ExLipBaB: Exact Lipschitz Constant Computation for Piecewise Linear Neural Networks

TL;DR

This paper introduces ExLipBaB, an algorithm for exact Lipschitz constant computation applicable to various piecewise linear neural networks, enhancing accuracy in robustness and generalization assessments.

Contribution

It generalizes the LipBaB algorithm to handle arbitrary piecewise linear activations and norms, enabling exact Lipschitz constant computation beyond ReLU networks.

Findings

Supports diverse activations like ReLU, LeakyReLU, GroupSort, MaxPool.

Applicable to different p-norms for Lipschitz calculations.

Facilitates precise robustness guarantees and benchmarking.

Abstract

It has been shown that a neural network's Lipschitz constant can be leveraged to derive robustness guarantees, to improve generalizability via regularization or even to construct invertible networks. Therefore, a number of methods varying in the tightness of their bounds and their computational cost have been developed to approximate the Lipschitz constant for different classes of networks. However, comparatively little research exists on methods for exact computation, which has been shown to be NP-hard. Nonetheless, there are applications where one might readily accept the computational cost of an exact method. These applications could include the benchmarking of new methods or the computation of robustness guarantees for small models on sensitive data. Unfortunately, existing exact algorithms restrict themselves to only ReLU-activated networks, which are known to come with severe downsides in the context of Lipschitz-constrained networks. We therefore propose a generalization of the LipBaB algorithm to compute exact Lipschitz constants for arbitrary piecewise linear neural networks and $p$-norms. With our method, networks may contain traditional activations like ReLU or LeakyReLU, activations like GroupSort or the related MinMax and FullSort, which have been of increasing interest in the context of Lipschitz constrained networks, or even other piecewise linear functions like MaxPool.