# Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification

1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
2 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
IRIT - Institut de recherche en informatique de Toulouse
Abstract : Deep equilibrium models are based on implicitly defined functional relations and have shown competitive performance compared with the traditional deep networks. Monotone operator equilibrium networks (monDEQ) retain interesting performance with additional theoretical guaranties. Existing certification tools for classical deep networks cannot directly be applied to monDEQs for which much fewer tools exist. We introduce a semialgebraic representation for ReLU based monDEQs which allows to approximate the corresponding input output relation by semidefinite programming (SDP). We present several applications to network certification and obtain SDP models for the following problems : robustness certification, Lipschitz constant estimation, ellipsoidal uncertainty propagation. We use these models to certify robustness of monDEQs w.r.t. a general $L_q$ norm. Experimental results show that the proposed models outperform existing approaches for monDEQ certification. Furthermore, our investigations suggest that monDEQs are much more robust to $L_2$ perturbations than $L_{\infty}$ perturbations.
Document type :
Preprints, Working Papers, ...

https://hal.laas.fr/hal-03265346
Contributor : Tong Chen <>
Submitted on : Sunday, June 20, 2021 - 1:44:02 AM
Last modification on : Friday, June 25, 2021 - 3:35:47 AM

### Identifiers

• HAL Id : hal-03265346, version 1
• ARXIV : 2106.01453

### Citation

Tong Chen, Jean-Bernard Lasserre, Victor Magron, Edouard Pauwels. Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification. 2021. ⟨hal-03265346⟩

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