Integral Quadratic Constraints for Repeated ReLU
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912972215943168 |
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| author | Noori, Sahel Vahedi Hu, Bin Dullerud, Geir Seiler, Peter |
| author_facet | Noori, Sahel Vahedi Hu, Bin Dullerud, Geir Seiler, Peter |
| contents | This paper presents a new dynamic integral quadratic constraint (IQC) for the repeated Rectified Linear Unit (ReLU). These dynamic IQCs can be used to analyze stability and induced $\ell_2$-gain performance of discrete-time, recurrent neural networks (RNNs) with ReLU activation functions. These analysis conditions can be incorporated into learning-based controller synthesis methods, which currently rely on static IQCs. We show that our proposed dynamic IQCs for repeated ReLU form a superset of the dynamic IQCs for repeated, slope-restricted nonlinearities. We also prove that the $\ell_2$-gain bounds are nonincreasing with respect to the horizon used in the dynamic IQC filter. A numerical example using a simple (academic) RNN shows that our proposed IQCs lead to less conservative bounds than existing IQCs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17137 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Integral Quadratic Constraints for Repeated ReLU Noori, Sahel Vahedi Hu, Bin Dullerud, Geir Seiler, Peter Systems and Control This paper presents a new dynamic integral quadratic constraint (IQC) for the repeated Rectified Linear Unit (ReLU). These dynamic IQCs can be used to analyze stability and induced $\ell_2$-gain performance of discrete-time, recurrent neural networks (RNNs) with ReLU activation functions. These analysis conditions can be incorporated into learning-based controller synthesis methods, which currently rely on static IQCs. We show that our proposed dynamic IQCs for repeated ReLU form a superset of the dynamic IQCs for repeated, slope-restricted nonlinearities. We also prove that the $\ell_2$-gain bounds are nonincreasing with respect to the horizon used in the dynamic IQC filter. A numerical example using a simple (academic) RNN shows that our proposed IQCs lead to less conservative bounds than existing IQCs. |
| title | Integral Quadratic Constraints for Repeated ReLU |
| topic | Systems and Control |
| url | https://arxiv.org/abs/2603.17137 |