Recurrent neural networks (RNNs) are particularly well-suited for modeling long-term dependencies in sequential data, but are notoriously hard to train because the error backpropagated in time either vanishes or explodes at an exponential rate. While a number of works attempt to mitigate this effect through gated recurrent units, well-chosen parametric constraints, and skip-connections, we develop a novel perspective that seeks to evolve the hidden state on the equilibrium manifold of an ordinary differential equation (ODE). We propose a family of novel RNNs, namely {\em Equilibriated Recurrent Neural Networks} (ERNNs) that overcome the gradient decay or explosion effect and lead to recurrent models that evolve on the equilibrium manifold. We show that equilibrium points are stable, leading to fast convergence of the discretized ODE to fixed points. Furthermore, ERNNs account for long-term dependencies, and can efficiently recall informative aspects of data from the distant past. We show that ERNNs achieve state-of-the-art accuracy on many challenging data sets with 3-10x speedups, 1.5-3x model size reduction, and with similar prediction cost relative to vanilla RNNs.
Shizuoka University（静冈大学）；Shonan Institute of Technology（湘南工业大学）
A common problem in training neural networks is the vanishing and/or exploding gradient problem which is more prominently seen in training of Recurrent Neural Networks (RNNs). Thus several algorithms have been proposed for training RNNs. This paper proposes a novel adaptive stochastic Nesterov accelerated quasiNewton (aSNAQ) method for training RNNs. The proposed method aSNAQ is an accelerated method that uses the Nesterov's gradient term along with second order curvature information. The performance of the proposed method is evaluated in Tensorflow on benchmark sequence modeling problems. The…
Federal University of Minas Gerais（米纳斯吉拉斯联邦大学）；Eindhoven University of Technology（埃因霍温理工大学）；Uppsala University（乌普萨拉大学）
The exploding and vanishing gradient problem has been the major conceptual principle behind most architecture and training improvements in recurrent neural networks (RNNs) during the last decade. In this paper, we argue that this principle, while powerful, might need some refinement to explain recent developments. We refine the concept of exploding gradients by reformulating the problem in terms of the cost function smoothness, which gives insight into higher-order derivatives and the existence of regions with many close local minima. We also clarify the distinction between vanishing gradients…
Unitary recurrent neural networks (URNNs) have been proposed as a method to overcome the vanishing and exploding gradient problem in modeling data with long-term dependencies. A basic question is how restrictive is the unitary constraint on the possible input-output mappings of such a network? This work shows that for any contractive RNN with ReLU activations, there is a URNN with at most twice the number of hidden states and the identical input-output mapping. Hence, with ReLU activations, URNNs are as expressive as general RNNs. In contrast, for certain smooth activations, it is shown that t…
We propose a new STAckable Recurrent cell (STAR) for recurrent neural networks (RNNs), which has fewer parameters than widely used LSTM and GRU while being more robust against vanishing or exploding gradients. Stacking recurrent units into deep architectures suffers from two major limitations: (i) many recurrent cells (e.g., LSTMs) are costly in terms of parameters and computation resources; and (ii) deep RNNs are prone to vanishing or exploding gradients during training. We investigate the training of multi-layer RNNs and examine the magnitude of the gradients as they propagate through the ne…
University of California, Santa Cruz（加州大学圣克鲁兹分校）；University of California, Irvine（加州大学欧文分校）；Columbia University（哥伦比亚大学）
Recurrent Neural Networks (RNNs) frequently exhibit complicated dynamics, and their sensitivity to the initialization process often renders them notoriously hard to train. Recent works have shed light on such phenomena analyzing when exploding or vanishing gradients may occur, either of which is detrimental for training dynamics. In this paper, we point to a formal connection between RNNs and chaotic dynamical systems and prove a qualitatively stronger phenomenon about RNNs than what exploding gradients seem to suggest. Our main result proves that under standard initialization (e.g., He, Xavie…
New York University（纽约大学）；Rice University（莱斯大学）；Baylor College of Medicine（贝勒医学院）
Training recurrent neural networks (RNNs) is a hard problem due to degeneracies in the optimization landscape, a problem also known as vanishing/exploding gradients. Short of designing new RNN architectures, previous methods for dealing with this problem usually boil down to orthogonalization of the recurrent dynamics, either at initialization or during the entire training period. The basic motivation behind these methods is that orthogonal transformations are isometries of the Euclidean space, hence they preserve (Euclidean) norms and effectively deal with vanishing/exploding gradients. Howev…
Contemporary wisdom based on empirical studies suggests that standard recurrent neural networks (RNNs) do not perform well on tasks requiring long-term memory. However, precise reasoning for this behavior is still unknown. This paper provides a rigorous explanation of this property in the special case of linear RNNs. Although this work is limited to linear RNNs, even these systems have traditionally been difficult to analyze due to their non-linear parameterization. Using recently-developed kernel regime analysis, our main result shows that linear RNNs learned from random initializations are f…
In recent years, using orthogonal matrices has been shown to be a promising approach in improving Recurrent Neural Networks (RNNs) with training, stability, and convergence, particularly, to control gradients. While Gated Recurrent Unit (GRU) and Long Short Term Memory (LSTM) architectures address the vanishing gradient problem by using a variety of gates and memory cells, they are still prone to the exploding gradient problem. In this work, we analyze the gradients in GRU and propose the usage of orthogonal matrices to prevent exploding gradient problems and enhance long-term memory. We study…
Central Institute of Mental Health（中央心理健康研究所）；Heidelberg University（海德堡大学）
Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or - more recently - imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, …
Recurrent neural networks (RNNs) have been successfully used on a wide range of sequential data problems. A well known difficulty in using RNNs is the \textit{vanishing or exploding gradient} problem. Recently, there have been several different RNN architectures that try to mitigate this issue by maintaining an orthogonal or unitary recurrent weight matrix. One such architecture is the scaled Cayley orthogonal recurrent neural network (scoRNN) which parameterizes the orthogonal recurrent weight matrix through a scaled Cayley transform. This parametrization contains a diagonal scaling matrix co…
Recurrent Neural Networks (RNNs) are designed to handle sequential data but suffer from vanishing or exploding gradients. Recent work on Unitary Recurrent Neural Networks (uRNNs) have been used to address this issue and in some cases, exceed the capabilities of Long Short-Term Memory networks (LSTMs). We propose a simpler and novel update scheme to maintain orthogonal recurrent weight matrices without using complex valued matrices. This is done by parametrizing with a skew-symmetric matrix using the Cayley transform. Such a parametrization is unable to represent matrices with negative one eige…
The design of recurrent neural networks (RNNs) to accurately process sequential inputs with long-time dependencies is very challenging on account of the exploding and vanishing gradient problem. To overcome this, we propose a novel RNN architecture which is based on a structure preserving discretization of a Hamiltonian system of second-order ordinary differential equations that models networks of oscillators. The resulting RNN is fast, invertible (in time), memory efficient and we derive rigorous bounds on the hidden state gradients to prove the mitigation of the exploding and vanishing gradi…
Lehrstuhl für Robotik und Echtzeitsysteme（机器人与实时系统教席）；Fakultät für Informatik（计算机科学学院）；Technische Universität München（慕尼黑工业大学）
Recurrent Neural Networks (RNNs) are rich models for the processing of sequential data. Recent work on advancing the state of the art has been focused on the optimization or modelling of RNNs, mostly motivated by adressing the problems of the vanishing and exploding gradients. The control of overfitting has seen considerably less attention. This paper contributes to that by analyzing fast dropout, a recent regularization method for generalized linear models and neural networks from a back-propagation inspired perspective. We show that fast dropout implements a quadratic form of an adaptive, pe…
Long Short-Term Memory (LSTM) is a recurrent neural network (RNN) architecture that has been designed to address the vanishing and exploding gradient problems of conventional RNNs. Unlike feedforward neural networks, RNNs have cyclic connections making them powerful for modeling sequences. They have been successfully used for sequence labeling and sequence prediction tasks, such as handwriting recognition, language modeling, phonetic labeling of acoustic frames. However, in contrast to the deep neural networks, the use of RNNs in speech recognition has been limited to phone recognition in smal…
IST Austria（奥地利科学技术研究所）；TU Wien（维也纳工业大学）；MIT（麻省理工学院）
Recurrent neural networks (RNNs) with continuous-time hidden states are a natural fit for modeling irregularly-sampled time series. These models, however, face difficulties when the input data possess long-term dependencies. We prove that similar to standard RNNs, the underlying reason for this issue is the vanishing or exploding of the gradient during training. This phenomenon is expressed by the ordinary differential equation (ODE) representation of the hidden state, regardless of the ODE solver's choice. We provide a solution by designing a new algorithm based on the long short-term memory …
Shandong University（山东大学）；University of Wollongong（伍伦贡大学）
Recurrent neural networks (RNNs) are known to be difficult to train due to the gradient vanishing and exploding problems and thus difficult to learn long-term patterns and construct deep networks. To address these problems, this paper proposes a new type of RNNs with the recurrent connection formulated as Hadamard product, referred to as independently recurrent neural network (IndRNN), where neurons in the same layer are independent of each other and connected across layers. Due to the better behaved gradient backpropagation, IndRNN with regulated recurrent weights effectively addresses the gr…
Ben-Gurion University（本-古里安大学）；ICSI（国际计算机科学研究所）；UC Berkeley（加州大学伯克利分校）；Technion – Israel Institute of Technology（以色列理工学院）
Recently, orthogonal recurrent neural networks (RNNs) have emerged as state-of-the-art models for learning long-term dependencies. This class of models mitigates the exploding and vanishing gradients problem by design. In this work, we employ tools and insights from differential geometry to offer a novel perspective on orthogonal RNNs. We show that orthogonal RNNs may be viewed as optimizing in the space of divergence-free vector fields. Specifically, based on a well-known result in differential geometry that relates vector fields and linear operators, we prove that every divergence-free vecto…
The Chinese University of Hong Kong（香港中文大学）；Microsoft Research（微软研究院）
Recurrent Neural Networks (RNNs) have become increasingly popular for the task of language understanding. In this task, a semantic tagger is deployed to associate a semantic label to each word in an input sequence. The success of RNN may be attributed to its ability to memorize long-term dependence that relates the current-time semantic label prediction to the observations many time instances away. However, the memory capacity of simple RNNs is limited because of the gradient vanishing and exploding problem. We propose to use an external memory to improve memorization capability of RNNs. We co…
University of Electronic Science and Technology of China（电子科技大学）；University of Wollongong（伍伦贡大学）
Recurrent neural networks (RNNs) have been widely used for processing sequential data. However, RNNs are commonly difficult to train due to the well-known gradient vanishing and exploding problems and hard to learn long-term patterns. Long short-term memory (LSTM) and gated recurrent unit (GRU) were developed to address these problems, but the use of hyperbolic tangent and the sigmoid action functions results in gradient decay over layers. Consequently, construction of an efficiently trainable deep network is challenging. In addition, all the neurons in an RNN layer are entangled together and …
University Of Virginia（弗吉尼亚大学）；Indiana University（印第安纳大学）；Boston University（波士顿大学）
Extracting temporal relationships over a range of scales is a hallmark of human perception and cognition -- and thus it is a critical feature of machine learning applied to real-world problems. Neural networks are either plagued by the exploding/vanishing gradient problem in recurrent neural networks (RNNs) or must adjust their parameters to learn the relevant time scales (e.g., in LSTMs). This paper introduces DeepSITH, a network comprising biologically-inspired Scale-Invariant Temporal History (SITH) modules in series with dense connections between layers. SITH modules respond to their input…
Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；Université de Montréal（蒙特利尔大学）；Ecole Polytechnique de Montréal（蒙特利尔高等理工学院）
A recent strategy to circumvent the exploding and vanishing gradient problem in RNNs, and to allow the stable propagation of signals over long time scales, is to constrain recurrent connectivity matrices to be orthogonal or unitary. This ensures eigenvalues with unit norm and thus stable dynamics and training. However this comes at the cost of reduced expressivity due to the limited variety of orthogonal transformations. We propose a novel connectivity structure based on the Schur decomposition and a splitting of the Schur form into normal and non-normal parts. This allows to parametrize matri…
Several variants of recurrent neural networks (RNNs) with orthogonal or unitary recurrent matrices have recently been developed to mitigate the vanishing/exploding gradient problem and to model long-term dependencies of sequences. However, with the eigenvalues of the recurrent matrix on the unit circle, the recurrent state retains all input information which may unnecessarily consume model capacity. In this paper, we address this issue by proposing an architecture that expands upon an orthogonal/unitary RNN with a state that is generated by a recurrent matrix with eigenvalues in the unit disc.…
Heidelberg University（海德堡大学）；Central Institute of Mental Health（中央心理健康研究所）
A main theoretical interest in biology and physics is to identify the nonlinear dynamical system (DS) that generated observed time series. Recurrent Neural Networks (RNNs) are, in principle, powerful enough to approximate any underlying DS, but in their vanilla form suffer from the exploding vs. vanishing gradients problem. Previous attempts to alleviate this problem resulted either in more complicated, mathematically less tractable RNN architectures, or strongly limited the dynamical expressiveness of the RNN. Here we address this issue by suggesting a simple regularization scheme for vanilla…
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…
pLSTM: parallelizable Linear Source Transition Mark networks
ELLIS Unit Linz（林茨ELLIS单位）；LIT AI Lab（LIT人工智能实验室）；Institute for Machine Learning（机器学习研究所）；Johannes Kepler University Linz（约翰·凯撒大学林茨分校）；NXAI GmbH（NXAI公司）
