Weight sensitive Boolean extraction produces compact expressions
Artificial neural networks are universal function approximators. The function actually implemented by a network is fully defined by its weights, but the representation in terms of weights is difficult for humans to understand or reason with. It is helpful to efficiently express the network's function in a symbolic form. Golea argues that extracting the minimum disjunctive normal form (DNF) of a network's function is difficult, but near-minimum DNF expressions may be extractable.
In an earlier work, we presented a technique for extracting a Boolean function from a single neuron efficiently. The computational complexity is linear in the size of the extracted expression. Our algorithm exploits the relative sizes of a neuron's weights to produce a natural and compact Boolean expression. We call this algorithm weight sensitive extraction. Tsukimoto and Morita recently presented an alternate technique using multilinear functions to extract Boolean functions in a disjunctive form. We show that the computational complexity of their algorithm is exponential in the length of each disjunct.
The two algorithms are compared in a series of experiments, and weight sensitive extraction is found to produce shorter expressions. We also examine our choice of weight ordering experimentally by using simulated annealing to find an order which produces a near-global minimum-length expression. We find that the order used in our earlier paper produces near-minimal expressions. Even in the cases where simulated annealing produces a better order, the difference in the length of the extracted expressions is only about one tenth of the expressions' length. This indicates that weight sensitivity is an important consideration in designing extraction algorithms to produce compact expressions.
Keywordsneural network rule extraction boolean expression weight sensitivity disjunctive normal form
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- 5.H. Tsukimoto and C. Morita, “An algorithm for extracting propositions from trained neural networks using multilinear functions,” in Rules and networks (R. Andrews and J. Diederich, eds.), pp. 103–114, Queensland University of Technology Neurocomputing Research Centre, Queensland University of Technology, April 1996.Google Scholar
- 7.R. Kane, L. Tchoumatchenko, and M. Milgram, “Extraction of knowledge from data using constrained neural networks,” in Machine Learning: European conference on machhine learning proceedings (P. B. Brazdil, ed.), vol. 667 of Lecture Notes in Artificial Intelligence, pp. 420–425, Springer-Verlag, 1993.Google Scholar
- 8.G. Towell and J. W. Shavlik, “Interpretation of artificial neural networks: mapping knowledge-based neural networks into rules,” in Advances in neural information processing systems, vol. 4, pp. 977–984, 1992.Google Scholar
- 9.M. Golea, “On the complexity of rule-extraction from neural networks and network querying,” in Rules and networks (R. Andrews and J. Diederich, eds.), pp. 51–59, Queensland University of Technology Neurocomputing Research Centre, Queensland University of Technology, April 1996.Google Scholar
- 10.M. L. Minski and S. A. Papert, Perceptrons: An introduction to computational geometry. Cambridge, MA: MIT press, 1988. Expanded version.Google Scholar
- 11.M. H. Hassoun, Fundamentals of Artificial Neural Networks, pp. 424–428. The MIT Press, 1995.Google Scholar