Robustness Analysis of Continuous-Depth Models with Lagrangian Techniques

TL;DR

This paper introduces Lagrangian verification techniques for assessing the robustness of continuous-depth models, providing formal guarantees and demonstrating superior performance over existing methods through experiments.

Contribution

It unifies deterministic and statistical Lagrangian verification methods for continuous-depth models, offering new algorithms and guarantees for robustness analysis.

Findings

Lagrangian techniques outperform existing methods like LRT, Flow*, and CAPD.

The algorithms provide tight reachtube over-approximations with formal guarantees.

Experiments validate the effectiveness of the proposed methods in robustness analysis.

Abstract

This paper presents, in a unified fashion, deterministic as well as statistical Lagrangian-verification techniques. They formally quantify the behavioral robustness of any time-continuous process, formulated as a continuous-depth model. To this end, we review LRT-NG, SLR, and GoTube, algorithms for constructing a tight reachtube, that is, an over-approximation of the set of states reachable within a given time-horizon, and provide guarantees for the reachtube bounds. We compare the usage of the variational equations, associated to the system equations, the mean value theorem, and the Lipschitz constants, in achieving deterministic and statistical guarantees. In LRT-NG, the Lipschitz constant is used as a bloating factor of the initial perturbation, to compute the radius of an ellipsoid in an optimal metric, which over-approximates the set of reachable states. In SLR and GoTube, we get statistical guarantees, by using the Lipschitz constants to compute local balls around samples. These are needed to calculate the probability of having found an upper bound, of the true maximum perturbation at every timestep. Our experiments demonstrate the superior performance of Lagrangian techniques, when compared to LRT, Flow*, and CAPD, and illustrate their use in the robustness analysis of various continuous-depth models.