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Exact Verification of ReLU Neural Control Barrier Functions
Machine Learning · Submitted 13 Oct 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions.
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The abstract, sentence by sentence
Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions. The key challenge in doing so is that, due to the piecewise linearity of the ReLU function, the NCBF will be nondifferentiable at certain points, thus invalidating traditional safety verification methods that assume a smooth barrier function. We resolve this issue by leveraging a generalization of Nagumo's theorem for proving invariance of sets with nonsmooth boundaries to derive necessary and sufficient conditions for safety. Based on this condition, we propose an algorithm for safety verification of NCBFs that first decomposes the NCBF into piecewise linear segments and then solves a nonlinear program to verify safety of each segment as well as the intersections of the linear segments. We mitigate the complexity by only considering the boundary of the safe region and by pruning the segments with Interval Bound Propagation (IBP) and linear relaxation. We evaluate our approach through numerical studies with comparison to state-of-the-art SMT-based methods. Our code is available at https://github.com/HongchaoZhang-HZ/exactverif-reluncbf-nips23.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. | background 1.00 | 0.04 | 0.02 | asserted 1.00 |
| s1 | In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s2 | Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. | background 1.00 | 0.05 | 0.02 | asserted 0.97 |
| s3 | However, verifying that a learned CBF guarantees safety remains a challenging research problem. | gap 0.97 | 0.05 | 0.02 | asserted 1.00 |
| s4 | This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions. | contribution 1.00 | 0.94 | 0.59 | asserted 1.00 |
| s5 | The key challenge in doing so is that, due to the piecewise linearity of the ReLU function, the NCBF will be nondifferentiable at certain points, thus invalidating traditional safety verification methods that assume a smooth barrier function. | gap 0.55 | 0.16 | 0.03 | asserted 1.00 |
| s6 | We resolve this issue by leveraging a generalization of Nagumo's theorem for proving invariance of sets with nonsmooth boundaries to derive necessary and sufficient conditions for safety. | method 0.86 | 0.92 | 0.22 | asserted 1.00 |
| s7 | Based on this condition, we propose an algorithm for safety verification of NCBFs that first decomposes the NCBF into piecewise linear segments and then solves a nonlinear program to verify safety of each segment as well as the intersections of the linear segments. | method 0.95 | 0.95 | 0.18 | asserted 1.00 |
| s8 | We mitigate the complexity by only considering the boundary of the safe region and by pruning the segments with Interval Bound Propagation (IBP) and linear relaxation. | method 0.98 | 0.92 | 0.05 | asserted 1.00 |
| s9 | We evaluate our approach through numerical studies with comparison to state-of-the-art SMT-based methods. | method 0.52 | 0.95 | 0.06 | asserted 0.99 |
| s10 | Our code is available at https://github.com/HongchaoZhang-HZ/exactverif-reluncbf-nips23. | contribution 0.67 | 0.50 | 0.02 | asserted 0.97 |
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The paper
Original abstract
Control Barrier Functions (CBFs) are a popular approach for safe control of nonlinear systems. In CBF-based control, the desired safety properties of the system are mapped to nonnegativity of a CBF, and the control input is chosen to ensure that the CBF remains nonnegative for all time. Recently, machine learning methods that represent CBFs as neural networks (neural control barrier functions, or NCBFs) have shown great promise due to the universal representability of neural networks. However, verifying that a learned CBF guarantees safety remains a challenging research problem. This paper presents novel exact conditions and algorithms for verifying safety of feedforward NCBFs with ReLU activation functions. The key challenge in doing so is that, due to the piecewise linearity of the ReLU function, the NCBF will be nondifferentiable at certain points, thus invalidating traditional safety verification methods that assume a smooth barrier function. We resolve this issue by leveraging a generalization of Nagumo's theorem for proving invariance of sets with nonsmooth boundaries to derive necessary and sufficient conditions for safety. Based on this condition, we propose an algorithm for safety verification of NCBFs that first decomposes the NCBF into piecewise linear segments and then solves a nonlinear program to verify safety of each segment as well as the intersections of the linear segments. We mitigate the complexity by only considering the boundary of the safe region and by pruning the segments with Interval Bound Propagation (IBP) and linear relaxation. We evaluate our approach through numerical studies with comparison to state-of-the-art SMT-based methods. Our code is available at https://github.com/HongchaoZhang-HZ/exactverif-reluncbf-nips23.
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References
How the literature uses it
2 arXiv papers cite this in 2 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 1 paper
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Despite the great success of deep reinforcement learning (RL) in recent years [ref] , ensuring safety (i.e., constraint satisfaction) is one key challenge when deploying them to real-world applications [ref] .
Gradient Shaping for Multi-Constraint Safe Reinforcement Learning · Introduction · 0.90
What it cites · 3 sentences without a confident label
Cites as background 4 papers
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Another related body of work deals with the problem of verifying neural networks including SMT [ref] , output reachable set verification [ref] , polynomial approximations of the barrier function [ref] , verify input/output relationships [ref] and ReLU neural networks focused verification [ref] .
Complete Verification via Multi-Neuron Relaxation Guided Branch-and-Bound · Introduction · 0.99
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Sum-of-squares (SOS) optimization and other techniques derived from algebraic geometry have been widely used for safety verification of polynomial barrier functions [ref] .
Safety Index Synthesis via Sum-of-Squares Programming · Introduction · 0.85
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Sum-of-squares (SOS) optimization and other techniques derived from algebraic geometry have been widely used for safety verification of polynomial barrier functions [ref] .
SOS Construction of Compatible Control Lyapunov and Barrier Functions · Introduction · 0.85
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