Deep Neural Networks are successful but highly computationally expensive learning systems. One of the main sources of time and energy drains is the well known backpropagation (backprop) algorithm, which roughly accounts for 2/3 of the computational complexity of training. In this work we propose a method for reducing the computational cost of backprop, which we named dithered backprop. It consists in applying a stochastic quantization scheme to intermediate results of the method. The particular quantisation scheme, called non-subtractive dither (NSD), induces sparsity which can be exploited by computing efficient sparse matrix multiplications. Experiments on popular image classification tasks show that it induces 92% sparsity on average across a wide set of models at no or negligible accuracy drop in comparison to state-of-the-art approaches, thus significantly reducing the computational complexity of the backward pass. Moreover, we show that our method is fully compatible to state-of-the-art training methods that reduce the bit-precision of training down to 8-bits, as such being able to further reduce the computational requirements. Finally we discuss and show potential benefits of applying dithered backprop in a distributed training setting, where both communication as well as compute efficiency may increase simultaneously with the number of participant nodes.
Backpropagation algorithm is indispensable for the training of feedforward neural networks. It requires propagating error gradients sequentially from the output layer all the way back to the input layer. The backward locking in backpropagation algorithm constrains us from updating network layers in parallel and fully leveraging the computing resources. Recently, several algorithms have been proposed for breaking the backward locking. However, their performances degrade seriously when networks are deep. In this paper, we propose decoupled parallel backpropagation algorithm for deep learning opt…
Neural network learning is usually time-consuming since backpropagation needs to compute full gradients and backpropagate them across multiple layers. Despite its success of existing works in accelerating propagation through sparseness, the relevant theoretical characteristics remain under-researched and empirical studies found that they suffer from the loss of information contained in unpropagated gradients. To tackle these problems, this paper presents a unified sparse backpropagation framework and provides a detailed analysis of its theoretical characteristics. Analysis reveals that when ap…
School of Information Science and Engineering（信息科学与工程学院）；Lanzhou University（兰州大学）
We introduce the chi-square test neural network: a single hidden layer backpropagation neural network using chi-square test theorem to redefine the cost function and the error function. The weights and thresholds are modified using standard backpropagation algorithm. The proposed approach has the advantage of making consistent data distribution over training and testing sets. It can be used for binary classification. The experimental results on real world data sets indicate that the proposed algorithm can significantly improve the classification accuracy comparing to related approaches.
Arguably the biggest challenge in applying neural networks is tuning the hyperparameters, in particular the learning rate. The sensitivity to the learning rate is due to the reliance on backpropagation to train the network. In this paper we present the first application of Implicit Stochastic Gradient Descent (ISGD) to train neural networks, a method known in convex optimization to be unconditionally stable and robust to the learning rate. Our key contribution is a novel layer-wise approximation of ISGD which makes its updates tractable for neural networks. Experiments show that our method is …
Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure -- updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm th…
In large domains, Monte-Carlo tree search (MCTS) is required to estimate the values of the states as efficiently and accurately as possible. However, the standard update rule in backpropagation assumes a stationary distribution for the returns, and particularly in min-max trees, convergence to the true value can be slow because of averaging. We present two methods, Softmax MCTS and Monotone MCTS, which generalize previous attempts to improve upon the backpropagation strategy. We demonstrate that both methods reduce to finding optimal monotone functions, which we do so by performing Bayesian op…
Convergence rate of training algorithms for neural networks is heavily affected by initialization of weights. In this paper, an original algorithm for initialization of weights in backpropagation neural net is presented with application to character recognition. The initialization method is mainly based on a customization of the Kalman filter, translating it into Bayesian statistics terms. A metrological approach is used in this context considering weights as measurements modeled by mutually dependent normal random variables. The algorithm performance is demonstrated by reporting and discussin…
