Deep networks are commonly used to model dynamical systems, predicting how the state of a system will evolve over time (either autonomously or in response to control inputs). Despite the predictive power of these systems, it has been difficult to make formal claims about the basic properties of the learned systems. In this paper, we propose an approach for learning dynamical systems that are guaranteed to be stable over the entire state space. The approach works by jointly learning a dynamics model and Lyapunov function that guarantees non-expansiveness of the dynamics under the learned Lyapunov function. We show that such learning systems are able to model simple dynamical systems and can be combined with additional deep generative models to learn complex dynamics, such as video textures, in a fully end-to-end fashion.
Stable behaviour of infinitely wide deep neural networks
University of Torino（都灵大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）；Bocconi University（博科尼大学）；Cogent Labs（Cogent实验室（注：根据机构粒度要求，若实验室本身是独立机构则提取，此处Cogent Labs作为独立实验室提取））
University of Torino（都灵大学）；Collegio Carlo Alberto（卡洛·阿尔贝托学院）；Bocconi University（博科尼大学）；Cogent Labs（Cogent实验室）
In modern deep learning, there is a recent and growing literature on the interplay between large-width asymptotic properties of deep Gaussian neural networks (NNs), i.e. deep NNs with Gaussian-distributed weights, and Gaussian stochastic processes (SPs). Such an interplay has proved to be critical in Bayesian inference under Gaussian SP priors, kernel regression for infinitely wide deep NNs trained via gradient descent, and information propagation within infinitely wide NNs. Motivated by empirical analyses that show the potential of replacing Gaussian distributions with Stable distributions fo…
We apply concepts from manifold regularization to develop new regularization techniques for training locally stable deep neural networks. Our regularizers are based on a sparsification of the graph Laplacian which holds with high probability when the data is sparse in high dimensions, as is common in deep learning. Empirically, our networks exhibit stability in a diverse set of perturbation models, including $\ell_2$, $\ell_\infty$, and Wasserstein-based perturbations; in particular, we achieve 40% adversarial accuracy on CIFAR-10 against an adaptive PGD attack using $\ell_\infty$ perturbation…
Huazhong University of Science and Technology（华中科技大学）；Illinois Institute of Technology（伊利诺伊理工学院）
Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, lots of log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios which could have high error and low convergence issues. In this work, we overcome some of these challenges and identify sto…
Politecnico di Milano（米兰理工大学）；NNAISENSE SA（NNAISENSE公司）
This paper introduces Non-Autonomous Input-Output Stable Network(NAIS-Net), a very deep architecture where each stacked processing block is derived from a time-invariant non-autonomous dynamical system. Non-autonomy is implemented by skip connections from the block input to each of the unrolled processing stages and allows stability to be enforced so that blocks can be unrolled adaptively to a pattern-dependent processing depth. NAIS-Net induces non-trivial, Lipschitz input-output maps, even for an infinite unroll length. We prove that the network is globally asymptotically stable so that for …
TU Darmstadt（达姆施塔特工业大学）；University of Verona（维罗纳大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）
We introduce ImitationFlow, a novel Deep generative model that allows learning complex globally stable, stochastic, nonlinear dynamics. Our approach extends the Normalizing Flows framework to learn stable Stochastic Differential Equations. We prove the Lyapunov stability for a class of Stochastic Differential Equations and we propose a learning algorithm to learn them from a set of demonstrated trajectories. Our model extends the set of stable dynamical systems that can be represented by state-of-the-art approaches, eliminates the Gaussian assumption on the demonstrations, and outperforms the …
Graduate School of Medicine（医学研究生院）；Kyoto University（京都大学）
Learning stable dynamics from observed time-series data is an essential problem in robotics, physical modeling, and systems biology. Many of these dynamics are represented as an inputs-output system to communicate with the external environment. In this study, we focus on input-output stable systems, exhibiting robustness against unexpected stimuli and noise. We propose a method to learn nonlinear systems guaranteeing the input-output stability. Our proposed method utilizes the differentiable projection onto the space satisfying the Hamilton-Jacobi inequality to realize the input-output stabili…
Fraunhofer IAIS（弗劳恩霍夫应用信息技术研究所）；IIT, Bombay（孟买印度理工学院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
A recent line of work has established intriguing connections between the generalization/compression properties of a deep neural network (DNN) model and the so-called layer weights' stable ranks. Intuitively, the latter are indicators of the effective number of parameters in the net. In this work, we address some natural questions regarding the space of DNNs conditioned on the layers' stable rank, where we study feed-forward dynamics, initialization, training and expressivity. To this end, we first propose a random DNN model with a new sampling scheme based on stable rank. Then, we show how fee…
Deep Graph Networks (DGNs) currently dominate the research landscape of learning from graphs, due to their efficiency and ability to implement an adaptive message-passing scheme between the nodes. However, DGNs are typically limited in their ability to propagate and preserve long-term dependencies between nodes, i.e., they suffer from the over-squashing phenomena. This reduces their effectiveness, since predictive problems may require to capture interactions at different, and possibly large, radii in order to be effectively solved. In this work, we present Anti-Symmetric Deep Graph Networks (A…
Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end, neural ODEs regularized with neural Lyapunov functions are a promising approach when states are fully observed. For practical applications however, partial observations are the norm. As we will demonstrate, initialization of unobserved augmented states can become a key problem for neural ODEs. To alleviate this issue, we propose to augment the system's state …
Lehigh University（理海大学）；Sandia National Laboratories（桑迪亚国家实验室）；IBM Research（IBM研究院）
Neural operators have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel …
NVIDIA Research（英伟达研究院）；Technion - Israel Institute of Technology（以色列理工学院）；Indian Institute of Technology, Indore（印度理工学院印多尔分校）
In recent years, there has been significant progress in applying deep reinforcement learning (RL) for solving challenging problems across a wide variety of domains. Nevertheless, convergence of various methods has been shown to suffer from inconsistencies, due to algorithmic instability and variance, as well as stochasticity in the benchmark environments. Particularly, despite the fact that the agent's performance may be improving on average, it may abruptly deteriorate at late stages of training. In this work, we study methods for enhancing the agent's learning process, by providing conservat…
University of British Columbia（不列颠哥伦比亚大学）；Honeywell Process Solutions（霍尼韦尔过程解决方案）；Backstrom Systems Engineering Ltd.（巴克斯特罗姆系统工程有限公司）
We introduce a method for learning provably stable deep neural network based dynamic models from observed data. Specifically, we consider discrete-time stochastic dynamic models, as they are of particular interest in practical applications such as estimation and control. However, these aspects exacerbate the challenge of guaranteeing stability. Our method works by embedding a Lyapunov neural network into the dynamic model, thereby inherently satisfying the stability criterion. To this end, we propose two approaches and apply them in both the deterministic and stochastic settings: one exploits …
Compression has emerged as one of the essential deep learning research topics, especially for the edge devices that have limited computation power and storage capacity. Among the main compression techniques, low-rank compression via matrix factorization has been known to have two problems. First, an extensive tuning is required. Second, the resulting compression performance is typically not impressive. In this work, we propose a low-rank compression method that utilizes a modified beam-search for an automatic rank selection and a modified stable rank for a compression-friendly training. The re…
ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximations
Software Development Center, Industrial and Commercial Bank of China（中国工商银行软件开发中心）；Hisilicon Semiconductor and Component Business Dept.(2012 Labs), Huawei Technologies Co., Ltd（华为技术有限公司半导体及组件业务部）；NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing（数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Deep neural networks are typically initialized with random weights, with variances chosen to facilitate signal propagation and stable gradients. It is also believed that diversity of features is an important property of these initializations. We construct a deep convolutional network with identical features by initializing almost all the weights to $0$. The architecture also enables perfect signal propagation and stable gradients, and achieves high accuracy on standard benchmarks. This indicates that random, diverse initializations are \textit{not} necessary for training neural networks. An es…
National Institute of Informatics（国立信息学研究所）；University of Waterloo（滑铁卢大学）
Prioritized experience replay (PER) samples important transitions, rather than uniformly, to improve the performance of a deep reinforcement learning agent. We claim that such prioritization has to be balanced with sample diversity for making the DQN stabilized and preventing forgetting. Our proposed improvement over PER, called Predictive PER (PPER), takes three countermeasures (TDInit, TDClip, TDPred) to (i) eliminate priority outliers and explosions and (ii) improve the sample diversity and distributions, weighted by priorities, both leading to stabilizing the DQN. The most notable among th…
Approaches based on deep neural networks have achieved striking performance when testing data and training data share similar distribution, but can significantly fail otherwise. Therefore, eliminating the impact of distribution shifts between training and testing data is crucial for building performance-promising deep models. Conventional methods assume either the known heterogeneity of training data (e.g. domain labels) or the approximately equal capacities of different domains. In this paper, we consider a more challenging case where neither of the above assumptions holds. We propose to addr…
Supervised regression to demonstrations has been demonstrated to be a stable way to train deep policy networks. We are motivated to study how we can take full advantage of supervised loss functions for stably training deep reinforcement learning agents. This is a challenging task because it is unclear how the training data could be collected to enable policy improvement. In this work, we propose Self-Supervised Reinforcement Learning (SSRL), a simple algorithm that optimizes policies with purely supervised losses. We demonstrate that, without policy gradient or value estimation, an iterative p…
University of Cambridge（剑桥大学）；University of Oslo（奥斯陆大学）
Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities, however, there does not exist any algorithm, even randomised, that can train (or compute) such a NN. For any positive integers $K > 2$ and $L$, there are …
Tsinghua University（清华大学）；Shanghai Jiao Tong University（上海交通大学）；University of California, Berkeley（加州大学伯克利分校）；Microsoft Research Asia（微软亚洲研究院）；Microsoft（微软）
One of the key challenges in deploying RL to real-world applications is to adapt to variations of unknown environment contexts, such as changing terrains in robotic tasks and fluctuated bandwidth in congestion control. Existing works on adaptation to unknown environment contexts either assume the contexts are the same for the whole episode or assume the context variables are Markovian. However, in many real-world applications, the environment context usually stays stable for a stochastic period and then changes in an abrupt and unpredictable manner within an episode, resulting in a segment str…
Deep reinforcement learning has been applied more and more widely nowadays, especially in various complex control tasks. Effective exploration for noisy networks is one of the most important issues in deep reinforcement learning. Noisy networks tend to produce stable outputs for agents. However, this tendency is not always enough to find a stable policy for an agent, which decreases efficiency and stability during the learning process. Based on NoisyNets, this paper proposes an algorithm called NROWAN-DQN, i.e., Noise Reduction and Online Weight Adjustment NoisyNet-DQN. Firstly, we develop a n…
Computing stable configurations of confined smectic liquid crystals with a deep variational framework
School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Columbia University（哥伦比亚大学）；Center for Applied Mathematics, Renmin University of China（中国人民大学应用数学中心）；School of Mathematical Sciences, Beijing International Center for Mathematical Research, Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学数学科学学院、北京国际数学研究中心、定量生物学中心、机器学习研究中心）
Reinforcement learning with function approximation can be unstable and even divergent, especially when combined with off-policy learning and Bellman updates. In deep reinforcement learning, these issues have been dealt with empirically by adapting and regularizing the representation, in particular with auxiliary tasks. This suggests that representation learning may provide a means to guarantee stability. In this paper, we formally show that there are indeed nontrivial state representations under which the canonical TD algorithm is stable, even when learning off-policy. We analyze representatio…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；RWTH-Aachen University（亚琛工业大学）；Simula Research Laboratory（西穆拉研究实验室）；University of San Diego（圣地亚哥大学）
In this paper we approach the problem of unique and stable identifiability of generic deep artificial neural networks with pyramidal shape and smooth activation functions from a finite number of input-output samples. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as $\mathcal O(D^2 \times m)$ nonadaptive inp…
Continental-NTU Corporate Lab（大陆-南洋理工大学联合实验室）；Nanyang Technological University（南洋理工大学）；Singapore University of Social Sciences（新跃社科大学）
Deep neural networks (DNNs) are well-known to be vulnerable to adversarial attacks, where malicious human-imperceptible perturbations are included in the input to the deep network to fool it into making a wrong classification. Recent studies have demonstrated that neural Ordinary Differential Equations (ODEs) are intrinsically more robust against adversarial attacks compared to vanilla DNNs. In this work, we propose a stable neural ODE with Lyapunov-stable equilibrium points for defending against adversarial attacks (SODEF). By ensuring that the equilibrium points of the ODE solution used as p…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究院）；Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences（俄罗斯科学院韦尔纳茨基地球化学与分析化学研究所）；Lebedev Physical Institute, Russian Academy of Sciences（俄罗斯科学院列别杰夫物理研究所）
We present a new approach for the generation of stable structures of nanoclusters using deep learning methods. Our method consists in constructing an artificial potential energy surface, with local minima corresponding to the most stable structures and which is much smoother than "real" potential in the intermediate regions of the configuration space. To build the surface, graph convolutional networks are used. The method can extrapolates the potential surface to cases of structures with larger number of atoms than was used in training. Thus, having a sufficient number of low-energy structures…
