Computing Optimal Decision Sets with SAT | SpringerLink
Skip to main content
Springer Nature Link
Log in

Computing Optimal Decision Sets with SAT

  • Conference paper
  • First Online:
Principles and Practice of Constraint Programming (CP 2020)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12333))

Abstract

As machine learning is increasingly used to help make decisions, there is a demand for these decisions to be explainable. Arguably, the most explainable machine learning models use decision rules. This paper focuses on decision sets, a type of model with unordered rules, which explains each prediction with a single rule. In order to be easy for humans to understand, these rules must be concise. Earlier work on generating optimal decision sets first minimizes the number of rules, and then minimizes the number of literals, but the resulting rules can often be very large. Here we consider a better measure, namely the total size of the decision set in terms of literals. So we are not driven to a small set of rules which require a large number of literals. We provide the first approach to determine minimum-size decision sets that achieve minimum empirical risk and then investigate sparse alternatives where we trade accuracy for size. By finding optimal solutions we show we can build decision set classifiers that are almost as accurate as the best heuristic methods, but far more concise, and hence more explainable.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 17159
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 21449
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    https://www.starexec.org/.

  2. 2.

    The prototype adapts all the developed models to the case of multiple classes, which is motivated by the practical importance of non-binary classification.

  3. 3.

    This average value is the highest possible accuracy that can be achieved on these datasets whatever machine learning model is considered.

  4. 4.

    In a unit-size decision set, the literal is meant to assign a constant class. This can be seen as applying a default rule.

References

  1. Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks and their applications. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 167–180. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_18

    Chapter  MATH  Google Scholar 

  2. Audemard, G., Lagniez, J.-M., Simon, L.: Improving glucose for incremental SAT solving with assumptions: application to MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 309–317. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39071-5_23

    Chapter  MATH  Google Scholar 

  3. Australian Government.: Artificial Intelligence Roadmap, November 2019. https://data61.csiro.au/en/Our-Research/Our-Work/AI-Roadmap

  4. Baehrens, D., Schroeter, T., Harmeling, S., Kawanabe, M., Hansen, K., Müller, K.: How to explain individual classification decisions. J. Mach. Learn. Res. 11, 1803–1831 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Bailleux, O., Boufkhad, Y.: Efficient CNF encoding of boolean cardinality constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45193-8_8

    Chapter  MATH  Google Scholar 

  6. Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998)

    Article  MathSciNet  Google Scholar 

  7. Batcher, K.E.: Sorting networks and their applications. In: AFIPS, vol. 32, pp. 307–314 (1968)

    Google Scholar 

  8. Bessiere, C., Hebrard, E., O’Sullivan, B.: Minimising decision tree size as combinatorial optimisation. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 173–187. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04244-7_16

    Chapter  Google Scholar 

  9. Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. IOS Press, Amsterdam (2009)

    MATH  Google Scholar 

  10. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Wadsworth, Belmont (1984)

    MATH  Google Scholar 

  11. Clark, P., Boswell, R.: Rule induction with CN2: some recent improvements. In: Kodratoff, Y. (ed.) EWSL 1991. LNCS, vol. 482, pp. 151–163. Springer, Heidelberg (1991). https://doi.org/10.1007/BFb0017011

    Chapter  Google Scholar 

  12. Clark, P., Niblett, T.: The CN2 induction algorithm. Mach. Learn. 3, 261–283 (1989)

    Google Scholar 

  13. Cohen, W.: Fast effective rule induction. In: ICML, pp. 115–123 (1995)

    Google Scholar 

  14. Darwiche, A.: Three modern roles for logic in AI. In: PODS, pp. 229–243. ACM (2020)

    Google Scholar 

  15. Dash, S., Günlük, O., Wei, D.: Boolean decision rules via column generation. In: NeurIPS, pp. 4660–4670 (2018)

    Google Scholar 

  16. Doshi-Velez, F., Kim, B.: A roadmap for a rigorous science of interpretability. arXiv preprint arXiv:1702.08608 (2017)

