In this paper, we consider the computational complexity of formally verifying the behavior of Rectified Linear Unit (ReLU) Neural Networks (NNs), where verification entails determining whether the NN satisfies convex polytopic specifications. Specifically, we show that for two different NN architectures -- shallow NNs and Two-Level Lattice (TLL) NNs -- the verification problem with (convex) polytopic constraints is polynomial in the number of neurons in the NN to be verified, when all other aspects of the verification problem held fixed. We achieve these complexity results by exhibiting explicit (but similar) verification algorithms for each type of architecture. Both algorithms efficiently translate the NN parameters into a partitioning of the NN's input space by means of hyperplanes; this has the effect of partitioning the original verification problem into polynomially many sub-verification problems derived from the geometry of the neurons. We show that these sub-problems may be chosen so that the NN is purely affine within each, and hence each sub-problem is solvable in polynomial time by means of a Linear Program (LP). Thus, a polynomial-time algorithm for the original verification problem can be obtained using known algorithms for enumerating the regions in a hyperplane arrangement. Finally, we adapt our proposed algorithms to the verification of dynamical systems, specifically when these NN architectures are used as state-feedback controllers for LTI systems. We further evaluate the viability of this approach numerically.
Fraunhofer Institute for Cognitive Systems（弗劳恩霍夫认知系统研究所）
In this paper, we consider the imperfection within machine learning-based 2D object detection and its impact on safety. We address a special sub-type of performance limitations: the prediction bounding box cannot be perfectly aligned with the ground truth, but the computed Intersection-over-Union metric is always larger than a given threshold. Under such type of performance limitation, we formally prove the minimum required bounding box enlargement factor to cover the ground truth. We then demonstrate that the factor can be mathematically adjusted to a smaller value, provided that the motion p…
Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates
Stanford University（斯坦福大学）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；IT University of Copenhagen（哥本哈根信息技术大学）；Google（谷歌公司）；GE Aerospace Research（GE航空航天研究院）
The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Stanford University（斯坦福大学）
Deep learning has emerged as an effective approach for creating modern software systems, with neural networks often surpassing hand-crafted systems. Unfortunately, neural networks are known to suffer from various safety and security issues. Formal verification is a promising avenue for tackling this difficulty, by formally certifying that networks are correct. We propose an SMT-based technique for verifying Binarized Neural Networks - a popular kind of neural network, where some weights have been binarized in order to render the neural network more memory and energy efficient, and quicker to e…
We present Revel, a partially neural reinforcement learning (RL) framework for provably safe exploration in continuous state and action spaces. A key challenge for provably safe deep RL is that repeatedly verifying neural networks within a learning loop is computationally infeasible. We address this challenge using two policy classes: a general, neurosymbolic class with approximate gradients and a more restricted class of symbolic policies that allows efficient verification. Our learning algorithm is a mirror descent over policies: in each iteration, it safely lifts a symbolic policy into the …
We consider the problem of formally verifying almost-sure (a.s.) asymptotic stability in discrete-time nonlinear stochastic control systems. While verifying stability in deterministic control systems is extensively studied in the literature, verifying stability in stochastic control systems is an open problem. The few existing works on this topic either consider only specialized forms of stochasticity or make restrictive assumptions on the system, rendering them inapplicable to learning algorithms with neural network policies. In this work, we present an approach for general nonlinear stochast…
East China Normal University（华东师范大学）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）
Occlusion is a prevalent and easily realizable semantic perturbation to deep neural networks (DNNs). It can fool a DNN into misclassifying an input image by occluding some segments, possibly resulting in severe errors. Therefore, DNNs planted in safety-critical systems should be verified to be robust against occlusions prior to deployment. However, most existing robustness verification approaches for DNNs are focused on non-semantic perturbations and are not suited to the occlusion case. In this paper, we propose the first efficient, SMT-based approach for formally verifying the occlusion robu…
The topic of provable deep neural network robustness has raised considerable interest in recent years. Most research has focused on adversarial robustness, which studies the robustness of perceptive models in the neighbourhood of particular samples. However, other works have proved global properties of smaller neural networks. Yet, formally verifying perception remains uncharted. This is due notably to the lack of relevant properties to verify, as the distribution of possible inputs cannot be formally specified. We propose to take advantage of the simulators often used either to train machine …
In order to properly train a machine learning model, data must be properly collected. To guarantee a proper data collection, verifying that the collected data set holds certain properties is a possible solution. For example, guaranteeing that the data set contains samples across the whole input space, or that the data set is balanced w.r.t. different classes. We present a formal approach for verifying a set of arbitrarily stated properties over a data set. The proposed approach relies on the transformation of the data set into a first order logic formula, which can be later verified w.r.t. the…
This survey presents an overview of verification techniques for autonomous systems, with a focus on safety-critical autonomous cyber-physical systems (CPS) and subcomponents thereof. Autonomy in CPS is enabling by recent advances in artificial intelligence (AI) and machine learning (ML) through approaches such as deep neural networks (DNNs), embedded in so-called learning enabled components (LECs) that accomplish tasks from classification to control. Recently, the formal methods and formal verification community has developed methods to characterize behaviors in these LECs with eventual goals …
We improve the scalability of Branch and Bound (BaB) algorithms for formally proving input-output properties of neural networks. First, we propose novel bounding algorithms based on Lagrangian Decomposition. Previous works have used off-the-shelf solvers to solve relaxations at each node of the BaB tree, or constructed weaker relaxations that can be solved efficiently, but lead to unnecessarily weak bounds. Our formulation restricts the optimization to a subspace of the dual domain that is guaranteed to contain the optimum, resulting in accelerated convergence. Furthermore, it allows for a mas…
Towards Efficient Verification of Quantized Neural Networks
Stanford University（斯坦福大学）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；IT University of Copenhagen（哥本哈根信息技术大学）；National University of Singapore（新加坡国立大学）
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verificat…
In this paper, we consider the problem of formally verifying a Neural Network (NN) based autonomous landing system. In such a system, a NN controller processes images from a camera to guide the aircraft while approaching the runway. A central challenge for the safety and liveness verification of vision-based closed-loop systems is the lack of mathematical models that captures the relation between the system states (e.g., position of the aircraft) and the images processed by the vision-based NN controller. Another challenge is the limited abilities of state-of-the-art NN model checkers. Such mo…
Massachusetts Institute of Technology（麻省理工学院）；UC Davis（加州大学戴维斯分校）；Harvard University（哈佛大学）；UT Austin（德克萨斯大学奥斯汀分校）
Verifying the robustness property of a general Rectified Linear Unit (ReLU) network is an NP-complete problem [Katz, Barrett, Dill, Julian and Kochenderfer CAV17]. Although finding the exact minimum adversarial distortion is hard, giving a certified lower bound of the minimum distortion is possible. Current available methods of computing such a bound are either time-consuming or delivering low quality bounds that are too loose to be useful. In this paper, we exploit the special structure of ReLU networks and provide two computationally efficient algorithms Fast-Lin and Fast-Lip that are able t…
