This paper extends the star set reachability approach to verify the robustness of feed-forward neural networks (FNNs) with sigmoidal activation functions such as Sigmoid and TanH. The main drawbacks of the star set approach in Sigmoid/TanH FNN verification are scalability, feasibility, and optimality issues in some cases due to the linear programming solver usage. We overcome this challenge by proposing a relaxed star (RStar) with symbolic intervals, which allows the usage of the back-substitution technique in DeepPoly to find bounds when overapproximating activation functions while maintaining the valuable features of a star set. RStar can overapproximate a sigmoidal activation function using four linear constraints (RStar4) or two linear constraints (RStar2), or only the output bounds (RStar0). We implement our RStar reachability algorithms in NNV and compare them to DeepPoly via robustness verification of image classification DNNs benchmarks. The experimental results show that the original star approach (i.e., no relaxation) is the least conservative of all methods yet the slowest. RStar4 is computationally much faster than the original star method and is the second least conservative approach. It certifies up to 40% more images against adversarial attacks than DeepPoly and on average 51 times faster than the star set. Last but not least, RStar0 is the most conservative method, which could only verify two cases for the CIFAR10 small Sigmoid network,
Verisig: verifying safety properties of hybrid systems with neural network controllers
This paper presents Verisig, a hybrid system approach to verifying safety properties of closed-loop systems using neural networks as controllers. We focus on sigmoid-based networks and exploit the fact that the sigmoid is the solution to a quadratic differential equation, which allows us to transform the neural network into an equivalent hybrid system. By composing the network's hybrid system with the plant's, we transform the problem into a hybrid system verification problem which can be solved using state-of-the-art reachability tools. We show that reachability is decidable for networks with one hidden layer and decidable for general networks if Schanuel's conjecture is true. We evaluate the applicability and scalability of Verisig in two case studies, one from reinforcement learning and one in which the neural network is used to approximate a model predictive controller.
more »
« less
- Award ID(s):
- 1837210
- NSF-PAR ID:
- 10110635
- Date Published:
- Journal Name:
- HSCC 2019
- Page Range / eLocation ID:
- 169-178
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Ξ΄ = 0.014. However, it is the fastest method that can verify neural networks up to 3528 times faster than the star set and up to 46 times faster than DeepPoly in our evaluation. -
This article addresses the problem of verifying the safety of autonomous systems with neural network (NN) controllers. We focus on NNs with sigmoid/tanh activations and use the fact that the sigmoid/tanh is the solution to a quadratic differential equation. This allows us to convert the NN into an equivalent hybrid system and cast the problem as a hybrid system verification problem, which can be solved by existing tools. Furthermore, we improve the scalability of the proposed method by approximating the sigmoid with a Taylor series with worst-case error bounds. Finally, we provide an evaluation over four benchmarks, including comparisons with alternative approaches based on mixed integer linear programming as well as on star sets.more » « less
-
In this paper, a computationally efficient data-driven hybrid automaton model is proposed to capture unknown complex dynamical system behaviors using multiple neural networks. The sampled data of the system is divided by valid partitions into groups corresponding to their topologies and based on which, transition guards are defined. Then, a collection of small-scale neural networks that are computationally efficient are trained as the local dynamical description for their corresponding topologies. After modeling the system with a neural-network-based hybrid automaton, the set-valued reachability analysis with low computation cost is provided based on interval analysis and a split and combined process. At last, a numerical example of the limit cycle is presented to illustrate that the developed models can significantly reduce the computational cost in reachable set computation without sacrificing any modeling precision.more » « less
-
null (Ed.)Abstract Neural state classification (NSC) is a recently proposed method for runtime predictive monitoring of hybrid automata (HA) using deep neural networks (DNNs). NSC trains a DNN as an approximate reachability predictor that labels an HA state x as positive if an unsafe state is reachable from x within a given time bound, and labels x as negative otherwise. NSC predictors have very high accuracy, yet are prone to prediction errors that can negatively impact reliability. To overcome this limitation, we present neural predictive monitoring (NPM), a technique that complements NSC predictions with estimates of the predictive uncertainty. These measures yield principled criteria for the rejection of predictions likely to be incorrect, without knowing the true reachability values. We also present an active learning method that significantly reduces the NSC predictorβs error rate and the percentage of rejected predictions. We develop two versions of NPM based, respectively, on the use of frequentist and Bayesian techniques to learn the predictor and the rejection rule. Both versions are highly efficient, with computation times on the order of milliseconds, and effective, managing in our experimental evaluation to successfully reject almost all incorrect predictions. In our experiments on a benchmark suite of six hybrid systems, we found that the frequentist approach consistently outperforms the Bayesian one. We also observed that the Bayesian approach is less practical, requiring a careful and problem-specific choice of hyperparameters.more » « less
-
Silva, A. and (Ed.)We present π²πΌπΎππΎπ’ππΎπΌππΎπ, a tool for verifying scenarios involving vehicles executing complex plans in large cluttered workspaces. π²πΌπΎππΎπ’ππΎπΌππΎπ converts the scenario verification problem to a standard hybrid system verification problem, and solves it effectively by exploiting structural properties in the plan and the vehicle dynamics. π²πΌπΎππΎπ’ππΎπΌππΎπ uses symmetry abstractions, a novel refinement algorithm, and importantly, is built to boost the performance of any existing reachability analysis tool as a plug-in subroutine. We evaluated π²πΌπΎππΎπ’ππΎπΌππΎπ on several scenarios involving ground and aerial vehicles with nonlinear dynamics and neural network controllers, employing different kinds of symmetries, using different reachability subroutines, and following plans with hundreds of waypoints in complex workspaces. Compared to two leading tools, DryVR and Flow*, π²πΌπΎππΎπ’ππΎπΌππΎπ shows 14Γ average speedup in verification time, even while using those very tools as reachability subroutines.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3302504.3311806
