Deep neural networks and mixed integer linear optimization
- 385 Downloads
Deep Neural Networks (DNNs) are very popular these days, and are the subject of a very intense investigation. A DNN is made up of layers of internal units (or neurons), each of which computes an affine combination of the output of the units in the previous layer, applies a nonlinear operator, and outputs the corresponding value (also known as activation). A commonly-used nonlinear operator is the so-called rectified linear unit (ReLU), whose output is just the maximum between its input value and zero. In this (and other similar cases like max pooling, where the max operation involves more than one input value), for fixed parameters one can model the DNN as a 0-1 Mixed Integer Linear Program (0-1 MILP) where the continuous variables correspond to the output values of each unit, and a binary variable is associated with each ReLU to model its yes/no nature. In this paper we discuss the peculiarity of this kind of 0-1 MILP models, and describe an effective bound-tightening technique intended to ease its solution. We also present possible applications of the 0-1 MILP model arising in feature visualization and in the construction of adversarial examples. Computational results are reported, aimed at investigating (on small DNNs) the computational performance of a state-of-the-art MILP solver when applied to a known test case, namely, hand-written digit recognition.
KeywordsDeep neural networks Mixed-integer programming Deep learning Mathematical optimization Computational experiments
The research of the first author was partially funded by the Vienna Science and Technology Fund (WWTF) through project ICT15-014, any by MiUR, Italy, through project PRIN2015 “Nonlinear and Combinatorial Aspects of Complex Networks”. The research of the second author was funded by the Institute for Data Valorization (IVADO), Montreal. We thank Yoshua Bengio and Andrea Lodi for helpful discussions.
- 2.Cheng, C.-H., Nührenberg, G., Ruess, H. (2017). Maximum resilience of artificial neural networks. In D’Souza, D., & Narayan Kumar, K. (Eds.) Automated technology for verification and analysis (pp. 251–268). Cham: Springer International Publishing.Google Scholar
- 4.Erhan, D., Bengio, Y, Courville, A., Vincent, P. (2009). Visualizing higher-layer features of a deep network.Google Scholar
- 8.Goodfellow, I, Bengio, Y, Courville, A. (2016). Deep Learning. MIT Press. http://www.deeplearningbook.org.
- 9.ILOG IBM. Cplex 12.7 user’s manual (2017).Google Scholar
- 11.Nair, V., & Hinton, G.E. (2010). Rectified linear units improve restricted Boltzmann machines. In Fürnkranz, J, & Joachims, T (Eds.) Proceedings of the 27th International Conference on Machine Learning (ICML-10) (pp. 807–814): Omnipress.Google Scholar
- 13.Serra, T., Tjandraatmadja, C., Ramalingam, S. (2017). Bounding and counting linear regions of deep neural networks. CoRR arXiv:1711.02114.
- 14.Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I.J., Fergus, R. (2013). Intriguing properties of neural networks. CoRR arXiv:1312.6199.
- 15.Tjeng, V., & Tedrake, R. (2017). Verifying neural networks with mixed integer programming. CoRR arXiv:1711.07356.