The two most important algorithms in artificial intelligence are backpropagation and belief propagation. In spite of their importance, the connection between them is poorly characterized. We show that when an input to backpropagation is converted into an input to belief propagation so that (loopy) belief propagation can be run on it, then the result of belief propagation encodes the result of backpropagation; thus backpropagation is recovered as a special case of belief propagation. In other words, we prove for apparently the first time that belief propagation generalizes backpropagation. Our …
The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to 2nd order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear quadratic operation. A single 2nd order neurons already has a strong nonlinear modeling ability, such as implementing basic fuzzy logic operations. In this paper, we develop a general backpropagation (BP) algorithm to train the network consisting of 2nd-order neurons. The numerical…
University of Technology Sydney（悉尼科技大学）；Northwestern University（西北大学）；Xiamen University（厦门大学）
In recent years, the mean field theory has been applied to the study of neural networks and has achieved a great deal of success. The theory has been applied to various neural network structures, including CNNs, RNNs, Residual networks, and Batch normalization. Inevitably, recent work has also covered the use of dropout. The mean field theory shows that the existence of depth scales that limit the maximum depth of signal propagation and gradient backpropagation. However, the gradient backpropagation is derived under the gradient independence assumption that weights used during feed forward are…
Donders Institute for Brain, Cognition and Behaviour（唐德斯大脑、认知与行为研究所）；Radboud University（拉德堡德大学）
Traditional backpropagation of error, though a highly successful algorithm for learning in artificial neural network models, includes features which are biologically implausible for learning in real neural circuits. An alternative called target propagation proposes to solve this implausibility by using a top-down model of neural activity to convert an error at the output of a neural network into layer-wise and plausible 'targets' for every unit. These targets can then be used to produce weight updates for network training. However, thus far, target propagation has been heuristically proposed w…
Many transformations in deep learning architectures are sparsely connected. When such transformations cannot be designed by hand, they can be learned, even through plain backpropagation, for instance in attention mechanisms. However, during learning, such sparse structures are often represented in a dense form, as we do not know beforehand which elements will eventually become non-zero. We introduce the adaptive, sparse hyperlayer, a method for learning a sparse transformation, paramatrized sparsely: as index-tuples with associated values. To overcome the lack of gradients from such a discrete…
Hulu Innovation Lab（Hulu创新实验室）；School of Electronics Engineering and Computer Science, Peking University（北京大学电子工程与计算机科学学院）
In real-world scenarios, it is appealing to learn a model carrying out stochastic operations internally, known as stochastic computation graphs (SCGs), rather than learning a deterministic mapping. However, standard backpropagation is not applicable to SCGs. We attempt to address this issue from the angle of cost propagation, with local surrogate costs, called Q-functions, constructed and learned for each stochastic node in an SCG. Then, the SCG can be trained based on these surrogate costs using standard backpropagation. We propose the entire framework as a solution to generalize backpropagat…
Preferred Networks, Inc.（Preferred Networks公司）；The University of Tokyo（东京大学）
Recomputation algorithms collectively refer to a family of methods that aims to reduce the memory consumption of the backpropagation by selectively discarding the intermediate results of the forward propagation and recomputing the discarded results as needed. In this paper, we will propose a novel and efficient recomputation method that can be applied to a wider range of neural nets than previous methods. We use the language of graph theory to formalize the general recomputation problem of minimizing the computational overhead under a fixed memory budget constraint, and provide a dynamic progr…
We describe recurrent neural networks (RNNs), which have attracted great attention on sequential tasks, such as handwriting recognition, speech recognition and image to text. However, compared to general feedforward neural networks, RNNs have feedback loops, which makes it a little hard to understand the backpropagation step. Thus, we focus on basics, especially the error backpropagation to compute gradients with respect to model parameters. Further, we go into detail on how error backpropagation algorithm is applied on long short-term memory (LSTM) by unfolding the memory unit.
