Strong mixed-integer programming formulations for trained neural networks
We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial i...
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| Published in | Mathematical programming Vol. 183; no. 1-2; pp. 3 - 39 |
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| Main Authors | , , , , |
| Format | Journal Article |
| Language | English |
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Berlin/Heidelberg
Springer Berlin Heidelberg
01.09.2020
Springer Nature B.V |
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| Abstract | We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of
d
affine functions over arbitrary polyhedral input domains. We apply this result to derive MIP formulations for a number of the most popular nonlinear operations (e.g. ReLU and max pooling) that are strictly stronger than other approaches from the literature. We corroborate this computationally, showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks. |
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| AbstractList | We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains. We apply this result to derive MIP formulations for a number of the most popular nonlinear operations (e.g. ReLU and max pooling) that are strictly stronger than other approaches from the literature. We corroborate this computationally, showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks. We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks. These formulations can be used for a number of important tasks, such as verifying that an image classification network is robust to adversarial inputs, or solving decision problems where the objective function is a machine learning model. We present a generic framework, which may be of independent interest, that provides a way to construct sharp or ideal formulations for the maximum of d affine functions over arbitrary polyhedral input domains. We apply this result to derive MIP formulations for a number of the most popular nonlinear operations (e.g. ReLU and max pooling) that are strictly stronger than other approaches from the literature. We corroborate this computationally, showing that our formulations are able to offer substantial improvements in solve time on verification tasks for image classification networks. |
| Author | Vielma, Juan Pablo Anderson, Ross Ma, Will Huchette, Joey Tjandraatmadja, Christian |
| Author_xml | – sequence: 1 givenname: Ross surname: Anderson fullname: Anderson, Ross organization: Google Inc – sequence: 2 givenname: Joey orcidid: 0000-0003-3552-0316 surname: Huchette fullname: Huchette, Joey email: joehuchette@rice.edu organization: Rice University – sequence: 3 givenname: Will orcidid: 0000-0002-2420-4468 surname: Ma fullname: Ma, Will organization: Columbia University – sequence: 4 givenname: Christian surname: Tjandraatmadja fullname: Tjandraatmadja, Christian organization: Google Inc – sequence: 5 givenname: Juan Pablo surname: Vielma fullname: Vielma, Juan Pablo organization: Google Inc, Massachusetts Institute of Technology |
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| Copyright | Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020 Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2020. |
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| Snippet | We present strong mixed-integer programming (MIP) formulations for high-dimensional piecewise linear functions that correspond to trained neural networks.... |
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| SubjectTerms | Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Image classification Integer programming Linear functions Linear programming Machine learning Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Neural networks Numerical Analysis Theoretical |
| Title | Strong mixed-integer programming formulations for trained neural networks |
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