  17. EU Data Protection Regulation.: Regulation (EU) 2016/679 of the European Parliament and of the Council (2016). http://eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32016R0679&from=en

  18. Evans, R., Grefenstette, E.: Learning explanatory rules from noisy data. J. Artif. Intell. Res. 61, 1–64 (2018)

    Article  MathSciNet  Google Scholar 

  19. Fürnkranz, J., Gamberger, D., Lavrac, N.: Foundations of Rule Learning. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-540-75197-7

    Book  MATH  Google Scholar 

  20. Ghosh B., Meel, K.S.: IMLI: an incremental framework for MAXSAT-based learning of interpretable classification rules. In: AIES, pp. 203–210. ACM (2019)

    Google Scholar 

  21. Goodman, B., Flaxman, S.R.: European Union regulations on algorithmic decision-making and a “right to explanation”. AI Mag. 38(3), 50–57 (2017)

    Article  Google Scholar 

  22. Guidotti, R., Monreale, A., Giannotti, F., Pedreschi, D., Ruggieri, S., Turini, F.: Factual and counterfactual explanations for black box decision making. IEEE Intell. Syst. 34(6), 14–23 (2019)

    Article  Google Scholar 

  23. Guidotti, R., Monreale, A., Ruggieri, S., Turini, F., Giannotti, F., Pedreschi, D.: A survey of methods for explaining black box models. ACM Comput. Surv. 51(5), 93:1–93:42 (2019)

    Article  Google Scholar 

  24. Gunning, D., Aha, D.: DARPA’s explainable artificial intelligence (XAI) program. AI Mag. 40(2), 44–58 (2019)

    Article  Google Scholar 

  25. Han, J., Kamber, M., Pei, J.: Data Mining: Concepts and Techniques, 3rd edn. Morgan Kaufmann, San Francisco (2012)

    MATH  Google Scholar 

  26. Ignatiev, A.: Towards trustable explainable AI. In: IJCAI, pp. 5154–5158 (2020)

    Google Scholar 

  27. Ignatiev, A., Morgado, A., Marques-Silva, J.: PySAT: a python toolkit for prototyping with SAT oracles. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 428–437. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_26

    Chapter  MATH  Google Scholar 

  28. Ignatiev, A., Morgado, A., Marques-Silva, J.: RC2: an efficient MaxSAT solver. J. Satisf. Boolean Model. Comput. 11(1), 53–64 (2019)

    Google Scholar 

  29. Ignatiev, A., Narodytska, N., Marques-Silva, J.: Abduction-based explanations for machine learning models. In: AAAI, pp. 1511–1519 (2019)

    Google Scholar 

  30. Ignatiev, A., Narodytska, N., Marques-Silva, J.: On relating explanations and adversarial examples. In: NeurIPS, pp. 15857–15867 (2019)

    Google Scholar 

  31. Ignatiev, A., Pereira, F., Narodytska, N., Marques-Silva, J.: A SAT-based approach to learn explainable decision sets. In: IJCAR, pp. 627–645 (2018)

    Google Scholar 

  32. Jordan, M.I., Mitchell, T.M.: Machine learning: trends, perspectives, and prospects. Science 349(6245), 255–260 (2015)

    Article  MathSciNet  Google Scholar 

  33. Kamath, A.P., Karmarkar, N., Ramakrishnan, K.G., Resende, M.G.C.: A continuous approach to inductive inference. Math. Program. 57, 215–238 (1992)

    Article  MathSciNet  Google Scholar 

  34. Lakkaraju, H., Bach, S.H., Leskovec, J.: Interpretable decision sets: a joint framework for description and prediction. In: KDD, pp. 1675–1684 (2016)

    Google Scholar 

  35. LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015)

    Article  Google Scholar 

  36. Li, O., Liu, H., Chen, C., Rudin, C.: Deep learning for case-based reasoning through prototypes: a neural network that explains its predictions. In: AAAI, February 2018

