DRIP: Domain Refinement Iteration with Polytopes for Backward Reachability Analysis of Neural Feedback Loops

TL;DR

This paper introduces DRIP, a novel algorithm that refines domain relaxations for backward reachability analysis of neural network controlled systems, significantly improving safety guarantees by tightening set bounds.

Contribution

The work presents a new refinement loop for domain relaxation in backward reachability, enabling tighter bounds and closed-form polytope representations for neural network safety analysis.

Findings

DRIP significantly tightens backward reachability set bounds.

The method handles polytope domains directly, improving over prior LP-based approaches.

Demonstrated effectiveness on neural network controlled robot obstacle avoidance.

Abstract

Safety certification of data-driven control techniques remains a major open problem. This work investigates backward reachability as a framework for providing collision avoidance guarantees for systems controlled by neural network (NN) policies. Because NNs are typically not invertible, existing methods conservatively assume a domain over which to relax the NN, which causes loose over-approximations of the set of states that could lead the system into the obstacle (i.e., backprojection (BP) sets). To address this issue, we introduce DRIP, an algorithm with a refinement loop on the relaxation domain, which substantially tightens the BP set bounds. Furthermore, we introduce a formulation that enables directly obtaining closed-form representations of polytopes to bound the BP sets tighter than prior work, which required solving linear programs and using hyper-rectangles. Furthermore, this work extends the NN relaxation algorithm to handle polytope domains, which further tightens the bounds on BP sets. DRIP is demonstrated in numerical experiments on control systems, including a ground robot controlled by a learned NN obstacle avoidance policy.