Recurrent neural networks (RNNs) are commonly trained with the truncated backpropagation-through-time (TBPTT) algorithm. For the purposes of computational tractability, the TBPTT algorithm truncates the chain rule and calculates the gradient on a finite block of the overall data sequence. Such approximation could lead to significant inaccuracies, as the block length for the truncated backpropagation is typically limited to be much smaller than the overall sequence length. In contrast, Real-time recurrent learning (RTRL) is an online optimization algorithm which asymptotically follows the true gradient of the loss on the data sequence as the number of sequence time steps $t \rightarrow \infty$. RTRL forward propagates the derivatives of the RNN hidden/memory units with respect to the parameters and, using the forward derivatives, performs online updates of the parameters at each time step in the data sequence. RTRL's online forward propagation allows for exact optimization over extremely long data sequences, although it can be computationally costly for models with large numbers of parameters. We prove convergence of the RTRL algorithm for a class of RNNs. The convergence analysis establishes a fixed point for the joint distribution of the data sequence, RNN hidden layer, and the RNN hidden layer forward derivatives as the number of data samples from the sequence and the number of training steps tend to infinity. We prove convergence of the RTRL algorithm to a stationary point of the loss. Numerical studies illustrate our theoretical results. One potential application area for RTRL is the analysis of financial data, which typically involve long time series and models with small to medium numbers of parameters. This makes RTRL computationally tractable and a potentially appealing optimization method for training models. Thus, we include an example of RTRL applied to limit order book data.
Despite all the impressive advances of recurrent neural networks, sequential data is still in need of better modelling. Truncated backpropagation through time (TBPTT), the learning algorithm most widely used in practice, suffers from the truncation bias, which drastically limits its ability to learn long-term dependencies.The Real-Time Recurrent Learning algorithm (RTRL) addresses this issue, but its high computational requirements make it infeasible in practice. The Unbiased Online Recurrent Optimization algorithm (UORO) approximates RTRL with a smaller runtime and memory cost, but with the d…
DeepMind（深度思维）；Google（谷歌）；University College London（伦敦大学学院）
Current methods for training recurrent neural networks are based on backpropagation through time, which requires storing a complete history of network states, and prohibits updating the weights `online' (after every timestep). Real Time Recurrent Learning (RTRL) eliminates the need for history storage and allows for online weight updates, but does so at the expense of computational costs that are quartic in the state size. This renders RTRL training intractable for all but the smallest networks, even ones that are made highly sparse. We introduce the Sparse n-step Approximation (SnAp) to the R…
One of the central goals of Recurrent Neural Networks (RNNs) is to learn long-term dependencies in sequential data. Nevertheless, the most popular training method, Truncated Backpropagation through Time (TBPTT), categorically forbids learning dependencies beyond the truncation horizon. In contrast, the online training algorithm Real Time Recurrent Learning (RTRL) provides untruncated gradients, with the disadvantage of impractically large computational costs. Recently published approaches reduce these costs by providing noisy approximations of RTRL. We present a new approximation algorithm of …
Exploring the Promise and Limits of Real-Time Recurrent Learning
Center for Brain Science, Harvard University（哈佛大学脑科学中心）；The Swiss AI Lab, IDSIA, USI & SUPSI（瑞士IDSIA人工智能实验室（隶属于瑞士意大利语区大学和瑞士应用科学大学））；AI Initiative, KAUST（阿卜杜拉国王科技大学人工智能倡议）
Institute for Neural Computation, Ruhr University Bochum（波鸿鲁尔大学神经计算研究所）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）
Backpropagation through time (BPTT) is the standard algorithm for training recurrent neural networks (RNNs), which requires separate simulation phases for the forward and backward passes for inference and learning, respectively. Moreover, BPTT requires storing the complete history of network states between phases, with memory consumption growing proportional to the input sequence length. This makes BPTT unsuited for online learning and presents a challenge for implementation on low-resource real-time systems. Real-Time Recurrent Learning (RTRL) allows online learning, and the growth of require…
University of Florence（佛罗伦萨大学）；University of Siena（锡耶纳大学）
This paper proposes an in-depth re-thinking of neural computation that parallels apparently unrelated laws of physics, that are formulated in the variational framework of the least action principle. The theory holds for neural networks that are also based on any digraph, and the resulting computational scheme exhibits the intriguing property of being truly biologically plausible. The scheme, which is referred to as SpatioTemporal Local Propagation (STLP), is local in both space and time. Space locality comes from the expression of the network connections by an appropriate Lagrangian term, so a…
Recurrent neural networks have shown remarkable success in modeling sequences. However low resource situations still adversely affect the generalizability of these models. We introduce a new family of models, called Lattice Recurrent Units (LRU), to address the challenge of learning deep multi-layer recurrent models with limited resources. LRU models achieve this goal by creating distinct (but coupled) flow of information inside the units: a first flow along time dimension and a second flow along depth dimension. It also offers a symmetry in how information can flow horizontally and vertically…
Institute for Computer Science（计算机科学研究所）；University of Bonn（波恩大学）；School of Computing（计算学院）；National University of Singapore（新加坡国立大学）
Complex numbers have long been favoured for digital signal processing, yet complex representations rarely appear in deep learning architectures. RNNs, widely used to process time series and sequence information, could greatly benefit from complex representations. We present a novel complex gated recurrent cell, which is a hybrid cell combining complex-valued and norm-preserving state transitions with a gating mechanism. The resulting RNN exhibits excellent stability and convergence properties and performs competitively on the synthetic memory and adding task, as well as on the real-world tasks…
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…
National University of Defense Technology（国防科技大学）；Army Medical University (Third Military Medical University)（陆军军医大学（第三军医大学））
It is hard to train Recurrent Neural Network (RNN) with stable convergence and avoid gradient vanishing and exploding problems, as the weights in the recurrent unit are repeated from iteration to iteration. Moreover, RNN is sensitive to the initialization of weights and bias, which brings difficulties in training. The Alternating Direction Method of Multipliers (ADMM) has become a promising algorithm to train neural networks beyond traditional stochastic gradient algorithms with the gradient-free features and immunity to unsatisfactory conditions. However, ADMM could not be applied to train RN…
Université de Montréal（蒙特利尔大学）；University of Southern California（南加州大学）
Recurrent Neural Networks (RNNs) are powerful models for sequential data that have the potential to learn long-term dependencies. However, they are computationally expensive to train and difficult to parallelize. Recent work has shown that normalizing intermediate representations of neural networks can significantly improve convergence rates in feedforward neural networks . In particular, batch normalization, which uses mini-batch statistics to standardize features, was shown to significantly reduce training time. In this paper, we show that applying batch normalization to the hidden-to-hidden…
Rice University（莱斯大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of Utah（犹他大学）
Designing deep neural networks is an art that often involves an expensive search over candidate architectures. To overcome this for recurrent neural nets (RNNs), we establish a connection between the hidden state dynamics in an RNN and gradient descent (GD). We then integrate momentum into this framework and propose a new family of RNNs, called {\em MomentumRNNs}. We theoretically prove and numerically demonstrate that MomentumRNNs alleviate the vanishing gradient issue in training RNNs. We study the momentum long-short term memory (MomentumLSTM) and verify its advantages in convergence speed …
SMILE Lab, University of Electronic Science and Technology of China（电子科技大学SMILE实验室）；Dept. of Computer Science, Brown University（布朗大学计算机科学系）；School of Computing, University of Utah（犹他大学计算机学院）；Xjera Labs, Pte.Ltd（Xjera实验室私人有限公司）
Recurrent Neural Networks (RNNs) are powerful sequence modeling tools. However, when dealing with high dimensional inputs, the training of RNNs becomes computational expensive due to the large number of model parameters. This hinders RNNs from solving many important computer vision tasks, such as Action Recognition in Videos and Image Captioning. To overcome this problem, we propose a compact and flexible structure, namely Block-Term tensor decomposition, which greatly reduces the parameters of RNNs and improves their training efficiency. Compared with alternative low-rank approximations, such…
The University of Tokyo（东京大学）；National Institute for Quantum and Radiological Science and Technology（国立量子与放射科学技术研究所）
During the radiotherapy treatment of patients with lung cancer, the radiation delivered to healthy tissue around the tumor needs to be minimized, which is difficult because of respiratory motion and the latency of linear accelerator systems. In the proposed study, we first use the Lucas-Kanade pyramidal optical flow algorithm to perform deformable image registration of chest computed tomography scan images of four patients with lung cancer. We then track three internal points close to the lung tumor based on the previously computed deformation field and predict their position with a recurrent …
The novel Unbiased Online Recurrent Optimization (UORO) algorithm allows for online learning of general recurrent computational graphs such as recurrent network models. It works in a streaming fashion and avoids backtracking through past activations and inputs. UORO is computationally as costly as Truncated Backpropagation Through Time (truncated BPTT), a widespread algorithm for online learning of recurrent networks. UORO is a modification of NoBackTrack that bypasses the need for model sparsity and makes implementation easy in current deep learning frameworks, even for complex models. Like N…
Structural credit assignment for recurrent learning is challenging. An algorithm called RTRL can compute gradients for recurrent networks online but is computationally intractable for large networks. Alternatives, such as BPTT, are not online. In this work, we propose a credit-assignment algorithm -- \algoname{} -- that approximates the gradients for recurrent learning in real-time using $O(n)$ operations and memory per-step. Our method builds on the idea that for modular recurrent networks, composed of columns with scalar states, it is sufficient for a parameter to only track its influence on…
ELLIS Institute Tübingen（图宾根ELLIS研究所）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Tübingen AI Center（图宾根人工智能中心）；Liquid AI（Liquid AI公司）
For a class of martingales, this paper provides a framework on the uniform consistency with broad applicability. The main condition imposed is only related to the conditional variance of the martingale, which holds true for stationary mixing time series, stationary iterated random function, Harris recurrent Markov chains and $I(1)$ processes with innovations being a linear process. Using the established results, this paper investigates the uniform convergence of the Nadaraya-Watson estimator in a non-linear cointegrating regression model. Our results not only provide sharp convergence rate, bu…
The University of Tokyo（东京大学）；Japan Atomic Energy Commission（日本原子能委员会）；National Institutes for Quantum and Radiological Science and Technology（日本量子科学技术研究开发机构）
During lung radiotherapy, the position of infrared reflective objects on the chest can be recorded to estimate the tumor location. However, radiotherapy systems have a latency inherent to robot control limitations that impedes the radiation delivery precision. Prediction with online learning of recurrent neural networks (RNN) allows for adaptation to non-stationary respiratory signals, but classical methods such as RTRL and truncated BPTT are respectively slow and biased. This study investigates the capabilities of unbiased online recurrent optimization (UORO) to forecast respiratory motion an…
UC Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）；University of Michigan（密歇根大学）；Meta AI；New York University（纽约大学）；MIT（麻省理工学院）
Deep learning has achieved tremendous success by training increasingly large models, which are then compressed for practical deployment. We propose a drastically different approach to compact and optimal deep learning: We decouple the Degrees of freedom (DoF) and the actual number of parameters of a model, optimize a small DoF with predefined random linear constraints for a large model of arbitrary architecture, in one-stage end-to-end learning. Specifically, we create a recurrent parameter generator (RPG), which repeatedly fetches parameters from a ring and unpacks them onto a large model wit…
Real-time respiratory motion forecasting with online learning of recurrent neural networks for accurate targeting in externally guided radiotherapy
The University of Tokyo（东京大学）；Japan Atomic Energy Commission（日本原子能委员会）；National Institutes for Quantum and Radiological Science and Technology（量子与辐射科学和技术国家研究所）
Neuromorphic hardware strives to emulate brain-like neural networks and thus holds the promise for scalable, low-power information processing on temporal data streams. Yet, to solve real-world problems, these networks need to be trained. However, training on neuromorphic substrates creates significant challenges due to the offline character and the required non-local computations of gradient-based learning algorithms. This article provides a mathematical framework for the design of practical online learning algorithms for neuromorphic substrates. Specifically, we show a direct connection betwe…
Technische Universität Darmstadt（达姆施塔特工业大学）；Concordia University（康考迪亚大学）
A regression problem with dependent data is considered. Regularity assumptions on the dependency of the data are introduced, and it is shown that under suitable structural assumptions on the regression function a deep recurrent neural network estimate is able to circumvent the curse of dimensionality.
China University of Political Science and Law（中国政法大学）；Institute of High Performance Computing（高性能计算研究所）
A deep learning approach has been widely applied in sequence modeling problems. In terms of automatic speech recognition (ASR), its performance has significantly been improved by increasing large speech corpus and deeper neural network. Especially, recurrent neural network and deep convolutional neural network have been applied in ASR successfully. Given the arising problem of training speed, we build a novel deep recurrent convolutional network for acoustic modeling and then apply deep residual learning to it. Our experiments show that it has not only faster convergence speed but better recog…
Gradient control plays an important role in feed-forward networks applied to various computer vision tasks. Previous work has shown that Recurrent Highway Networks minimize the problem of vanishing or exploding gradients. They achieve this by setting the eigenvalues of the temporal Jacobian to 1 across the time steps. In this work, batch normalized recurrent highway networks are proposed to control the gradient flow in an improved way for network convergence. Specifically, the introduced model can be formed by batch normalizing the inputs at each recurrence loop. The proposed model is tested o…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Adobe Systems, Inc.（奥多比系统公司）
In this paper, we propose a new Recurrent Neural Network (RNN) architecture. The novelty is simple: We use diagonal recurrent matrices instead of full. This results in better test likelihood and faster convergence compared to regular full RNNs in most of our experiments. We show the benefits of using diagonal recurrent matrices with popularly used LSTM and GRU architectures as well as with the vanilla RNN architecture, on four standard symbolic music datasets.
Advancements in parallel processing have lead to a surge in multilayer perceptrons' (MLP) applications and deep learning in the past decades. Recurrent Neural Networks (RNNs) give additional representational power to feedforward MLPs by providing a way to treat sequential data. However, RNNs are hard to train using conventional error backpropagation methods because of the difficulty in relating inputs over many time-steps. Regularization approaches from MLP sphere, like dropout and noisy weight training, have been insufficiently applied and tested on simple RNNs. Moreover, solutions have been …
Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université de Paris（巴黎大学）；Collège de France（法兰西公学院）；LightOn（LightOn公司）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Reservoir Computing is a class of simple yet efficient Recurrent Neural Networks where internal weights are fixed at random and only a linear output layer is trained. In the large size limit, such random neural networks have a deep connection with kernel methods. Our contributions are threefold: a) We rigorously establish the recurrent kernel limit of Reservoir Computing and prove its convergence. b) We test our models on chaotic time series prediction, a classic but challenging benchmark in Reservoir Computing, and show how the Recurrent Kernel is competitive and computationally efficient whe…
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 …
We explore the robustness of recurrent neural networks when the computations within the network are noisy. One of the motivations for looking into this problem is to reduce the high power cost of conventional computing of neural network operations through the use of analog neuromorphic circuits. Traditional GPU/CPU-centered deep learning architectures exhibit bottlenecks in power-restricted applications, such as speech recognition in embedded systems. The use of specialized neuromorphic circuits, where analog signals passed through memory-cell arrays are sensed to accomplish matrix-vector mult…
Reinforcement learning (RL) agents performing complex tasks must be able to remember observations and actions across sizable time intervals. This is especially true during the initial learning stages, when exploratory behaviour can increase the delay between specific actions and their effects. Many new or popular approaches for learning these distant correlations employ backpropagation through time (BPTT), but this technique requires storing observation traces long enough to span the interval between cause and effect. Besides memory demands, learning dynamics like vanishing gradients and slow …
Gate functions in recurrent models, such as an LSTM and GRU, play a central role in learning various time scales in modeling time series data by using a bounded activation function. However, it is difficult to train gates to capture extremely long time scales due to gradient vanishing of the bounded function for large inputs, which is known as the saturation problem. We closely analyze the relation between saturation of the gate function and efficiency of the training. We prove that the gradient vanishing of the gate function can be mitigated by accelerating the convergence of the saturating f…
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 Recurren…
National Physical Laboratory（国家物理实验室）；Mathematical Institute（数学研究所）；University of Oxford（牛津大学）；IIIT Delhi（德里印度理工学院）
We propose to use stochastic Riemannian coordinate descent on the orthogonal group for recurrent neural network training. The algorithm rotates successively two columns of the recurrent matrix, an operation that can be efficiently implemented as a multiplication by a Givens matrix. In the case when the coordinate is selected uniformly at random at each iteration, we prove the convergence of the proposed algorithm under standard assumptions on the loss function, stepsize and minibatch noise. In addition, we numerically demonstrate that the Riemannian gradient in recurrent neural network trainin…
Recently a variety of methods have been developed to encode graphs into low-dimensional vectors that can be easily exploited by machine learning algorithms. The majority of these methods start by embedding the graph nodes into a low-dimensional vector space, followed by using some scheme to aggregate the node embeddings. In this work, we develop a new approach to learn graph-level representations, which includes a combination of unsupervised and supervised learning components. We start by learning a set of node representations in an unsupervised fashion. Graph nodes are mapped into node sequen…
Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）；Boston University（波士顿大学）
We propose a novel {\it Equilibrated Recurrent Neural Network} (ERNN) to combat the issues of inaccuracy and instability in conventional RNNs. Drawing upon the concept of autapse in neuroscience, we propose augmenting an RNN with a time-delayed self-feedback loop. Our sole purpose is to modify the dynamics of each internal RNN state and, at any time, enforce it to evolve close to the equilibrium point associated with the input signal at that time. We show that such self-feedback helps stabilize the hidden state transitions leading to fast convergence during training while efficiently learning …
University of Washington（华盛顿大学）；Applied Physics Laboratory, University of Washington（华盛顿大学应用物理实验室）
In this paper, we propose a novel recurrent neural network architecture for speech separation. This architecture is constructed by unfolding the iterations of a sequential iterative soft-thresholding algorithm (ISTA) that solves the optimization problem for sparse nonnegative matrix factorization (NMF) of spectrograms. We name this network architecture deep recurrent NMF (DR-NMF). The proposed DR-NMF network has three distinct advantages. First, DR-NMF provides better interpretability than other deep architectures, since the weights correspond to NMF model parameters, even after training. This…