    Google Scholar 

  37. Lipton, Z.C.: The mythos of model interpretability. Commun. ACM 61(10), 36–43 (2018)

    Article  Google Scholar 

  38. Lundberg, S.M., Lee, S.: A unified approach to interpreting model predictions. In: NIPS, pp. 4765–4774 (2017)

    Google Scholar 

  39. Maliotov, D., Meel, K.S.: MLIC: a MaxSAT-based framework for learning interpretable classification rules. In: Hooker, J. (ed.) CP 2018. LNCS, vol. 11008, pp. 312–327. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98334-9_21

    Chapter  Google Scholar 

  40. Mitchell, T.M.: Machine Learning. McGraw-Hill, New York (1997)

    MATH  Google Scholar 

  41. Mnih, V., et al.: Human-level control through deep reinforcement learning. Nature 518(7540), 529 (2015)

    Article  Google Scholar 

  42. Monroe, D.: AI, explain yourself. Commun. ACM 61(11), 11–13 (2018)

    Article  Google Scholar 

  43. Montavon, G., Samek, W., Müller, K.: Methods for interpreting and understanding deep neural networks. Digit. Signal Proc. 73, 1–15 (2018)

    Article  MathSciNet  Google Scholar 

  44. Narodytska, N., Ignatiev, A., Pereira, F., Marques-Silva, J.: Learning optimal decision trees with SAT. In: IJCAI, pp. 1362–1368 (2018)

    Google Scholar 

  45. Orange: A component-based data mining framework. https://orange.biolab.si/

  46. Pedregosa, F., et al.: Scikit-learn: machine learning in python. J. Mach. Learn. Res. 12, 2825–2830 (2011)

    MathSciNet  MATH  Google Scholar 

  47. Quinlan, J.R.: C4.5: Programs for Machine Learning. Morgan Kauffmann, San Mateo (1993)

    Google Scholar 

  48. Ribeiro, M.T., Singh, S., Guestrin, C.: “Why should I trust you?”: explaining the predictions of any classifier. In: KDD, pp. 1135–1144 (2016)

    Google Scholar 

  49. Ribeiro, M.T., Singh, S., Guestrin C.: Anchors: high-precision model-agnostic explanations. In: AAAI (2018)

    Google Scholar 

  50. Rivest, R.L.: Learning decision lists. Mach. Learn. 2(3), 229–246 (1987)

    Google Scholar 

  51. SAT-based miner of smallest size decision sets. https://github.com/alexeyignatiev/minds

  52. Shih, A., Choi, A., Darwiche, A.: A symbolic approach to explaining bayesian network classifiers. In: IJCAI, pp. 5103–5111 (2018)

    Google Scholar 

  53. Sinz, C.: Towards an optimal CNF encoding of boolean cardinality constraints. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005). https://doi.org/10.1007/11564751_73

    Chapter  MATH  Google Scholar 

  54. Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, Part II, pp. 115–125. Consultants Bureau, New York (1968)

    Google Scholar 

  55. UCI Machine Learning Repository. https://archive.ics.uci.edu/ml

  56. Ruleset covering algorithms for transparent machine learning. https://github.com/imoscovitz/wittgenstein

Download references

Acknowledgments

This work is partially supported by the Australian Research Council through Discovery Grant DP170103174.

Author information

Authors and Affiliations

  1. Faculty of Information Technology, Monash University, Melbourne, Australia

    Jinqiang Yu, Alexey Ignatiev, Peter J. Stuckey & Pierre Le Bodic

Authors
  1. Jinqiang Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexey Ignatiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peter J. Stuckey
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pierre Le Bodic
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexey Ignatiev .

Editor information

Editors and Affiliations

  1. Insight Centre for Data Analytics, School for Computer Science and Information Technology, University College Cork, Cork, Ireland

    Helmut Simonis

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Yu, J., Ignatiev, A., Stuckey, P.J., Le Bodic, P. (2020). Computing Optimal Decision Sets with SAT. In: Simonis, H. (eds) Principles and Practice of Constraint Programming. CP 2020. Lecture Notes in Computer Science(), vol 12333. Springer, Cham. https://doi.org/10.1007/978-3-030-58475-7_55

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-58475-7_55

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58474-0

  • Online ISBN: 978-3-030-58475-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation