Recurrent neural networks (RNNs) are a class of nonlinear dynamical systems often used to model sequence-to-sequence maps. RNNs have excellent expressive power but lack the stability or robustness guarantees that are necessary for many applications. In this paper, we formulate convex sets of RNNs with stability and robustness guarantees. The guarantees are derived using incremental quadratic constraints and can ensure global exponential stability of all solutions, and bounds on incremental $ \ell_2 $ gain (the Lipschitz constant of the learned sequence-to-sequence mapping). Using an implicit model structure, we construct a parametrization of RNNs that is jointly convex in the model parameters and stability certificate. We prove that this model structure includes all previously-proposed convex sets of stable RNNs as special cases, and also includes all stable linear dynamical systems. We illustrate the utility of the proposed model class in the context of non-linear system identification.
Data-space inversion (DSI) and related procedures represent a family of methods applicable for data assimilation in subsurface flow settings. These methods differ from model-based techniques in that they provide only posterior predictions for quantities (time series) of interest, not posterior models with calibrated parameters. DSI methods require a large number of flow simulations to first be performed on prior geological realizations. Given observed data, posterior predictions can then be generated directly. DSI operates in a Bayesian setting and provides posterior samples of the data vector…
When designing large-scale distributed controllers, the information-sharing constraints between sub-controllers, as defined by a communication topology interconnecting them, are as important as the controller itself. Controllers implemented using dense topologies typically outperform those implemented using sparse topologies, but it is also desirable to minimize the cost of controller deployment. Motivated by the above, we introduce a compact but expressive graph recurrent neural network (GRNN) parameterization of distributed controllers that is well suited for distributed controller and commu…
This paper investigates the use of extended Kalman filtering to train recurrent neural networks with rather general convex loss functions and regularization terms on the network parameters, including $\ell_1$-regularization. We show that the learning method is competitive with respect to stochastic gradient descent in a nonlinear system identification benchmark and in training a linear system with binary outputs. We also explore the use of the algorithm in data-driven nonlinear model predictive control and its relation with disturbance models for offset-free closed-loop tracking.
We propose a formulation for nonlinear recurrent models that includes simple parametric models of recurrent neural networks as a special case. The proposed formulation leads to a natural estimator in the form of a convex program. We provide a sample complexity for this estimator in the case of stable dynamics, where the nonlinear recursion has a certain contraction property, and under certain regularity conditions on the input distribution. We evaluate the performance of the estimator by simulation on synthetic data. These numerical experiments also suggest the extent at which the imposed theo…
University of Washington（华盛顿大学）；Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）
Recurrent neural networks are powerful models for processing sequential data, but they are generally plagued by vanishing and exploding gradient problems. Unitary recurrent neural networks (uRNNs), which use unitary recurrence matrices, have recently been proposed as a means to avoid these issues. However, in previous experiments, the recurrence matrices were restricted to be a product of parameterized unitary matrices, and an open question remains: when does such a parameterization fail to represent all unitary matrices, and how does this restricted representational capacity limit what can be…
We study the effects of mild over-parameterization on the optimization landscape of a simple ReLU neural network of the form $\mathbf{x}\mapsto\sum_{i=1}^k\max\{0,\mathbf{w}_i^{\top}\mathbf{x}\}$, in a well-studied teacher-student setting where the target values are generated by the same architecture, and when directly optimizing over the population squared loss with respect to Gaussian inputs. We prove that while the objective is strongly convex around the global minima when the teacher and student networks possess the same number of neurons, it is not even \emph{locally convex} after any amo…
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗系统）
In this work, we study over-parameterization as a necessary condition for having the ability for the models to extrapolate outside the convex hull of training set. We specifically, consider classification models, e.g., image classification and other applications of deep learning. Such models are classification functions that partition their domain and assign a class to each partition \cite{strang2019linear}. Partitions are defined by decision boundaries and so is the classification model/function. Convex hull of training set may occupy only a subset of the domain, but trained model may partiti…
Australian Center for Field Robotics（澳大利亚现场机器人中心）；Sydney Institute for Robotics and Intelligent Systems（悉尼机器人与智能系统研究所）；The University of Sydney（悉尼大学）
Stability of recurrent models is closely linked with trainability, generalizability and in some applications, safety. Methods that train stable recurrent neural networks, however, do so at a significant cost to expressibility. We propose an implicit model structure that allows for a convex parametrization of stable models using contraction analysis of non-linear systems. Using these stability conditions we propose a new approach to model initialization and then provide a number of empirical results comparing the performance of our proposed model set to previous stable RNNs and vanilla RNNs. By…
This paper proposes a novel algorithm for training recurrent neural network models of nonlinear dynamical systems from an input/output training dataset. Arbitrary convex and twice-differentiable loss functions and regularization terms are handled by sequential least squares and either a line-search (LS) or a trust-region method of Levenberg-Marquardt (LM) type for ensuring convergence. In addition, to handle non-smooth regularization terms such as $\ell_1$, $\ell_0$, and group-Lasso regularizers, as well as to impose possibly non-convex constraints such as integer and mixed-integer constraints…
There is an implicit assumption that by unfolding recurrent neural networks (RNN) in finite time, the misspecification of choosing a zero value for the initial hidden state is mitigated by later time steps. This assumption has been shown to work in practice and alternative initialization may be suggested but often overlooked. In this paper, we propose a method of parameterizing the initial hidden state of an RNN. The resulting architecture, referred to as a Contextual RNN, can be trained end-to-end. The performance on an associative retrieval task is found to improve by conditioning the RNN in…
Humboldt-Universität zu Berlin（柏林洪堡大学）；Bosch Center for AI（博世人工智能中心）
Recurrent neural networks (RNNs) are a popular choice for modeling sequential data. Modern RNN architectures assume constant time-intervals between observations. However, in many datasets (e.g. medical records) observation times are irregular and can carry important information. To address this challenge, we propose continuous recurrent units (CRUs) -- a neural architecture that can naturally handle irregular intervals between observations. The CRU assumes a hidden state, which evolves according to a linear stochastic differential equation and is integrated into an encoder-decoder framework. T…
Australian Centre for Field Robotics（澳大利亚野外机器人中心）；The University of Sydney（悉尼大学）
This paper presents a parameterization of nonlinear controllers for uncertain systems building on a recently developed neural network architecture, called the recurrent equilibrium network (REN), and a nonlinear version of the Youla parameterization. The proposed framework has "built-in" guarantees of stability, i.e., all policies in the search space result in a contracting (globally exponentially stable) closed-loop system. Thus, it requires very mild assumptions on the choice of cost function and the stability property can be generalized to unseen data. Another useful feature of this approac…
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；University of Cambridge（剑桥大学）；Université de Montréal（蒙特利尔大学）
Recurrent neural networks (RNNs) are notoriously difficult to train. When the eigenvalues of the hidden to hidden weight matrix deviate from absolute value 1, optimization becomes difficult due to the well studied issue of vanishing and exploding gradients, especially when trying to learn long-term dependencies. To circumvent this problem, we propose a new architecture that learns a unitary weight matrix, with eigenvalues of absolute value exactly 1. The challenge we address is that of parametrizing unitary matrices in a way that does not require expensive computations (such as eigendecomposit…
Multidimensional recurrent neural networks (MDRNNs) have shown a remarkable performance in the area of speech and handwriting recognition. The performance of an MDRNN is improved by further increasing its depth, and the difficulty of learning the deeper network is overcome by using Hessian-free (HF) optimization. Given that connectionist temporal classification (CTC) is utilized as an objective of learning an MDRNN for sequence labeling, the non-convexity of CTC poses a problem when applying HF to the network. As a solution, a convex approximation of CTC is formulated and its relationship with…
Department of Electrical and Computer Engineering Michigan State University（密歇根州立大学电气与计算机工程系）；Circuits, Systems, and Neural Networks (CSANN) Lab（电路、系统与神经网络（CSANN）实验室）
Long Short-Term Memory (LSTM) Recurrent Neural networks (RNNs) rely on gating signals, each driven by a function of a weighted sum of at least 3 components: (i) one of an adaptive weight matrix multiplied by the incoming external input vector sequence, (ii) one adaptive weight matrix multiplied by the previous memory/state vector, and (iii) one adaptive bias vector. In effect, they augment the simple Recurrent Neural Networks (sRNNs) structure with the addition of a "memory cell" and the incorporation of at most 3 gating signals. The standard LSTM structure and components encompass redundancy …
École Polytechnique de Montréal（蒙特利尔理工学院）；Montreal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；CHUM Research Center（蒙特利尔大学医学中心研究中心）
It is well known that it is challenging to train deep neural networks and recurrent neural networks for tasks that exhibit long term dependencies. The vanishing or exploding gradient problem is a well known issue associated with these challenges. One approach to addressing vanishing and exploding gradients is to use either soft or hard constraints on weight matrices so as to encourage or enforce orthogonality. Orthogonal matrices preserve gradient norm during backpropagation and may therefore be a desirable property. This paper explores issues with optimization convergence, speed and gradient …
Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks~(RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that arise in its training. Specifically, we parameterize the transition matrix by its singular value decomposition(SVD), which allows us to explicitly track and control its singular values. We attain efficiency by using tools that are common in numerical linear algebra, namely Househ…
React to Surprises: Stable-by-Design Neural Feedback Control and the Youla-REN
Australian Centre for Robotics（澳大利亚机器人中心）；School of Aerospace, Mechanical and Mechatronic Engineering（航空航天、机械与机电工程学院）；The University of Sydney（悉尼大学）；Laboratory for Information and Decision Systems（信息与决策系统实验室）；Dept. Electrical Engineering and Computer Science（电气工程与计算机科学系）
Recurrent Neural Networks (RNNs) offer fast inference on long sequences but are hard to optimize and slow to train. Deep state-space models (SSMs) have recently been shown to perform remarkably well on long sequence modeling tasks, and have the added benefits of fast parallelizable training and RNN-like fast inference. However, while SSMs are superficially similar to RNNs, there are important differences that make it unclear where their performance boost over RNNs comes from. In this paper, we show that careful design of deep RNNs using standard signal propagation arguments can recover the imp…
The accuracy and complexity of kernel learning algorithms is determined by the set of kernels over which it is able to optimize. An ideal set of kernels should: admit a linear parameterization (tractability); be dense in the set of all kernels (accuracy); and every member should be universal so that the hypothesis space is infinite-dimensional (scalability). Currently, there is no class of kernel that meets all three criteria - e.g. Gaussians are not tractable or accurate; polynomials are not scalable. We propose a new class that meet all three criteria - the Tessellated Kernel (TK) class. Spe…
Microsoft Research AI（微软研究院AI）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；UT-Austin（德克萨斯大学奥斯汀分校）；University of Washington（华盛顿大学）；Harvard University（哈佛大学）
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely unsettled. In this work, we prove why stochastic gradient descent (SGD) can find $\textit{global minima}$ on the training objective of DNNs in $\textit{polynomial time}$. We only make two assumptions: the inputs are non-degenerate and the network is over-parameterized. The latter…
Deep unfolding methods---for example, the learned iterative shrinkage thresholding algorithm (LISTA)---design deep neural networks as learned variations of optimization methods. These networks have been shown to achieve faster convergence and higher accuracy than the original optimization methods. In this line of research, this paper develops a novel deep recurrent neural network (coined reweighted-RNN) by the unfolding of a reweighted $\ell_1$-$\ell_1$ minimization algorithm and applies it to the task of sequential signal reconstruction. To the best of our knowledge, this is the first deep un…
ETH Zürich（苏黎世联邦理工学院）；VMWare Research（威睿研究部）；UIUC（伊利诺伊大学厄巴纳-香槟分校）；Hitachi Power Grids Research（日立电网研究院）
We present a scalable and precise verifier for recurrent neural networks, called Prover based on two novel ideas: (i) a method to compute a set of polyhedral abstractions for the non-convex and nonlinear recurrent update functions by combining sampling, optimization, and Fermat's theorem, and (ii) a gradient descent based algorithm for abstraction refinement guided by the certification problem that combines multiple abstractions for each neuron. Using Prover, we present the first study of certifying a non-trivial use case of recurrent neural networks, namely speech classification. To achieve t…
Similar to convolution neural networks, recurrent neural networks (RNNs) typically suffer from over-parameterization. Quantizing bit-widths of weights and activations results in runtime efficiency on hardware, yet it often comes at the cost of reduced accuracy. This paper proposes a quantization approach that increases model size with bit-width reduction. This approach will allow networks to perform at their baseline accuracy while still maintaining the benefits of reduced precision and overall model size reduction.