We introduce backdrop, a flexible and simple-to-implement method, intuitively described as dropout acting only along the backpropagation pipeline. Backdrop is implemented via one or more masking layers which are inserted at specific points along the network. Each backdrop masking layer acts as the identity in the forward pass, but randomly masks parts of the backward gradient propagation. Intuitively, inserting a backdrop layer after any convolutional layer leads to stochastic gradients corresponding to features of that scale. Therefore, backdrop is well suited for problems in which the data h…
A common practice in most of deep convolutional neural architectures is to employ fully-connected layers followed by Softmax activation to minimize cross-entropy loss for the sake of classification. Recent studies show that substitution or addition of the Softmax objective to the cost functions of support vector machines or linear discriminant analysis is highly beneficial to improve the classification performance in hybrid neural networks. We propose a novel paradigm to link the optimization of several hybrid objectives through unified backpropagation. This highly alleviates the burden of ext…
South Westphalia University of Applied Science（西南法伦应用科学大学）
Quaternion valued neural networks experienced rising popularity and interest from researchers in the last years, whereby the derivatives with respect to quaternions needed for optimization are calculated as the sum of the partial derivatives with respect to the real and imaginary parts. However, we can show that product- and chain-rule does not hold with this approach. We solve this by employing the GHRCalculus and derive quaternion backpropagation based on this. Furthermore, we experimentally prove the functionality of the derived quaternion backpropagation.
Gradient descent and backpropagation have enabled neural networks to achieve remarkable results in many real-world applications. Despite ongoing success, training a neural network with gradient descent can be a slow and strenuous affair. We present a simple yet faster training algorithm called Zeroth-Order Relaxed Backpropagation (ZORB). Instead of calculating gradients, ZORB uses the pseudoinverse of targets to backpropagate information. ZORB is designed to reduce the time required to train deep neural networks without penalizing performance. To illustrate the speed up, we trained a feed-forw…
Probabilistic modeling enables combining domain knowledge with learning from data, thereby supporting learning from fewer training instances than purely data-driven methods. However, learning probabilistic models is difficult and has not achieved the level of performance of methods such as deep neural networks on many tasks. In this paper, we attempt to address this issue by presenting a method for learning the parameters of a probabilistic program using backpropagation. Our approach opens the possibility to building deep probabilistic programming models that are trained in a similar way to ne…
In recent years, an increasing number of neural network models have included derivatives with respect to inputs in their loss functions, resulting in so-called double backpropagation for first-order optimization. However, so far no general description of the involved derivatives exists. Here, we cover a wide array of special cases in a very general Hilbert space framework, which allows us to provide optimized backpropagation rules for many real-world scenarios. This includes the reduction of calculations for Frobenius-norm-penalties on Jacobians by roughly a third for locally linear activation…
Cerebras；Tesla（特斯拉）；Cerebras Systems, Inc（Cerebras Systems 公司）；Google（谷歌）
New hardware can substantially increase the speed and efficiency of deep neural network training. To guide the development of future hardware architectures, it is pertinent to explore the hardware and machine learning properties of alternative training algorithms. In this work we evaluate the use of small batch, fine-grained Pipelined Backpropagation, an asynchronous pipeline parallel training algorithm that has significant hardware advantages. We introduce two methods, Spike Compensation and Linear Weight Prediction, that effectively mitigate the downsides caused by the asynchronicity of Pipe…
By and large, Backpropagation (BP) is regarded as one of the most important neural computation algorithms at the basis of the progress in machine learning, including the recent advances in deep learning. However, its computational structure has been the source of many debates on its arguable biological plausibility. In this paper, it is shown that when framing supervised learning in the Lagrangian framework, while one can see a natural emergence of Backpropagation, biologically plausible local algorithms can also be devised that are based on the search for saddle points in the learning adjoint…
This work shows that a differentiable activation function is not necessary any more for error backpropagation. The derivative of the activation function can be replaced by an iterative temporal differencing using fixed random feedback alignment. Using fixed random synaptic feedback alignment with an iterative temporal differencing is transforming the traditional error backpropagation into a more biologically plausible approach for learning deep neural network architectures. This can be a big step toward the integration of STDP-based error backpropagation in deep learning.
