Lipschitz Bounded Equilibrium Networks

TL;DR

This paper proposes a new parameterization for equilibrium neural networks that maintains Lipschitz bounds during training, enhancing robustness and generalization, with theoretical guarantees and empirical validation on image classification tasks.

Contribution

Introduces a novel Lipschitz-bounded parameterization for equilibrium networks with relaxed conditions and theoretical insights linking to convex optimization and neural ODEs.

Findings

Lipschitz bounds are accurate and improve adversarial robustness.

The model includes standard networks as special cases.

Theoretical guarantees for well-posedness under natural conditions.

Abstract

This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. This model class includes standard multilayer and residual networks as special cases. The new parameterization admits a Lipschitz bound during training via unconstrained optimization: no projections or barrier functions are required. Lipschitz bounds are a common proxy for robustness and appear in many generalization bounds. Furthermore, compared to previous works we show well-posedness (existence of solutions) under less restrictive conditions on the network weights and more natural assumptions on the activation functions: that they are monotone and slope restricted. These results are proved by establishing novel connections with convex optimization, operator splitting on non-Euclidean spaces, and contracting neural ODEs. In image classification experiments we show that the Lipschitz bounds are very accurate and improve robustness to adversarial attacks.