This paper studies the problem of range analysis for feedforward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reachability analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present specific difficulties: approaches that exploit derivatives do not apply in general, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-based methods and abstract interpretation that have been very successful in coping with similar difficulties in classical program verification. We present an approach that abstracts ReLU feedforward neural networks using tropical polyhedra. We show that tropical polyhedra can efficiently abstract ReLU activation function, while being able to control the loss of precision due to linear computations. We show how the connection between ReLU networks and tropical rational functions can provide approaches for range analysis of ReLU neural networks.
Neural networks have been widely used to solve complex real-world problems. Due to the complicate, nonlinear, non-convex nature of neural networks, formal safety guarantees for the output behaviors of neural networks will be crucial for their applications in safety-critical systems.In this paper, the output reachable set computation and safety verification problems for a class of neural networks consisting of Rectified Linear Unit (ReLU) activation functions are addressed. A layer-by-layer approach is developed to compute output reachable set. The computation is formulated in the form of a set…
We present a method for computing exact reachable sets for deep neural networks with rectified linear unit (ReLU) activation. Our method is well-suited for use in rigorous safety analysis of robotic perception and control systems with deep neural network components. Our algorithm can compute both forward and backward reachable sets for a ReLU network iterated over multiple time steps, as would be found in a perception-action loop in a robotic system. Our algorithm is unique in that it builds the reachable sets by incrementally enumerating polyhedral cells in the input space, rather than iterat…
Topological Signatures of ReLU Neural Network Activation Patterns
Applied Math and Computational Sciences University of Pennsylvania（应用数学与计算科学大学公园大学）；Department of Mathematics Colorado State University（数学系科罗拉多州立大学）；Departments of Mechanical Engineering & Computational Mathematics, Sciences and Engineering (CMSE) Michigan State University（机械工程与计算数学、科学与工程系密歇根州立大学）；Department of Mathematics University of Washington（数学系华盛顿大学）；Georgia Tech Research Institute（佐治亚理工研究学院）
Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. These decompositions can be represented by a dual graph with vertices corresponding to polyhedra and edges corresponding to polyhedra sharing a facet, which is a subgraph of a Hamming graph. This paper illustrates how one can utilize the dual graph to detect and analyze adversarial attacks in the context of digital images. When an image passes through a network containing ReLU nodes, the firing or non-firing at a node can be enco…
We establish, for the first time, connections between feedforward neural networks with ReLU activation and tropical geometry --- we show that the family of such neural networks is equivalent to the family of tropical rational maps. Among other things, we deduce that feedforward ReLU neural networks with one hidden layer can be characterized by zonotopes, which serve as building blocks for deeper networks; we relate decision boundaries of such neural networks to tropical hypersurfaces, a major object of study in tropical geometry; and we prove that linear regions of such neural networks corresp…
Data Analysis Systems（数据分析系统部）；Software Competence Center Hagenberg（哈根贝格软件能力中心）
We propose a new metric space of ReLU activation codes equipped with a truncated Hamming distance which establishes an isometry between its elements and polyhedral bodies in the input space which have recently been shown to be strongly related to safety, robustness, and confidence. This isometry allows the efficient computation of adjacency relations between the polyhedral bodies. Experiments on MNIST and CIFAR-10 indicate that information besides accuracy might be stored in the code space.
Cornell University（康奈尔大学）；National Technical University of Athens（雅典国立技术大学）
We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations as tropical polynomials, which generalize polynomials in the so-called $(\max, +)$ or tropical algebra, with possibly real-valued exponents. Motivated by the discussion in (arXiv:1402.1869), this approach enables us to refine their upper bounds on linear regions of layers with ReLU or leaky ReLU activations to $\min\left\{ 2^m, \sum_{j=0}^n \binom{m}{j} \right\}$, where $n, m$ are the numb…
We algorithmically determine the regions and facets of all dimensions of the canonical polyhedral complex, the universal object into which a ReLU network decomposes its input space. We show that the locations of the vertices of the canonical polyhedral complex along with their signs with respect to layer maps determine the full facet structure across all dimensions. We present an algorithm which calculates this full combinatorial structure, making use of our theorems that the dual complex to the canonical polyhedral complex is cubical and it possesses a multiplication compatible with its facet…
In this work, we examine the process of Tropical Polynomial Division, a geometric method which seeks to emulate the division of regular polynomials, when applied to those of the max-plus semiring. This is done via the approximation of the Newton Polytope of the dividend polynomial by that of the divisor. This process is afterwards generalized and applied in the context of neural networks with ReLU activations. In particular, we make use of the intuition it provides, in order to minimize a two-layer fully connected network, trained for a binary classification problem. This method is later evalu…
Freie Universität Berlin（柏林自由大学）；London School of Economics and Political Science（伦敦政治经济学院）；University of Twente（特文特大学）
We prove that the set of functions representable by ReLU neural networks with integer weights strictly increases with the network depth while allowing arbitrary width. More precisely, we show that $\lceil\log_2(n)\rceil$ hidden layers are indeed necessary to compute the maximum of $n$ numbers, matching known upper bounds. Our results are based on the known duality between neural networks and Newton polytopes via tropical geometry. The integrality assumption implies that these Newton polytopes are lattice polytopes. Then, our depth lower bounds follow from a parity argument on the normalized vo…
National Technical University of Athens（雅典国家技术大学）；Harvard University（哈佛大学）
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and extend both of them using weighted lattices and a max-$\star$ algebra with an arbitrary binary operation $\star$ that distributes over max, and outline applications to geometry, machine learning, and optimization. Further, we generalize tropical geometrical objects using weighted lattices. Finally, we provide the optimal solution of max-$\star$ equations usi…
King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；University of Oxford（牛津大学）；Zalando SE（Zalando公司）；itemis AG（itemis股份公司）
This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of these zonotopes are functions of the network parameters. This geometric c…
On the Depth of Monotone ReLU Neural Networks and ICNNs
Department of Computer Science, the University of Copenhagen（计算机科学系，哥本哈根大学）；University of Technology Nuremberg（纽伦堡技术大学）；Worcester Polytechnic Institute（沃思堡理工学院）；Department of Mathematics, Technion-IIT（数学系，技术学院-以色列理工学院）
We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of $M$ data points on the nondifferentiability of the ReLU divides the parameter space into at most $2^M$ regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, $O(M)$ (in)equa…
Polyhedral Geometry of Time-to-First-Spike Neural Networks
Ludwig-Maximilians-Universität München（慕尼黑大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Max Planck Institute for Mathematics in the Sciences, Leipzig（马克斯·普朗克数学科学研究所）；University of Tromsø（特罗姆瑟大学）
Towards Lower Bounds on the Depth of ReLU Neural Networks
London School of Economics and Political Science（伦敦政治经济学院）；Johns Hopkins University（约翰斯·霍普金斯大学）；Università degli Studi di Padova（帕多瓦大学）；Technische Universität Berlin（柏林工业大学）
We consider the problem of learning an unknown ReLU network with respect to Gaussian inputs and obtain the first nontrivial results for networks of depth more than two. We give an algorithm whose running time is a fixed polynomial in the ambient dimension and some (exponentially large) function of only the network's parameters. Our bounds depend on the number of hidden units, depth, spectral norm of the weight matrices, and Lipschitz constant of the overall network (we show that some dependence on the Lipschitz constant is necessary). We also give a bound that is doubly exponential in the size…