The backpropagation algorithm for neural networks is widely felt hard to understand, despite the existence of some well-written explanations and/or derivations. This paper provides a new derivation of this algorithm based on the concept of derivative amplification coefficients. First proposed by this author for fully connected cascade networks, this concept is found to well carry over to conventional feedforward neural networks and it paves the way for the use of mathematical induction in establishing a key result that enables backpropagation for derivative amplification coefficients. Then we …
Department of Computer Science, New York University（纽约大学计算机科学系）
Most successful machine intelligence systems rely on gradient-based learning, which is made possible by backpropagation. Some systems are designed to aid us in interpreting data when explicit goals cannot be provided. These unsupervised systems are commonly trained by backpropagating through a likelihood function. We introduce a tool that allows us to do this even when the likelihood is not explicitly set, by instead using the implicit likelihood of the model. Explicitly defining the likelihood often entails making heavy-handed assumptions that impede our ability to solve challenging tasks. On…
The backpropagation algorithm, which had been originally introduced in the 1970s, is the workhorse of learning in neural networks. This backpropagation algorithm makes use of the famous machine learning algorithm known as Gradient Descent, which is a first-order iterative optimization algorithm for finding the minimum of a function. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. In this paper, we develop an alternative to the backpropagation without…
Recurrent Backpropagation and Equilibrium Propagation are supervised learning algorithms for fixed point recurrent neural networks which differ in their second phase. In the first phase, both algorithms converge to a fixed point which corresponds to the configuration where the prediction is made. In the second phase, Equilibrium Propagation relaxes to another nearby fixed point corresponding to smaller prediction error, whereas Recurrent Backpropagation uses a side network to compute error derivatives iteratively. In this work we establish a close connection between these two algorithms. We sh…
University of Edinburgh（爱丁堡大学）；AndPlus LLC（AndPlus有限责任公司）
Neural networks have been able to achieve groundbreaking accuracy at tasks conventionally considered only doable by humans. Using stochastic gradient descent, optimization in many dimensions is made possible, albeit at a relatively high computational cost. By splitting training data into batches, networks can be distributed and trained vastly more efficiently and with minimal accuracy loss. We have explored the mathematics behind efficiently implementing tensor-based batch backpropagation algorithms. A common approach to batch training is iterating over batch items individually. Explicitly usi…
Truncated Backpropagation Through Time (truncated BPTT) is a widespread method for learning recurrent computational graphs. Truncated BPTT keeps the computational benefits of Backpropagation Through Time (BPTT) while relieving the need for a complete backtrack through the whole data sequence at every step. However, truncation favors short-term dependencies: the gradient estimate of truncated BPTT is biased, so that it does not benefit from the convergence guarantees from stochastic gradient theory. We introduce Anticipated Reweighted Truncated Backpropagation (ARTBP), an algorithm that keeps t…
Training Implicit Networks for Image Deblurring using Jacobian-Free Backpropagation
Colorado School of Mines（科罗拉多矿业学院）；Brown University（布朗大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of California, Berkeley（加利福尼亚大学伯克利分校）
University of Cambridge（剑桥大学）；MPI for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google Brain（谷歌大脑）；Google DeepMind（谷歌DeepMind）
Deep neural networks are powerful parametric models that can be trained efficiently using the backpropagation algorithm. Stochastic neural networks combine the power of large parametric functions with that of graphical models, which makes it possible to learn very complex distributions. However, as backpropagation is not directly applicable to stochastic networks that include discrete sampling operations within their computational graph, training such networks remains difficult. We present MuProp, an unbiased gradient estimator for stochastic networks, designed to make this task easier. MuProp…
