# Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operators

## TL;DR

This paper develops a method to compute tight Lipschitz constants for layered neural networks using averaged operators, improving robustness assessment by capturing layer interactions more accurately.

## Contribution

It introduces a novel framework for deriving sharp Lipschitz bounds for layered networks with averaged operators, surpassing traditional product-based estimates.

## Key findings

- Tighter Lipschitz constants than traditional bounds.
- Applicable to standard convolutional neural networks.
- Enhanced robustness evaluation for neural network models.

## Abstract

Obtaining sharp Lipschitz constants for feed-forward neural networks is essential to assess their robustness in the face of perturbations of their inputs. We derive such constants in the context of a general layered network model involving compositions of nonexpansive averaged operators and affine operators. By exploiting this architecture, our analysis finely captures the interactions between the layers, yielding tighter Lipschitz constants than those resulting from the product of individual bounds for groups of layers. The proposed framework is shown to cover in particular many practical instances encountered in feed-forward neural networks. Our Lipschitz constant estimates are further improved in the case of structures employing scalar nonlinear functions, which include standard convolutional networks as special cases.