Exploiting Connections between Lipschitz Structures for Certifiably Robust Deep Equilibrium Models Aaron J. Havens

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.