We present backpropagation clipping, a novel variant of differentially private stochastic gradient descent (DP-SGD) for privacy-preserving deep learning. Our approach clips each trainable layer's inputs (during the forward pass) and its upstream gradients (during the backward pass) to ensure bounded global sensitivity for the layer's gradient; this combination replaces the gradient clipping step in existing DP-SGD variants. Our approach is simple to implement in existing deep learning frameworks. The results of our empirical evaluation demonstrate that backpropagation clipping provides higher …
This article presents matrix backpropagation algorithms for the QR decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m < n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel matrix backpropagation results for the pivoted (full-rank) QR decomposition and for the LQ decomposition of deep input matrices. Differentiable QR decomposition offers a numerically stable, computationally efficient method to solve least squares problems frequently encountered in machine learning and computer vision. Other use cases such as graph learning and network compre…
Technical University of Munich（慕尼黑工业大学）；University of Siegen（锡根大学）
We propose proximal backpropagation (ProxProp) as a novel algorithm that takes implicit instead of explicit gradient steps to update the network parameters during neural network training. Our algorithm is motivated by the step size limitation of explicit gradient descent, which poses an impediment for optimization. ProxProp is developed from a general point of view on the backpropagation algorithm, currently the most common technique to train neural networks via stochastic gradient descent and variants thereof. Specifically, we show that backpropagation of a prediction error is equivalent to s…
IST Austria（奥地利科学技术研究所）；ETH Zurich（苏黎世联邦理工学院）；Neural Magic, Inc.（Neural Magic公司）
We provide a new efficient version of the backpropagation algorithm, specialized to the case where the weights of the neural network being trained are sparse. Our algorithm is general, as it applies to arbitrary (unstructured) sparsity and common layer types (e.g., convolutional or linear). We provide a fast vectorized implementation on commodity CPUs, and show that it can yield speedups in end-to-end runtime experiments, both in transfer learning using already-sparsified networks, and in training sparse networks from scratch. Thus, our results provide the first support for sparse training on …
Local learning for stable backpropagation-free neural network training towards physical learning
Department of Materials Science and Engineering, Delft University of Technology（材料科学与工程系，代尔夫特理工大学）；Department of Precision and Microsystems Engineering, Delft University of Technology（精密与微系统工程系，代尔夫特理工大学）；Department of Intelligent Systems, Delft University of Technology（智能系统系，代尔夫特理工大学）
School of Computer and Communication Sciences（计算机与通信科学学院）；EPFL（洛桑联邦理工学院）
Eigendecomposition (ED) is widely used in deep networks. However, the backpropagation of its results tends to be numerically unstable, whether using ED directly or approximating it with the Power Iteration method, particularly when dealing with large matrices. While this can be mitigated by partitioning the data in small and arbitrary groups, doing so has no theoretical basis and makes its impossible to exploit the power of ED to the full. In this paper, we introduce a numerically stable and differentiable approach to leveraging eigenvectors in deep networks. It can handle large matrices witho…
Spiking Neural Networks (SNNs) operate with asynchronous discrete events (or spikes) which can potentially lead to higher energy-efficiency in neuromorphic hardware implementations. Many works have shown that an SNN for inference can be formed by copying the weights from a trained Artificial Neural Network (ANN) and setting the firing threshold for each layer as the maximum input received in that layer. These type of converted SNNs require a large number of time steps to achieve competitive accuracy which diminishes the energy savings. The number of time steps can be reduced by training SNNs w…
University of Toronto（多伦多大学）；Uber Advanced Technologies Group（Uber先进技术集团）
Deep residual networks (ResNets) have significantly pushed forward the state-of-the-art on image classification, increasing in performance as networks grow both deeper and wider. However, memory consumption becomes a bottleneck, as one needs to store the activations in order to calculate gradients using backpropagation. We present the Reversible Residual Network (RevNet), a variant of ResNets where each layer's activations can be reconstructed exactly from the next layer's. Therefore, the activations for most layers need not be stored in memory during backpropagation. We demonstrate the effect…
