Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929428052836352 |
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| author | McKenzie, Daniel Fung, Samy Wu Heaton, Howard |
| author_facet | McKenzie, Daniel Fung, Samy Wu Heaton, Howard |
| contents | In many applications, a combinatorial problem must be repeatedly solved with similar, but distinct parameters. Yet, the parameters $w$ are not directly observed; only contextual data $d$ that correlates with $w$ is available. It is tempting to use a neural network to predict $w$ given $d$. However, training such a model requires reconciling the discrete nature of combinatorial optimization with the gradient-based frameworks used to train neural networks. We study the case where the problem in question is an Integer Linear Program (ILP). We propose applying a three-operator splitting technique, also known as Davis-Yin splitting (DYS), to the quadratically regularized continuous relaxation of the ILP. We prove that the resulting scheme is compatible with the recently introduced Jacobian-free backpropagation (JFB). Our experiments on two representative ILPs: the shortest path problem and the knapsack problem, demonstrate that this combination-DYS on the forward pass, JFB on the backward pass-yields a scheme which scales more effectively to high-dimensional problems than existing schemes. All code associated with this paper is available at github.com/mines-opt-ml/fpo-dys. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13395 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting McKenzie, Daniel Fung, Samy Wu Heaton, Howard Machine Learning In many applications, a combinatorial problem must be repeatedly solved with similar, but distinct parameters. Yet, the parameters $w$ are not directly observed; only contextual data $d$ that correlates with $w$ is available. It is tempting to use a neural network to predict $w$ given $d$. However, training such a model requires reconciling the discrete nature of combinatorial optimization with the gradient-based frameworks used to train neural networks. We study the case where the problem in question is an Integer Linear Program (ILP). We propose applying a three-operator splitting technique, also known as Davis-Yin splitting (DYS), to the quadratically regularized continuous relaxation of the ILP. We prove that the resulting scheme is compatible with the recently introduced Jacobian-free backpropagation (JFB). Our experiments on two representative ILPs: the shortest path problem and the knapsack problem, demonstrate that this combination-DYS on the forward pass, JFB on the backward pass-yields a scheme which scales more effectively to high-dimensional problems than existing schemes. All code associated with this paper is available at github.com/mines-opt-ml/fpo-dys. |
| title | Differentiating Through Integer Linear Programs with Quadratic Regularization and Davis-Yin Splitting |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2301.13395 |