Sydney Institute for Robotics and Intelligent Systems (SIRIS)（悉尼机器人与智能系统研究所）；Australian Centre for Field Robotics (ACFR)（澳大利亚野外机器人中心）；University of Sydney（悉尼大学）
This paper introduces new parameterizations of equilibrium neural networks, i.e. networks defined by implicit equations. This model class includes standard multilayer and residual networks as special cases. The new parameterization admits a Lipschitz bound during training via unconstrained optimization: no projections or barrier functions are required. Lipschitz bounds are a common proxy for robustness and appear in many generalization bounds. Furthermore, compared to previous works we show well-posedness (existence of solutions) under less restrictive conditions on the network weights and mor…
Imperial College London（伦敦帝国理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；EDF R&D（法国电力公司研发部）
Data assimilation techniques are widely used to predict complex dynamical systems with uncertainties, based on time-series observation data. Error covariance matrices modelling is an important element in data assimilation algorithms which can considerably impact the forecasting accuracy. The estimation of these covariances, which usually relies on empirical assumptions and physical constraints, is often imprecise and computationally expensive especially for systems of large dimension. In this work, we propose a data-driven approach based on long short term memory (LSTM) recurrent neural networ…
Université Paris-Est（巴黎东部大学）；École des Ponts ParisTech（巴黎高科桥路学校）；Inria（法国国家信息与自动化研究所）
We propose to impose symmetry in neural network parameters to improve parameter usage and make use of dedicated convolution and matrix multiplication routines. Due to significant reduction in the number of parameters as a result of the symmetry constraints, one would expect a dramatic drop in accuracy. Surprisingly, we show that this is not the case, and, depending on network size, symmetry can have little or no negative effect on network accuracy, especially in deep overparameterized networks. We propose several ways to impose local symmetry in recurrent and convolutional neural networks, and…
Microsoft Research AI（微软研究院AI）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；UT-Austin（德克萨斯大学奥斯汀分校）；University of Washington（华盛顿大学）；Harvard University（哈佛大学）
How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory only covers networks with one hidden layer, so can we go deeper? In this paper, we focus on recurrent neural networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the $\textit{same}$ recurrent unit is repeatedly applied across the entire time horizon of l…
Northeastern University（东北大学）；University of Notre Dame（圣母大学）；MIT（麻省理工学院）
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on h…
Yale Center for Medical Informatics（耶鲁医学信息学中心）；VA Connecticut Healthcare System（康涅狄格州退伍军人医疗系统）
We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These models have orders of magnitude more parameters than their training samples, and they can achieve perfect accuracy on their training sets, even when training images are randomly labeled, or the contents of images are replaced with random noise. The training loss function of these models has infinite number of near zero minimizers, where only a small subset of …
This paper presents a new deep learning-based framework for robust nonlinear estimation and control using the concept of a Neural Contraction Metric (NCM). The NCM uses a deep long short-term memory recurrent neural network for a global approximation of an optimal contraction metric, the existence of which is a necessary and sufficient condition for exponential stability of nonlinear systems. The optimality stems from the fact that the contraction metrics sampled offline are the solutions of a convex optimization problem to minimize an upper bound of the steady-state Euclidean distance between…
This paper proposes a family of online second order methods for possibly non-convex stochastic optimizations based on the theory of preconditioned stochastic gradient descent (PSGD), which can be regarded as an enhance stochastic Newton method with the ability to handle gradient noise and non-convexity simultaneously. We have improved the implementations of the original PSGD in several ways, e.g., new forms of preconditioners, more accurate Hessian vector product calculations, and better numerical stability with vanishing or ill-conditioned Hessian, etc.. We also have unrevealed the relationsh…
Columbia University（哥伦比亚大学）；Singapore University of Technology and Design（新加坡科技设计大学）
We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem …
Borealis AI；University of Toronto（多伦多大学）；Vector Institute（Vector研究所）
Most neural networks are trained using first-order optimization methods, which are sensitive to the parameterization of the model. Natural gradient descent is invariant to smooth reparameterizations because it is defined in a coordinate-free way, but tractable approximations are typically defined in terms of coordinate systems, and hence may lose the invariance properties. We analyze the invariance properties of the Kronecker-Factored Approximate Curvature (K-FAC) algorithm by constructing the algorithm in a coordinate-free way. We explicitly construct a Riemannian metric under which the natur…
Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training
Department of Computer Science, Vanderbilt University, Nashville TN, USA（范德比尔特大学计算机科学系）；National Renewable Energy Laboratory, Golden CO, USA（国家可再生能源实验室）
King’s College London（伦敦国王学院）；Nanjing University of Information Science and Technology（南京信息工程大学）
This letter investigates a channel assignment problem in uplink wireless communication systems. Our goal is to maximize the sum rate of all users subject to integer channel assignment constraints. A convex optimization based algorithm is provided to obtain the optimal channel assignment, where the closed-form solution is obtained in each step. Due to high computational complexity in the convex optimization based algorithm, machine learning approaches are employed to obtain computational efficient solutions. More specifically, the data are generated by using convex optimization based algorithm …
In learning with recurrent or very deep feed-forward networks, employing unitary matrices in each layer can be very effective at maintaining long-range stability. However, restricting network parameters to be unitary typically comes at the cost of expensive parameterizations or increased training runtime. We propose instead an efficient method based on rank-$k$ updates -- or their rank-$k$ approximation -- that maintains performance at a nearly optimal training runtime. We introduce two variants of this method, named Direct (projUNN-D) and Tangent (projUNN-T) projected Unitary Neural Networks,…
