Abstract
Integrating machine learning with automated reasoning is one of the major goals of modern AI systems. In this paper, we propose a non-fully-differentiable architecture that is able to extract preferences from data and push it into (weighted) Constraint Networks (aka Cost Function Networks or CFN) by learning cost functions. Our approach combines a (scalable) convex optimization approach with empirical hyper-parameter tuning to learn cost functions from a list of high-quality solutions. The proposed architecture has the ability to learn from noisy solutions and its output is just a CFN model. This model can be analyzed, empirically hardened, completed with side-constraints, and directly fed to a Weighted Constraint Satisfaction Problem solver.
To explore the performances and range of applicability of this architecture, we compare it with two recent neural-net friendly learning systems designed to “learn to reason” on the Sudoku problem and also show how it can be used to learn and integrate preferences into an existing CP model, in the context of Configuration systems.
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- 1.
This \(\log \) representation is often using in MRFs and the co-domain of factors is called “energy”.
- 2.
The weights learned in the convex relaxation are floating-point numbers. A precise integer approximation generates large integer costs which are usually not the sweet spot of the most efficient, core-based, Max-SAT solvers [28].
- 3.
- 4.
We removed the first variable corresponding to the date of each sale product.
- 5.
Solution counting was done by Backtracking with Tree Decomposition algorithm [15] using min-fill heuristic implemented in toulbar2 v1.0.1 with options -ub=1 -a -O=-3 -B=1 -hbfs: -nopre Reported tree-width was 10 for medium and 12 for big instance.
- 6.
We ensure our CFN and the XCSP2.1 XML file for the constraints use the same variable domains with the same increasing value ordering.
- 7.
We used toulbar2 v1.0.1 with a limit of 50,000 backtracks and no preprocessing.
- 8.
We implemented an incremental version of the toulbar2 solving procedure using its Python interface in order to load the problem and preprocess it only once.
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Acknowledgments
We thank the GenoToul (Toulouse, France) Bioinformatics and IFB Core (Evry, France) platforms for their computational support. We also thank the reviewers for their critics: the paper did improve, we think. This work has been supported by the French ANR through grants ANR-16-CE40-0028 and ANR-19-PIA3-0004.
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Brouard, C., de Givry, S., Schiex, T. (2020). Pushing Data into CP Models Using Graphical Model Learning and Solving. 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_47
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