Abstract
Several hardness results are presented for the parent assignment problem: Given m observations of n attributes x 1, ..., x n , find the best parents for x n , that is, a subset of the preceding attributes so as to minimize a fixed cost function. This attribute or feature selection task plays an important role, e.g., in structure learning in Bayesian networks, yet little is known about its computational complexity. In this paper we prove that, under the commonly adopted full-multinomial likelihood model, the MDL, BIC, or AIC cost cannot be approximated in polynomial time to a ratio less than 2 unless there exists a polynomial-time algorithm for determining whether a directed graph with n nodes has a dominating set of size logn, a LOGSNP-complete problem for which no polynomial-time algorithm is known; as we also show, it is unlikely that these penalized maximum likelihood costs can be approximated to within any constant ratio. For the NML (normalized maximum likelihood) cost we prove an NP-completeness result. These results both justify the application of existing methods and motivate research on heuristic and super-polynomial-time algorithms.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Cooper, G.F., Herskovits, E.: A Bayesian method for the induction of probabilistic networks from data. Machine Learning 9, 309–347 (1992)
Heckerman, D., Geiger, D., Chickering, D.M.: Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning 20, 197–243 (1995)
Suzuki, J.: Learning Bayesian belief networks based on the Minimun Description Length principle: An efficient algorithm using the b & b technique. In: Proceedings of the Thirteenth International Conference on Machine Learning (ICML), pp. 462–470 (1996)
Tian, J.: A branch-and-bound algorithm for MDL learning Bayesian networks. In: Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI), pp. 580–588. Morgan Kaufmann, San Francisco (2000)
Rissanen, J.: Modeling by shortest data description. Automatica 14, 465–471 (1978)
Bouckaert, R.R.: Probabilistic network construction using the minimum description length principle. In: Moral, S., Kruse, R., Clarke, E. (eds.) ECSQARU 1993. LNCS, vol. 747, pp. 41–48. Springer, Heidelberg (1993)
Bouckaert, R.R.: Properties of Bayesian belief network learning algorithms. In: de Mantaras, R.L., Poole, D. (eds.) Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence (UAI), pp. 102–109. Morgan Kaufmann, San Francisco (1994)
Papadimitriou, C., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. Journal of Computer and System Sciences 53, 161–170 (1996)
Akaike, H.: A new look at the statistical model identification. IEEE Transactions on Automatic Control 19, 716–722 (1974)
Shtarkov, Y.M.: Universal sequential coding of single messages. Problems of Information Transmission 23, 3–17 (1987)
Kontkanen, P., Buntine, W., Myllymäki, P., Rissanen, J., Tirri, H.: Efficient computation of stochastic complexity. In: Bishop, C.M., Frey, B.J. (eds.) Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics (AISTAT), Key West, FL, pp. 181–188 (2003)
Schwarz, G.: Estimating the dimension of a model. Annals of Statistics 6, 461–464 (1978)
Cai, L., Juedes, D., Kanj, I.: The inapproximability of non-NP-hard optimization problems. Theoretical Computer Science 289, 553–571 (2002)
Garey, M., Johnson, D.: Computers and Intractability - A Guide to the Theory of NP-completeness. W. H. Freeman & Co., San Fransisco (1971)
Chickering, D.M., Meek, C.: Finding optimal Bayesian networks. In: Proceedings of Eighteenth Conference on Uncertainty in Artificial Intelligence (UAI), pp. 94–102. Morgan Kaufmann, Edmonton (2002)
Koller, D., Sahami, M.: Toward optimal feature selection. In: Proceedings of the Thirteenth International Conference on Machine Learning (ICML), pp. 284–292. Morgan Kaufmann, San Francisco (1996)
Charikar, M., Guruswami, V., Kumar, R., Rajagopalan, S., Sahai, A.: Combinatorial feature selection problems. In: Proceedings of the 41st IEEE Symposium on Foundations of Computer Science (FOCS), pp. 631–640. IEEE, Los Alamitos (2000)
Amaldi, E., Kann, V.: On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems. Theoretical Computer Science 209, 237–260 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koivisto, M. (2006). Parent Assignment Is Hard for the MDL, AIC, and NML Costs. In: Lugosi, G., Simon, H.U. (eds) Learning Theory. COLT 2006. Lecture Notes in Computer Science(), vol 4005. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11776420_23
Download citation
DOI: https://doi.org/10.1007/11776420_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35294-5
Online ISBN: 978-3-540-35296-9
eBook Packages: Computer ScienceComputer Science (R0)