. 17.
    Diakonikolas, I., Kane, D.M., Nelson, J.: Bounded independence fools degree-2 threshold functions. In: Proceedings of 51st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 11–20 (2010)Google Scholar
  18. 18.
    Dowson, D.C., Landau, B.V.: The Frechet distance between multivariate normal distributions. J. Multivar. Anal. 12(3), 450–455 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Diakonikolas, I., O’Donnell, R., Servedio, R., Wu, Y.: Hardness results for agnostically learning low-degree polynomial threshold functions. In: SODA, pp. 1590–1606 (2011)Google Scholar
  20. 20.
    Diakonikolas, I., Servedio, R., Tan, L.-Y., Wan, A.: A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions. In: CCC, pp. 211–222 (2010)Google Scholar
  21. 21.
    Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, London (1968)zbMATHGoogle Scholar
  22. 22.
    Friedman, J., Wigderson, A.: On the second eigenvalue of hypergraphs. Combinatorica 15(1), 43–65 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goldmann, M., Håstad, J., Razborov, A.: Majority gates vs. general weighted threshold gates. Comput. Complex. 2, 277–300 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gopalan, P., Klivans, A., Meka, R., Stefankovic, D., Vempala, S., Vigoda, E.: An fptas for #knapsack and related counting problems. In: FOCS, pp. 817–826 (2011)Google Scholar
  25. 25.
    Golub, G., Van Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  26. 26.
    Gopalan, P., Meka, R., Reingold, O.: DNF sparsification and a faster deterministic counting algorithm. Comput. Complex. 22(2), 275–310 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gopalan, P., O’Donnell, R., Wu, Y., Zuckerman, D.: Fooling functions of halfspaces under product distributions. In: IEEE Conference on Computational Complexity (CCC), pp. 223–234 (2010)Google Scholar
  28. 28.
    Håstad, J.: On the size of weights for threshold gates. SIAM J. Discret. Math. 7(3), 484–492 (1994)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Janson, S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  30. 30.
    Kane, D.M.: The Gaussian surface area and noise sensitivity of degree-d polynomial threshold functions. In: CCC, pp. 205–210 (2010)Google Scholar
  31. 31.
    Kane, D.M.: k-independent gaussians fool polynomial threshold functions. In: IEEE Conference on Computational Complexity, pp. 252–261 (2011)Google Scholar
  32. 32.
    Kane, D.M.: A small PRG for polynomial threshold functions of gaussians. In: FOCS, pp. 257–266 (2011)Google Scholar
  33. 33.
    Kane, D.M.: The correct exponent for the Gotsman–Linial conjecture. arXiv:1210.1283 (2012)
  34. 34.
    Kane, D.M.: A pseudorandom generator for polynomial threshold functions of gaussian with subpolynomial seed length. arXiv:1210.1280 (2012)
  35. 35.
    Kane, D.M.: A structure theorem for poorly anticoncentrated gaussian chaoses and applications to the study of polynomial threshold functions. In: FOCS, pp. 91–100 (2012)Google Scholar
  36. 36.
    Kalai, A., Klivans, A., Mansour, Y., Servedio, R.: Agnostically learning halfspaces. SIAM J. Comput. 37(6), 1777–1805 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Karnin, Z.S., Rabani, Y., Shpilka, A.: Explicit dimension reduction and its applications. SIAM J. Comput. 41(1), 219–249 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Latala, R.: Estimates of moments and tails of gaussian chaoses. Ann. Probab. 34(6), 2315–2331 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Latala, R.: Personal communication (2013)Google Scholar
  40. 40.
    Ledoux, M.: Personal communication (2013)Google Scholar
  41. 41.
    Luby, M., Velickovic, B.: On deterministic approximation of DNF. Algorithmica 16(4/5), 415–433 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Luby, M., Velickovic, B., Wigderson, A.: Deterministic approximate counting of depth-2 circuits. In: Proceedings of the 2nd ISTCS, pp. 18–24 (1993)Google Scholar
  43. 43.
    Myhill, J., Kautz, W.: On the size of weights required for linear-input switching functions. IRE Trans. Electron. Comput. EC10(2), 288–290 (1961)CrossRefGoogle Scholar
  44. 44.
    Mossel, E., O’Donnell, R., Oleszkiewicz, K.K.: Noise stability of functions with low influences: invariance and optimality. Ann. Math. 171, 295–341 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Minsky, M., Papert, S.: Perceptrons: An Introduction to Computational Geometry. MIT Press, Cambridge (1968)zbMATHGoogle Scholar
  46. 46.
    Muroga, S., Toda, I., Takasu, S.: Theory of majority switching elements. J. Frankl. Inst. 271, 376–418 (1961)CrossRefzbMATHGoogle Scholar
  47. 47.
    Muroga, S.: Threshold Logic and Its Applications. Wiley-Interscience, New York (1971)zbMATHGoogle Scholar
  48. 48.
    Meka, R., Zuckerman, D.: Pseudorandom generators for polynomial threshold functions. http://arxiv.org/abs/0910.4122 (2009)
  49. 49.
    Meka, R., Zuckerman, D.: Pseudorandom generators for polynomial threshold functions. In: STOC, pp. 427–436 (2010)Google Scholar
  50. 50.
    Naor, J., Naor, M.: Small-bias probability spaces: efficient constructions and applications. SIAM J. Comput. 22(4), 838–856 (1993). (Earlier version in STOC’90) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nourdin, I.: Lectures on gaussian approximations with Malliavin calculus. Technical report. http://arxiv.org/abs/1203.4147v3, 28 June 2012
  52. 52.
    Nourdin, I.: Personal communication (2013)Google Scholar
  53. 53.
    Nualart, D., Peccati, G.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Nourdin, I., Peccati, G.: Stein’s method meets Malliavin calculus: a short survey with new estimates. Technical report. http://arxiv.org/abs/0906.4419v2, 17 Sep 2009
  55. 55.
    Nourdin, I., Peccati, G., Réveillac, A.: Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré Probab. Stat. 46(1), 45–58 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Oleszkiewicz, K.: Personal communication (2013)Google Scholar
  57. 57.
    Orponen, P.: Neural networks and complexity theory. In: Proceedings of the 17th International Symposium on Mathematical Foundations of Computer Science, pp. 50–61 (1992)Google Scholar
  58. 58.
    Podolskii, V.V.: Perceptrons of large weight. Probl. Inf. Transm. 45(1), 46–53 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Servedio, R.: Every linear threshold function has a low-weight approximator. Comput. Complex. 16(2), 180–209 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Sherstov, A.A.: Halfspace matrices. Comput. Complex. 17(2), 149–178 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Sherstov, A.: The intersection of two halfspaces has high threshold degree. In: Proceedings of 50th IEEE Symposium on Foundations of Computer Science (FOCS) (2009)Google Scholar
  62. 62.
    Shalev-Shwartz, S., Shamir, O., Sridharan, K.: Learning kernel-based halfspaces with the 0–1 loss. SIAM J. Comput. 40(6), 1623–1646 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Trevisan, L.: A note on approximate counting for \(k\)-DNF. In: Proceedings of the Eighth International Workshop on Randomization and Computation, pp. 417–426 (2004)Google Scholar
  64. 64.
    Viola, E.: The sum of \(d\) small-bias generators fools polynomials of degree \(d\). Comput. Complex. 18(2), 209–217 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA
  2. 2.Columbia UniversityNew YorkUSA

Personalised recommendations