Reachability Analysis of Neural Feedback Loops using Sparse Polynomial Optimisation
Reachability Analysis of Neural Feedback Loops using Sparse Polynomial Optimisation
复制标题
DOI：
10.1109/cdc51059.2022.9992719
复制
发表时间：
2022-12
期刊：
影响因子：
--
通讯作者：
M. Newton;A. Papachristodoulou
中科院分区：
文献类型：
--
作者：
M. Newton;A. Papachristodoulou
神经网络最近在控制反馈系统中的使用有所增加。然而，事实证明，为这些反馈系统提供鲁棒性保证具有挑战性，为了解决这些问题，当前有大量的研究。神经网络的最大缺点之一是它们对对抗性输入非常敏感。鉴于反馈系统通常会受到外部扰动，因此必须克服围绕神经网络的这个问题，然后才能将其用于安全关键型应用。解决这个问题的一种方法是通过限制神经网络控制器中的激活函数来计算可达集的外部近似值。我们的方法是在稀疏多项式优化框架中结合 Positivstellensatz 使用这些边界。稀疏性属性能够利用神经网络的自然级联结构来实现易于处理的求解时间。 Positivstellensatz 能够通过断言半代数集的空性来提供比类似方法更准确的界限。我们通过示例表明，我们的方法可以以合理的计算时间提供比类似方法更严格的界限。由于我们使用多项式优化框架，我们的方法还能够处理非线性多项式动力学，而其他方法需要使用替代解决方案来合并非线性。
Neural networks have seen a recent increased use in control feedback systems. However, providing robustness guarantees on these feedback systems has proven challenging and to combat these issues, there is a significant amount of current research. One of the biggest shortcoming of neural networks is how sensitive they are to adversarial inputs. Given that feedback systems are usually subject to external perturbations, this issue surrounding neural networks must be overcome before they can be used in safety-critical applications. One method to tackle this problem is to compute outer-approximations of the reachable sets, through bounding the activation functions in the neural network controller. Our approach is to use these bounds in a sparse polynomial optimisation framework in conjunction with the Positivstellensatz. The sparsity property is able to exploit the natural cascading structure of the neural network to allow for tractable solve times. The Positivstellensatz is able to provide more accurate bounds over similar methods by asserting the emptiness of a semi-algebraic set. We show through examples that our method can provide tighter bounds over similar methods, with reasonable computational time. Our approach is also able to deal with non-linear polynomial dynamics due to the polynomial optimisation framework we use, while other methods need to use alternative work-around solutions to incorporate the non-linearities.