#### Robust Implicit Networks via Non-Euclidean Contractions

##### Saber Jafarpour, Alexander Davydov, Anton V. Proskurnikov, Francesco Bullo

Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed point equation. They generalize classic feedforward models and are equivalent to infinite-depth weight-tied feedforward networks. While implicit models show improved accuracy and significant reduction in memory consumption, they can suffer from ill-posedness and convergence instability. This paper provides a new framework to design well-posed and robust implicit neural networks based upon contraction theory for the non-Euclidean norm $\ell_\infty$. Our framework includes (i) a novel condition for well-posedness based on one-sided Lipschitz constants, (ii) an average iteration for computing fixed-points, and (iii) explicit estimates on input-output Lipschitz constants. Additionally, we design a training problem with the well-posedness condition and the average iteration as constraints and, to achieve robust models, with the input-output Lipschitz constant as a regularizer. Our $\ell_\infty$ well-posedness condition leads to a larger polytopic training search space than existing conditions and our average iteration enjoys accelerated convergence. Finally, we perform several numerical experiments for function estimation and digit classification through the MNIST data set. Our numerical results demonstrate improved accuracy and robustness of the implicit models with smaller input-output Lipschitz bounds.

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