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CLR-DRNets: Curriculum Learning with Restarts to Solve Visual Combinatorial Games

Authors Yiwei Bai, Di Chen, Carla P. Gomes

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Author Details

Yiwei Bai
  • Cornell University, Ithaca, NY, USA
Di Chen
  • Cornell University, Ithaca, NY, USA
Carla P. Gomes
  • Cornell University, Ithaca, NY, USA


We want to thank Wenting Zhao and anonymous reviewers for their valuable feedback.

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Yiwei Bai, Di Chen, and Carla P. Gomes. CLR-DRNets: Curriculum Learning with Restarts to Solve Visual Combinatorial Games. In 27th International Conference on Principles and Practice of Constraint Programming (CP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 210, pp. 17:1-17:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We introduce a curriculum learning framework for challenging tasks that require a combination of pattern recognition and combinatorial reasoning, such as single-player visual combinatorial games. Our work harnesses Deep Reasoning Nets (DRNets) [Chen et al., 2020], a framework that combines deep learning with constraint reasoning for unsupervised pattern demixing. We propose CLR-DRNets (pronounced Clear-DRNets), a curriculum-learning-with-restarts framework to boost the performance of DRNets. CLR-DRNets incrementally increase the difficulty of the training instances and use restarts, a new model selection method that selects multiple models from the same training trajectory to learn a set of diverse heuristics and apply them at inference time. An enhanced reasoning module is also proposed for CLR-DRNets to improve the ability of reasoning and generalize to unseen instances. We consider Visual Sudoku, i.e., Sudoku with hand-written digits or letters, and Visual Mixed Sudoku, a substantially more challenging task that requires the demixing and completion of two overlapping Visual Sudokus. We propose an enhanced reasoning module for the DRNets framework for encoding these visual games We show how CLR-DRNets considerably outperform DRNets and other approaches on these visual combinatorial games.

Subject Classification

ACM Subject Classification
  • Computing methodologies → Machine learning
  • Unsupervised Learning
  • Combinatorial Optimization


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  1. Brandon Amos and J Zico Kolter. Optnet: Differentiable optimization as a layer in neural networks. arXiv preprint, 2017. URL:
  2. Armin Biere and Andreas Fröhlich. Evaluating CDCL restart schemes. In Daniel Le Berre and Matti Järvisalo, editors, Proceedings of Pragmatics of SAT 2015, Austin, Texas, USA, September 23, 2015 / Pragmatics of SAT 2018, Oxford, UK, July 7, 2018, volume 59 of EPiC Series in Computing, pages 1-17. EasyChair, 2018. Google Scholar
  3. Céline Brouard, Simon de Givry, and Thomas Schiex. Pushing data into cp models using graphical model learning and solving. In International Conference on Principles and Practice of Constraint Programming, pages 811-827. Springer, 2020. Google Scholar
  4. Di Chen, Yiwei Bai, Wenting Zhao, Sebastian Ament, John Gregoire, and Carla Gomes. Deep reasoning networks for unsupervised pattern de-mixing with constraint reasoning. In International Conference on Machine Learning, pages 1500-1509. PMLR, 2020. Google Scholar
  5. Gregory Cohen, Saeed Afshar, Jonathan Tapson, and Andre Van Schaik. Emnist: Extending mnist to handwritten letters. In 2017 International Joint Conference on Neural Networks (IJCNN), pages 2921-2926. IEEE, 2017. Google Scholar
  6. Travis Dick, Eric Wong, and Christoph Dann. How many random restarts are enough. Technical report, Technical report, 2014. Google Scholar
  7. Dieqiao Feng, Carla P. Gomes, and Bart Selman. Solving hard AI planning instances using curriculum-driven deep reinforcement learning. In Christian Bessiere, editor, Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020, pages 2198-2205., 2020. URL:
  8. Matteo Gagliolo and Jürgen Schmidhuber. Learning restart strategies. In IJCAI, pages 792-797, 2007. Google Scholar
  9. Artur Garcez, Tarek R Besold, L d Raedt, Peter Földiak, Pascal Hitzler, Thomas Icard, Kai-Uwe Kühnberger, Luis C Lamb, Risto Miikkulainen, and Daniel L Silver. Neural-symbolic learning and reasoning: contributions and challenges, 2015. Google Scholar
  10. Carla P Gomes, Bart Selman, Henry Kautz, et al. Boosting combinatorial search through randomization. AAAI/IAAI, 98:431-437, 1998. Google Scholar
  11. Akhilesh Gotmare, Nitish Shirish Keskar, Caiming Xiong, and Richard Socher. A closer look at deep learning heuristics: Learning rate restarts, warmup and distillation. arXiv preprint, 2018. URL:
  12. Guy Hacohen and Daphna Weinshall. On the power of curriculum learning in training deep networks. arXiv preprint, 2019. URL:
  13. Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016. Google Scholar
  14. Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735-1780, 1997. Google Scholar
  15. Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278-2324, 1998. Google Scholar
  16. Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc De Raedt. Deepproblog: Neural probabilistic logic programming. In Advances in Neural Information Processing Systems, pages 3749-3759, 2018. Google Scholar
  17. Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. arXiv preprint, 2014. URL:
  18. Maxime Mulamba, Jayanta Mandi, Rocsildes Canoy, and Tias Guns. Hybrid classification and reasoning for image-based constraint solving. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pages 364-380. Springer, 2020. Google Scholar
  19. Sanmit Narvekar, Bei Peng, Matteo Leonetti, Jivko Sinapov, Matthew E Taylor, and Peter Stone. Curriculum learning for reinforcement learning domains: A framework and survey. arXiv preprint, 2020. URL:
  20. Rasmus Palm, Ulrich Paquet, and Ole Winther. Recurrent relational networks. In Advances in Neural Information Processing Systems, pages 3368-3378, 2018. Google Scholar
  21. Zhipeng Ren, Daoyi Dong, Huaxiong Li, and Chunlin Chen. Self-paced prioritized curriculum learning with coverage penalty in deep reinforcement learning. IEEE transactions on neural networks and learning systems, 29(6):2216-2226, 2018. Google Scholar
  22. Po-Wei Wang, Priya Donti, Bryan Wilder, and Zico Kolter. Satnet: Bridging deep learning and logical reasoning using a differentiable satisfiability solver. In International Conference on Machine Learning, pages 6545-6554, 2019. Google Scholar
  23. Daphna Weinshall, Gad Cohen, and Dan Amir. Curriculum learning by transfer learning: Theory and experiments with deep networks. arXiv preprint, 2018. URL:
  24. Fan Yang, Zhilin Yang, and William W Cohen. Differentiable learning of logical rules for knowledge base reasoning. In Advances in Neural Information Processing Systems, pages 2319-2328, 2017. Google Scholar
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