Recently, deep equilibrium models (DEQs) have drawn increasing attention from the machine learning community. However, DEQs are much less understood in terms of certified robustness than their explicit network counterparts. In this paper, we advance the understanding of certified robustness of DEQs via exploiting the connections between various Lipschitz network parameteriza-tions for both explicit and implicit models. Importantly, we show that various popular Lipschitz network structures, including convex potential layers (CPL), SDP-based Lipschitz layers (SLL), almost orthogonal layers (AOL), Sandwich layers, and monotone DEQs (MonDEQ) can all be reparameterized as special cases of the Lipschitz-bounded equilibrium networks (LBEN) without changing the prescribed Lipschitz constant in the original network parameterization. A key feature of our reparameterization technique is that it preserves the Lip-schitz prescription used in different structures. This opens the possibility of achieving improved certified robustness of DEQs via a combination of network reparameterization, structure-preserving regularization, and LBEN-based fine-tuning.

Implicit-depth neural networks have grown as powerful alternatives to traditional networks in various applications in recent years. However, these models often lack guarantees of existence and uniqueness, raising stability, performance, and reproducibility issues. In this paper, we present a new analysis of the existence and uniqueness of fixed points for implicit-depth neural networks based on the concept of subhomogeneous operators and the nonlinear Perron-Frobenius theory. Compared to previous similar analyses, our theory allows for weaker assumptions on the parameter matrices, thus yielding a more flexible framework for well-defined implicit networks. We illustrate the performance of the resulting subhomogeneous networks on feedforward, convolutional, and graph neural network examples.

Monotone Operator Equilibrium Models (monDEQs) represent a class of models combining the powerful deep equilibrium paradigm with convergence guarantees. Further, their inherent robustness to adversarial perturbations makes investigating their certifiability a promising research direction. Unfortunately, existing approaches are either imprecise or severely limited in scalability. In this work, we propose the first scalable and precise monDEQ verifier, based on two key ideas: (i) a novel convex relaxation enabling efficient inclusion checks, and (ii) non-trivial mathematical insights characterizing the fixpoint operations at the heart of monDEQs on sets rather than concrete inputs. An extensive evaluation of our verifier on the challenging $\ell_\infty$ perturbations demonstrates that it exceeds state-of-the-art performance in terms of speed (two orders of magnitude) and scalability (an order of magnitude) while yielding 25% higher certified accuracies on the same networks.