Recurrent neural networks (RNNs) have gained a great deal of attention in solving sequential learning problems. The learning of long-term dependencies, however, remains challenging due to the problem of a vanishing or exploding hidden states gradient. By exploring further the recently established connections between RNNs and dynamical systems we propose a novel RNN architecture, which we call a Hamiltonian recurrent neural network (Hamiltonian RNN), based on a symplectic discretization of an appropriately chosen Hamiltonian system. The key benefit of this approach is that the corresponding RNN…
IBM Thomas J. Watson Research Center（IBM托马斯·J·沃森研究中心）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Learning with recurrent neural networks (RNNs) on long sequences is a notoriously difficult task. There are three major challenges: 1) complex dependencies, 2) vanishing and exploding gradients, and 3) efficient parallelization. In this paper, we introduce a simple yet effective RNN connection structure, the DilatedRNN, which simultaneously tackles all of these challenges. The proposed architecture is characterized by multi-resolution dilated recurrent skip connections and can be combined flexibly with diverse RNN cells. Moreover, the DilatedRNN reduces the number of parameters needed and enha…
ETH Zürich（苏黎世联邦理工学院）；Seminar for Applied Mathematics (SAM)（应用数学研讨班（SAM））；Department of Mathematics（数学系）
Circuits of biological neurons, such as in the functional parts of the brain can be modeled as networks of coupled oscillators. Inspired by the ability of these systems to express a rich set of outputs while keeping (gradients of) state variables bounded, we propose a novel architecture for recurrent neural networks. Our proposed RNN is based on a time-discretization of a system of second-order ordinary differential equations, modeling networks of controlled nonlinear oscillators. We prove precise bounds on the gradients of the hidden states, leading to the mitigation of the exploding and vani…
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…
Idiap Research Institute（伊迪亚普研究所）；EPFL（洛桑联邦理工学院）；Facebook AI Research（Facebook人工智能研究院）
Our work addresses two important issues with recurrent neural networks: (1) they are over-parameterized, and (2) the recurrence matrix is ill-conditioned. The former increases the sample complexity of learning and the training time. The latter causes the vanishing and exploding gradient problem. We present a flexible recurrent neural network model called Kronecker Recurrent Units (KRU). KRU achieves parameter efficiency in RNNs through a Kronecker factored recurrent matrix. It overcomes the ill-conditioning of the recurrent matrix by enforcing soft unitary constraints on the factors. Thanks to…
Data61, CSIRO（澳大利亚联邦科学与工业研究组织Data61）；The University of Melbourne（墨尔本大学）
The problem of learning long-term dependencies in sequences using Recurrent Neural Networks (RNNs) is still a major challenge. Recent methods have been suggested to solve this problem by constraining the transition matrix to be unitary during training which ensures that its norm is equal to one and prevents exploding gradients. These methods either have limited expressiveness or scale poorly with the size of the network when compared with the simple RNN case, especially when using stochastic gradient descent with a small mini-batch size. Our contributions are as follows; we first show that con…
Institute of MMS of NASU（乌克兰国家科学院建模系统研究所）；Center for Cybernetics（控制论中心）
Vanishing (and exploding) gradients effect is a common problem for recurrent neural networks with nonlinear activation functions which use backpropagation method for calculation of derivatives. Deep feedforward neural networks with many hidden layers also suffer from this effect. In this paper we propose a novel universal technique that makes the norm of the gradient stay in the suitable range. We construct a way to estimate a contribution of each training example to the norm of the long-term components of the target function s gradient. Using this subroutine we can construct mini-batches for …
LIT AI Lab（LIT人工智能实验室）；Institute of Bioinformatics（生物信息学研究所）；Johannes Kepler University Linz（约翰内斯·开普勒大学林茨分校）
Deep Learning has revolutionized vision via convolutional neural networks (CNNs) and natural language processing via recurrent neural networks (RNNs). However, success stories of Deep Learning with standard feed-forward neural networks (FNNs) are rare. FNNs that perform well are typically shallow and, therefore cannot exploit many levels of abstract representations. We introduce self-normalizing neural networks (SNNs) to enable high-level abstract representations. While batch normalization requires explicit normalization, neuron activations of SNNs automatically converge towards zero mean and …
Department of Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学恩佐·法拉利工程系）；University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学）；Engineering “Enzo Ferrari”, University of Modena and Reggio Emilia（摩德纳和雷焦艾米利亚大学恩佐·法拉利工程系）；Department of Engineering “Enzo Ferrari”（恩佐·法拉利工程系）
Recurrent Neural Networks with Long Short-Term Memory (LSTM) make use of gating mechanisms to mitigate exploding and vanishing gradients when learning long-term dependencies. For this reason, LSTMs and other gated RNNs are widely adopted, being the standard de facto for many sequence modeling tasks. Although the memory cell inside the LSTM contains essential information, it is not allowed to influence the gating mechanism directly. In this work, we improve the gate potential by including information coming from the internal cell state. The proposed modification, named Working Memory Connection…