This work combines Convolutional Neural Networks (CNNs), clustering via Self-Organizing Maps (SOMs) and Hebbian Learning to propose the building blocks of Convolutional Self-Organizing Neural Networks (CSNNs), which learn representations in an unsupervised and Backpropagation-free manner. Our approach replaces the learning of traditional convolutional layers from CNNs with the competitive learning procedure of SOMs and simultaneously learns local masks between those layers with separate Hebbian-like learning rules to overcome the problem of disentangling factors of variation when filters are l…
University of Minnesota（明尼苏达大学）；University of Surrey（萨里大学）；University of Applied Sciences, Stralsund（施特拉尔松德应用科学大学）
In this work we propose a new supervised learning method for temporally-encoded multilayer spiking networks to perform classification. The method employs a reinforcement signal that mimics backpropagation but is far less computationally intensive. The weight update calculation at each layer requires only local data apart from this signal. We also employ a rule capable of producing specific output spike trains; by setting the target spike time equal to the actual spike time with a slight negative offset for key high-value neurons the actual spike time becomes as early as possible. In simulated …
The Swiss AI Lab IDSIA（瑞士人工智能实验室IDSIA）；University of Lugano (USI)（卢加诺大学）；SUPSI（瑞士意大利语区高等专业学院）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）
Many concepts have been proposed for meta learning with neural networks (NNs), e.g., NNs that learn to reprogram fast weights, Hebbian plasticity, learned learning rules, and meta recurrent NNs. Our Variable Shared Meta Learning (VSML) unifies the above and demonstrates that simple weight-sharing and sparsity in an NN is sufficient to express powerful learning algorithms (LAs) in a reusable fashion. A simple implementation of VSML where the weights of a neural network are replaced by tiny LSTMs allows for implementing the backpropagation LA solely by running in forward-mode. It can even meta l…
Efficient Deep Learning with Decorrelated Backpropagation
Department of Machine Learning and Neural Computing（机器学习与神经计算系）；Donders Institute for Brain, Cognition and Behaviour（大脑、认知与行为研究所）；Radboud University（拉德堡德大学）
University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Rensselaer Polytechnic Institute（伦斯勒理工学院）
We study a class of deep neural networks with networks that form a directed acyclic graph (DAG). For backpropagation defined by gradient descent with adaptive momentum, we show weights converge for a large class of nonlinear activation functions. The proof generalizes the results of Wu et al. (2008) who showed convergence for a feed forward network with one hidden layer. For an example of the effectiveness of DAG architectures, we describe an example of compression through an autoencoder, and compare against sequential feed forward networks under several metrics.
Colorado School of Mines（科罗拉多矿业学院）；Typal Research, Typal LLC；Alibaba Group (US), Damo Academy（阿里巴巴集团（美国），达摩院）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
A promising trend in deep learning replaces traditional feedforward networks with implicit networks. Unlike traditional networks, implicit networks solve a fixed point equation to compute inferences. Solving for the fixed point varies in complexity, depending on provided data and an error tolerance. Importantly, implicit networks may be trained with fixed memory costs in stark contrast to feedforward networks, whose memory requirements scale linearly with depth. However, there is no free lunch -- backpropagation through implicit networks often requires solving a costly Jacobian-based equation …
The vanishing and exploding gradient problems are well-studied obstacles that make it difficult for recurrent neural networks to learn long-term time dependencies. We propose a reparameterization of standard recurrent neural networks to update linear transformations in a provably norm-preserving way through Givens rotations. Additionally, we use the absolute value function as an element-wise non-linearity to preserve the norm of backpropagated signals over the entire network. We show that this reparameterization reduces the number of parameters and maintains the same algorithmic complexity as …
Boolean Variation and Boolean Logic BackPropagation
Intelligent Computing and Communication Research（智能计算与通信研究）；Advanced Wireless Technology Laboratory（先进无线技术实验室）；Paris Research Center, Huawei Technologies（华为技术巴黎研究中心）
