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Deep equilibrium models are based on implicitly defined functional relationsand have shown competitive performance compared with the traditional deepnetworks. Monotone operator equilibrium networks (monDEQ) retain interestingperformance with additional theoretical guaranties. Existing certificationtools for classical deep networks cannot directly be applied to monDEQs forwhich much fewer tools exist. We introduce a semialgebraic representation forReLU based monDEQs which allows to approximate the corresponding input outputrelation by semidefinite programming (SDP). We present several applications tonetwork certification and obtain SDP models for the following problems :robustness certification, Lipschitz constant estimation, ellipsoidaluncertainty propagation. We use these models to certify robustness of monDEQsw.r.t. a general $L_q$ norm. Experimental results show that the proposed modelsoutperform existing approaches for monDEQ certification. Furthermore, ourinvestigations suggest that monDEQs are much more robust to $L_2$ perturbationsthan $L_{\infty}$ perturbations.

Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification