This paper relates parameter distance to gradient breakdown for a broad class of nonlinear compositional functions. The analysis leads to a new distance function called deep relative trust and a descent lemma for neural networks. Since the resulting learning rule seems to require little to no learning rate tuning, it may unlock a simpler workflow for training deeper and more complex neural networks. The Python code used in this paper is here: https://github.com/jxbz/fromage.
We propose a general framework for increasing local stability of Artificial Neural Nets (ANNs) using Robust Optimization (RO). We achieve this through an alternating minimization-maximization procedure, in which the loss of the network is minimized over perturbed examples that are generated at each parameter update. We show that adversarial training of ANNs is in fact robustification of the network optimization, and that our proposed framework generalizes previous approaches for increasing local stability of ANNs. Experimental results reveal that our approach increases the robustness of the ne…
Florida State University（佛罗里达州立大学）；IFM Lab（IFM实验室）
Graph distance metric learning serves as the foundation for many graph learning problems, e.g., graph clustering, graph classification and graph matching. Existing research works on graph distance metric (or graph kernels) learning fail to maintain the basic properties of such metrics, e.g., non-negative, identity of indiscernibles, symmetry and triangle inequality, respectively. In this paper, we will introduce a new graph neural network based distance metric learning approaches, namely GB-DISTANCE (GRAPH-BERT based Neural Distance). Solely based on the attention mechanism, GB-DISTANCE can le…
Neural networks for bifurcation and linear stability analysis of steady states in partial differential equations
organization= Department of Mathematics, Khalifa University of Science \& Technology , city= Abu Dhabi , postcode= PO Box 127788 , country= United Arab Emirates
Stability and Discretization Error of State Space Model Neural Operators
École polytechnique（巴黎政治经济学院）；Université Claude Bernard Lyon 1（里昂1大学）；CNRS（法国国家科学研究中心）；LAGEPP UMR 5007；Université Lyon 1（里昂1大学）；INSA Lyon（里昂国立应用科学学院）；LIRIS（里昂图像与信号研究所）
An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper boun…
Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks in terms of both the input data and the (trained) network weights. As trained network weights are typically very rough when seen as functions of the layer, we propose to derive stability bounds in terms of the total $p$-variation of trained weights for any $p\in[1,3]$. Unlike the $C^1$-theory underlying the neural ODE literature, our estimates remain bounded even in the limiting case of weights behaving like Brownian motions, as suggested in [arXiv:2105.12245]. Mathematically, we interpre…
Algebraic neural networks (AlgNNs) are composed of a cascade of layers each one associated to and algebraic signal model, and information is mapped between layers by means of a nonlinearity function. AlgNNs provide a generalization of neural network architectures where formal convolution operators are used, like for instance traditional neural networks (CNNs) and graph neural networks (GNNs). In this paper we study stability of AlgNNs on the framework of algebraic signal processing. We show how any architecture that uses a formal notion of convolution can be stable beyond particular choices of…
National Institute of Technology Durgapur（杜尔加布尔国家理工学院）
Supervised classification is the most active and emerging research trends in today's scenario. In this view, Artificial Neural Network (ANN) techniques have been widely employed and growing interest to the researchers day by day. ANN training aims to find the proper setting of parameters such as weights ($\textbf{W}$) and biases ($b$) to properly classify the given data samples. The training process is formulated in an error minimization problem which consists of many local optima in the search landscape. In this paper, an enhanced Particle Swarm Optimization is proposed to minimize the error …
Graph Neural Networks (GNNs) show impressive performance in many practical scenarios, which can be largely attributed to their stability properties. Empirically, GNNs can scale well on large size graphs, but this is contradicted by the fact that existing stability bounds grow with the number of nodes. Graphs with well-defined limits can be seen as samples from manifolds. Hence, in this paper, we analyze the stability properties of convolutional neural networks on manifolds to understand the stability of GNNs on large graphs. Specifically, we focus on stability to relative perturbations of the …
Carnegie Mellon University（卡内基梅隆大学）；Bosch AI（博世人工智能）；Determined AI
We empirically demonstrate that full-batch gradient descent on neural network training objectives typically operates in a regime we call the Edge of Stability. In this regime, the maximum eigenvalue of the training loss Hessian hovers just above the numerical value $2 / \text{(step size)}$, and the training loss behaves non-monotonically over short timescales, yet consistently decreases over long timescales. Since this behavior is inconsistent with several widespread presumptions in the field of optimization, our findings raise questions as to whether these presumptions are relevant to neural …
Fraunhofer Heinrich Hertz Institute（弗劳恩霍夫海因里希·赫兹研究所）
We study the recently introduced stability training as a general-purpose method to increase the robustness of deep neural networks against input perturbations. In particular, we explore its use as an alternative to data augmentation and validate its performance against a number of distortion types and transformations including adversarial examples. In our image classification experiments using ImageNet data stability training performs on a par or even outperforms data augmentation for specific transformations, while consistently offering improved robustness against a broader range of distortio…
Recent work have shown how the optimal state-feedback, obtained as the solution to the Hamilton-Jacobi-Bellman equations, can be approximated for several nonlinear, deterministic systems by deep neural networks. When imitation (supervised) learning is used to train the neural network on optimal state-action pairs, for instance as derived by applying Pontryagin's theory of optimal processes, the resulting model is referred here as the guidance and control network. In this work, we analyze the stability of nonlinear and deterministic systems controlled by such networks. We then propose a method …
University of Stuttgart（斯图加特大学）；RWTH Aachen University（亚琛工业大学）
In this paper, we analyze the stability of feedback interconnections of a linear time-invariant system with a neural network nonlinearity in discrete time. Our analysis is based on abstracting neural networks using integral quadratic constraints (IQCs), exploiting the sector-bounded and slope-restricted structure of the underlying activation functions. In contrast to existing approaches, we leverage the full potential of dynamic IQCs to describe the nonlinear activation functions in a less conservative fashion. To be precise, we consider multipliers based on the full-block Yakubovich / circle …
Graph neural networks (GNNs) are learning architectures that rely on knowledge of the graph structure to generate meaningful representations of large-scale network data. GNN stability is thus important as in real-world scenarios there are typically uncertainties associated with the graph. We analyze GNN stability using kernel objects called graphons. Graphons are both limits of convergent graph sequences and generating models for deterministic and stochastic graphs. Building upon the theory of graphon signal processing, we define graphon neural networks and analyze their stability to graphon p…
Graph Neural Networks (GNNs) are information processing architectures for signals supported on graphs. They are presented here as generalizations of convolutional neural networks (CNNs) in which individual layers contain banks of graph convolutional filters instead of banks of classical convolutional filters. Otherwise, GNNs operate as CNNs. Filters are composed with pointwise nonlinearities and stacked in layers. It is shown that GNN architectures exhibit equivariance to permutation and stability to graph deformations. These properties help explain the good performance of GNNs that can be obs…
Understanding generalization and robustness of machine learning models fundamentally relies on assuming an appropriate metric on the data space. Identifying such a metric is particularly challenging for non-Euclidean data such as graphs. Here, we propose a pseudometric for attributed graphs, the Tree Mover's Distance (TMD), and study its relation to generalization. Via a hierarchical optimal transport problem, TMD reflects the local distribution of node attributes as well as the distribution of local computation trees, which are known to be decisive for the learning behavior of graph neural ne…
We revisit on-average algorithmic stability of GD for training overparameterised shallow neural networks and prove new generalisation and excess risk bounds without the NTK or PL assumptions. In particular, we show oracle type bounds which reveal that the generalisation and excess risk of GD is controlled by an interpolating network with the shortest GD path from initialisation (in a sense, an interpolating network with the smallest relative norm). While this was known for kernelised interpolants, our proof applies directly to networks trained by GD without intermediate kernelisation. At the s…
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…
Inspired by convolutional neural networks on 1D and 2D data, graph convolutional neural networks (GCNNs) have been developed for various learning tasks on graph data, and have shown superior performance on real-world datasets. Despite their success, there is a dearth of theoretical explorations of GCNN models such as their generalization properties. In this paper, we take a first step towards developing a deeper theoretical understanding of GCNN models by analyzing the stability of single-layer GCNN models and deriving their generalization guarantees in a semi-supervised graph learning setting…
We explore the possibility to use physics-informed neural networks to drastically accelerate the solution of ordinary differential-algebraic equations that govern the power system dynamics. When it comes to transient stability assessment, the traditionally applied methods either carry a significant computational burden, require model simplifications, or use overly conservative surrogate models. Conventional neural networks can circumvent these limitations but are faced with high demand of high-quality training datasets, while they ignore the underlying governing equations. Physics-informed neu…
University of Oxford（牛津大学）；Swiss Data Science Center（瑞士数据科学中心）
Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to small perturbations in the input, is not yet well understood. Although there exists some results showing the stability of graph neural networks, most take the form of an upper bound on the magnitude of change due to a perturbation in the graph topology. However, the change in the graph topology captured in existing bounds tend not to be expressed in terms of…
NYUAD Research Institute, New York University Abu Dhabi（纽约大学阿布扎比分校NYUAD研究院）
In this paper, we present an approach to neural network mean-field-type control and its stochastic stability analysis by means of adversarial inputs (aka adversarial attacks). This is a class of data-driven mean-field-type control where the distribution of the variables such as the system states and control inputs are incorporated into the problem. Besides, we present a methodology to validate the feasibility of the approximations of the solutions via neural networks and evaluate their stability. Moreover, we enhance the stability by enlarging the training set with adversarial inputs to obtain…
The goal of this paper is to provide sufficient conditions for guaranteeing the Input-to-State Stability (ISS) and the Incremental Input-to-State Stability (δISS) of Gated Recurrent Units (GRUs) neural networks. These conditions, devised for both single-layer and multi-layer architectures, consist of nonlinear inequalities on network's weights. They can be employed to check the stability of trained networks, or can be enforced as constraints during the training procedure of a GRU. The resulting training procedure is tested on a Quadruple Tank nonlinear benchmark system, showing satisfactory mo…
We study algebraic neural networks (AlgNNs) with commutative algebras which unify diverse architectures such as Euclidean convolutional neural networks, graph neural networks, and group neural networks under the umbrella of algebraic signal processing. An AlgNN is a stacked layered information processing structure where each layer is conformed by an algebra, a vector space and a homomorphism between the algebra and the space of endomorphisms of the vector space. Signals are modeled as elements of the vector space and are processed by convolutional filters that are defined as the images of the …
University of Birmingham（伯明翰大学）；Alibaba Group（阿里巴巴集团）；State University of New York at Albany（纽约州立大学奥尔巴尼分校）
While significant theoretical progress has been achieved, unveiling the generalization mystery of overparameterized neural networks still remains largely elusive. In this paper, we study the generalization behavior of shallow neural networks (SNNs) by leveraging the concept of algorithmic stability. We consider gradient descent (GD) and stochastic gradient descent (SGD) to train SNNs, for both of which we develop consistent excess risk bounds by balancing the optimization and generalization via early-stopping. As compared to existing analysis on GD, our new analysis requires a relaxed overpara…
Stability certification and identifying a safe and stabilizing initial set are two important concerns in ensuring operational safety, stability, and robustness of dynamical systems. With the advent of machine-learning tools, these issues need to be addressed for the systems with machine-learned components in the feedback loop. To develop a general theory for stability and stabilizability of a neural network (NN)-controlled nonlinear system subject to bounded parametric variation, a Lyapunov-based stability certificate is proposed and is further used to devise a maximal Lipschitz bound for the …
Negative Imaginary Neural ODEs: Learning to Control Mechanical Systems with Stability Guarantees
Australian Centre for Robotics and School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney（澳大利亚机器人中心和航空航天、机械与机电工程学院，悉尼大学）
Differential equations are frequently used in engineering domains, such as modeling and control of industrial systems, where safety and performance guarantees are of paramount importance. Traditional physics-based modeling approaches require domain expertise and are often difficult to tune or adapt to new systems. In this paper, we show how to model discrete ordinary differential equations (ODE) with algebraic nonlinearities as deep neural networks with varying degrees of prior knowledge. We derive the stability guarantees of the network layers based on the implicit constraints imposed on the …
Learning for control of dynamical systems with formal guarantees remains a challenging task. This paper proposes a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing …
We examine the stability of loss-minimizing training processes that are used for deep neural networks (DNN) and other classifiers. While a classifier is optimized during training through a so-called loss function, the performance of classifiers is usually evaluated by some measure of accuracy, such as the overall accuracy which quantifies the proportion of objects that are well classified. This leads to the guiding question of stability: does decreasing loss through training always result in increased accuracy? We formalize the notion of stability, and provide examples of instability. Our main…
University of Stuttgart（斯图加特大学）；ETH Zurich（苏黎世联邦理工学院）
In this paper, we present a method to analyze local and global stability in offset-free setpoint tracking using neural network controllers and we provide ellipsoidal inner approximations of the corresponding region of attraction. We consider a feedback interconnection of a linear plant in connection with a neural network controller and an integrator, which allows for offset-free tracking of a desired piecewise constant reference that enters the controller as an external input. Exploiting the fact that activation functions used in neural networks are slope-restricted, we derive linear matrix in…
University of Pennsylvania（宾夕法尼亚大学）；Delft University of Technology（代尔夫特理工大学）
Graph convolutional neural networks (GCNNs) are nonlinear processing tools to learn representations from network data. A key property of GCNNs is their stability to graph perturbations. Current analysis considers deterministic perturbations but fails to provide relevant insights when topological changes are random. This paper investigates the stability of GCNNs to stochastic graph perturbations induced by link losses. In particular, it proves the expected output difference between the GCNN over random perturbed graphs and the GCNN over the nominal graph is upper bounded by a factor that is lin…
Zhejiang University（浙江大学）；Ningbo University（宁波大学）；Ningbo Artificial Intelligence Institute, Shanghai Jiao Tong University（上海交通大学宁波人工智能研究院）；Huzhou Institute of Zhejiang University（浙江大学湖州研究院）
In this paper, we propose a novel nonlinear observer based on neural networks, called neural observer, for observation tasks of linear time-invariant (LTI) systems and uncertain nonlinear systems. In particular, the neural observer designed for uncertain systems is inspired by the active disturbance rejection control, which can measure the uncertainty in real-time. The stability analysis (e.g., exponential convergence rate) of LTI and uncertain nonlinear systems (involving neural observers) are presented and guaranteed, where it is shown that the observation problems can be solved only using t…
In this paper we address the issue of output instability of deep neural networks: small perturbations in the visual input can significantly distort the feature embeddings and output of a neural network. Such instability affects many deep architectures with state-of-the-art performance on a wide range of computer vision tasks. We present a general stability training method to stabilize deep networks against small input distortions that result from various types of common image processing, such as compression, rescaling, and cropping. We validate our method by stabilizing the state-of-the-art In…
Potsdam Institute for Climate Impact Research（波茨坦气候影响研究所）；University of St.Gallen（圣加仑大学）
To mitigate climate change, the share of renewable needs to be increased. Renewable energies introduce new challenges to power grids due to decentralization, reduced inertia and volatility in production. The operation of sustainable power grids with a high penetration of renewable energies requires new methods to analyze the dynamic stability. We provide new datasets of dynamic stability of synthetic power grids and find that graph neural networks (GNNs) are surprisingly effective at predicting the highly non-linear target from topological information only. To illustrate the potential to scale…
In the application of neural networks, we need to select a suitable model based on the problem complexity and the dataset scale. To analyze the network's capacity, quantifying the information learned by the network is necessary. This paper proves that the distance between the neural network weights in different training stages can be used to estimate the information accumulated by the network in the training process directly. The experiment results verify the utility of this method. An application of this method related to the label corruption is shown at the end.
University of California, Santa Cruz（加州大学圣克鲁兹分校）；NASA Langley Research Center（美国国家航空航天局兰利研究中心）；Naval Postgraduate School（海军研究生院）
Recent research shows that supervised learning can be an effective tool for designing near-optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of neural network controllers is still not well understood. In particular, some neural networks with high test accuracy can fail to even locally stabilize the dynamic system. To address this challenge we propose several novel neural network architectures, which we show guarantee local asymptotic stability while retaining the approximation capacity to learn the optimal feedback policy semi-globally. The proposed …
University of California, Santa Cruz（加州大学圣克鲁兹分校）；Naval Postgraduate School（海军研究生院）
Recent research has shown that supervised learning can be an effective tool for designing optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of these neural network (NN) controllers is still not well understood. In this paper we use numerical simulations to demonstrate that typical test accuracy metrics do not effectively capture the ability of an NN controller to stabilize a system. In particular, some NNs with high test accuracy can fail to stabilize the dynamics. To address this we propose two NN architectures which locally approximate a linear qua…
Charles University（查理大学）；Faculty of Mathematics and Physics（数学与物理学院）；LMU Munich（慕尼黑大学）；Center for Information and Language Processing（信息与语言处理中心）
We propose the neural string edit distance model for string-pair matching and string transduction based on learnable string edit distance. We modify the original expectation-maximization learned edit distance algorithm into a differentiable loss function, allowing us to integrate it into a neural network providing a contextual representation of the input. We evaluate on cognate detection, transliteration, and grapheme-to-phoneme conversion, and show that we can trade off between performance and interpretability in a single framework. Using contextual representations, which are difficult to int…
