Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks

TL;DR

This paper introduces a convex optimization framework using semidefinite programming to accurately and efficiently estimate the Lipschitz constants of deep neural networks, aiding robustness and stability analysis.

Contribution

It presents a novel SDP-based method that interprets activation functions as gradients of convex potentials, improving accuracy and scalability in Lipschitz constant estimation.

Findings

Our bounds are more accurate than existing methods.

The approach scales well with network size.

Lipschitz bounds help assess robustness against adversarial attacks.

Abstract

Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning controllers. Existing methods in the literature for estimating the Lipschitz constant suffer from either lack of accuracy or poor scalability. In this paper, we present a convex optimization framework to compute guaranteed upper bounds on the Lipschitz constant of DNNs both accurately and efficiently. Our main idea is to interpret activation functions as gradients of convex potential functions. Hence, they satisfy certain properties that can be described by quadratic constraints. This particular description allows us to pose the Lipschitz constant estimation problem as a semidefinite program (SDP). The resulting SDP can be adapted to increase either the estimation accuracy (by capturing the interaction between activation functions of different layers) or scalability (by decomposition and parallel implementation). We illustrate the utility of our approach with a variety of experiments on randomly generated networks and on classifiers trained on the MNIST and Iris datasets. In particular, we experimentally demonstrate that our Lipschitz bounds are the most accurate compared to those in the literature. We also study the impact of adversarial training methods on the Lipschitz bounds of the resulting classifiers and show that our bounds can be used to efficiently provide robustness guarantees.