Abstract
A wide range of positive and negative results have been established for learning different classes of Boolean functions from uniformly distributed random examples. However, polynomial-time algorithms have thus far been obtained almost exclusively for various classes of monotone functions, while the computational hardness results obtained to date have all been for various classes of general (nonmonotone) functions. Motivated by this disparity between known positive results (for monotone functions) and negative results (for nonmonotone functions), we establish strong computational limitations on the efficient learnability of various classes of monotone functions.
We give several such hardness results which are provably almost optimal since they nearly match known positive results. Some of our results show cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than \(1/2 + 1/\sqrt{n}\); this accuracy bound is close to optimal by known positive results (Blum et al., FOCS ’98). Other results show that under a plausible cryptographic hardness assumption, a class of constant-depth, sub-polynomial-size circuits computing monotone functions is hard to learn; this result is close to optimal in terms of the circuit size parameter by known positive results as well (Servedio, Information and Computation ’04). Our main tool is a complexity-theoretic approach to hardness amplification via noise sensitivity of monotone functions that was pioneered by O’Donnell (JCSS ’04).
Supported by NSF award CNS-0716245, DARPA grant HR0011-07-1-0012, NSF CAREER award CCF-0347282, NSF CAREER CCF-03-047839, NSF award SBE-0245014, NSF award CCF-0523664, and a Sloan Foundation Fellowship.
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References
Allender, E., Hellerstein, L., McCabe, P., Pitassi, T., Saks, M.: Minimizing DNF Formulas and \(AC^0_d\) Circuits Given a Truth Table. In: CCC, pp. 237–251 (2006)
Blum, A., Burch, C., Langford, J.: On learning monotone boolean functions. In: 39th FOCS, pp. 408–415 (1998)
Berkowitz, S.J.: On some relationships between monotone and non-monotone circuit complexity. Technical Report, University of Toronto (1982)
Blum, A., Furst, M., Kearns, M., Lipton, R.: Cryptographic Primitives Based on Hard Learning Problems. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 278–291. Springer, Heidelberg (1993)
Bshouty, N., Tamon, C.: On the Fourier spectrum of monotone functions. Journal of the ACM 43(4), 747–770 (1996)
Healy, A., Vadhan, S., Viola, E.: Using Nondeterminism to Amplify Hardness. SIAM Journal on Computing 35(4), 903–931 (2006)
Kharitonov, M.: Cryptographic hardness of distribution-specific learning. In: 25th STOC, pp. 372–381 (1993)
Kharitonov, M.: Cryptographic lower bounds for learnability of Boolean functions on the uniform distribution. JCSS 50, 600–610 (1995)
Kahn, J., Kalai, G., Linial, N.: The influence of variables on boolean functions. In: 29th FOCS, pp. 68–80 (1988)
Kearns, M.J., Li, M., Valiant, L.G.: Learning boolean formulas. J. ACM 41(6), 1298–1328 (1994)
Klivans, A., O’Donnell, R., Servedio, R.: Learning intersections and thresholds of halfspaces. JCSS 68(4), 808–840 (2004)
Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform and learnability. Journal of the ACM 40(3), 607–620 (1993)
Mansour, Y.: Learning Boolean functions via the Fourier transform, pp. 391–424. Kluwer Academic Publishers, Dordrecht (1994)
Mossel, E., O’Donnell, R.: On the noise sensitivity of monotone functions. Random Struct. Algorithms 23(3), 333–350 (2003)
Mossel, E., O’Donnell, R., Servedio, R.: Learning functions of k relevant variables. J. Comput. & Syst. Sci. 69(3), 421–434 (2004)
Nepomnjaščil̆, V.A.: Rudimentary predicates and Turing calculations. Math Dokl. 11, 1462–1465 (1970)
Naor, M., Reingold, O.: Number-theoretic constructions of efficient pseudo-random functions. Journal of the ACM 51(2), 231–262 (2004)
O’Donnell, R.: Hardness amplification within NP. JCSS 69(1), 68–94 (2004)
O’Donnell, R., Servedio, R.: Learning monotone decision trees in polynomial time. SIAM Journal on Computing 37(3), 827–844 (2007)
Razborov, A.: Lower bounds on the monotone network complexity of the logical permanent. Mat. Zametki 37, 887–900 (1985)
Servedio, R.: On learning monotone DNF under product distributions. Information and Computation 193(1), 57–74 (2004)
Valiant, L.: A theory of the learnable. CACM 27(11), 1134–1142 (1984)
Verbeurgt, K.: Learning DNF under the uniform distribution in quasi-polynomial time. In: 3rd COLT, pp. 314–326 (1990)
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Dachman-Soled, D., Lee, H.K., Malkin, T., Servedio, R.A., Wan, A., Wee, H. (2008). Optimal Cryptographic Hardness of Learning Monotone Functions. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_4
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