Robustness Analysis of Neural Networks via Efficient Partitioning: Theory and Applications in Control Systems

Neural networks (NNs) are now routinely implemented on systems that must operate in uncertain environments, but the tools for formally analyzing how this uncertainty propagates to NN outputs are not yet commonplace. Computing tight bounds on NN output sets (given an input set) provides a measure of confidence associated with the NN decisions and is essential to deploy NNs on safety-critical systems. Recent works approximate the propagation of sets through nonlinear activations or partition the uncertainty set to provide a guaranteed outer bound on the set of possible NN outputs. However, the bound looseness causes excessive conservatism and/or the computation is too slow for online analysis. This paper unifies propagation and partition approaches to provide a family of robustness analysis algorithms that give tighter bounds than existing works for the same amount of computation time (or reduced computational effort for a desired accuracy level). Moreover, we provide new partitioning techniques that are aware of their current bound estimates and desired boundary shape (e.g., lower bounds, weighted $\ell_\infty$-ball, convex hull), leading to further improvements in the computation-tightness tradeoff. The paper demonstrates the tighter bounds and reduced conservatism of the proposed robustness analysis framework with examples from model-free RL and forward kinematics learning.