Optimizing deep neural networks (DNNs) often suffers from the ill-conditioned problem. We observe that the scaling-based weight space symmetry property in rectified nonlinear network will cause this negative effect. Therefore, we propose to constrain the incoming weights of each neuron to be unit-norm, which is formulated as an optimization problem over Oblique manifold. A simple yet efficient method referred to as projection based weight normalization (PBWN) is also developed to solve this problem. PBWN executes standard gradient updates, followed by projecting the updated weight back to Oblique manifold. This proposed method has the property of regularization and collaborates well with the commonly used batch normalization technique. We conduct comprehensive experiments on several widely-used image datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet for supervised learning over the state-of-the-art convolutional neural networks, such as Inception, VGG and residual networks. The results show that our method is able to improve the performance of DNNs with different architectures consistently. We also apply our method to Ladder network for semi-supervised learning on permutation invariant MNIST dataset, and our method outperforms the state-of-the-art methods: we obtain test errors as 2.52 , 1.06 , and 0.91 with only 20, 50, and 100 labeled samples, respectively.
There are large amount of work introducing orthogonality to the weight matrix @cite_48 @cite_9 @cite_27 @cite_30 @cite_31 @cite_12 @cite_24 in deep neural networks to address the gradient vanish and explosion problem. Solving the problem with such orthogonality constraint is usually limited to the hidden-to-hidden transformation in Recurrent neural networks @cite_48 @cite_9 @cite_27 @cite_30 . Some work also consider orthogonal weight matrix in feed forward neural networks @cite_31 @cite_12 @cite_24 , while their solutions introduce expensive computation costs.
@cite_30 @cite_48 @cite_9 @cite_24 @cite_27 @cite_31 @cite_12
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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 stability when encouraging or enforcing orthogonality. To perform this analysis, we propose a weight matrix factorization and parameterization strategy through which we can bound matrix norms and therein control the degree of expansivity induced during backpropagation. We find that hard constraints on orthogonality can negatively affect the speed of convergence and model performance. 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 eigendecomposition) after each weight update. We construct an expressive unitary weight matrix by composing several structured matrices that act as building blocks with parameters to be learned. Optimization with this parameterization becomes feasible only when considering hidden states in the complex domain. We demonstrate the potential of this architecture by achieving state of the art results in several hard tasks involving very longterm dependencies. 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 eigendecomposition) after each weight update. We construct an expressive unitary weight matrix by composing several structured matrices that act as building blocks with parameters to be learned. Optimization with this parameterization becomes feasible only when considering hidden states in the complex domain. We demonstrate the potential of this architecture by achieving state of the art results in several hard tasks involving very long-term dependencies. Orthogonal matrix has shown advantages in training Recurrent Neural Networks (RNNs), but such matrix is limited to be square for the hidden-to-hidden transformation in RNNs. In this paper, we generalize such square orthogonal matrix to orthogonal rectangular matrix and formulating this problem in feed-forward Neural Networks (FNNs) as Optimization over Multiple Dependent Stiefel Manifolds (OMDSM). We show that the rectangular orthogonal matrix can stabilize the distribution of network activations and regularize FNNs. We also propose a novel orthogonal weight normalization method to solve OMDSM. Particularly, it constructs orthogonal transformation over proxy parameters to ensure the weight matrix is orthogonal and back-propagates gradient information through the transformation during training. To guarantee stability, we minimize the distortions between proxy parameters and canonical weights over all tractable orthogonal transformations. In addition, we design an orthogonal linear module (OLM) to learn orthogonal filter banks in practice, which can be used as an alternative to standard linear module. Extensive experiments demonstrate that by simply substituting OLM for standard linear module without revising any experimental protocols, our method largely improves the performance of the state-of-the-art networks, including Inception and residual networks on CIFAR and ImageNet datasets. In particular, we have reduced the test error of wide residual network on CIFAR-100 from 20.04 to 18.61 with such simple substitution. Our code is available online for result reproduction. 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 a standard recurrent neural network, while outperforming standard recurrent neural networks with orthogonal initializations and Long Short-Term Memory networks on the copy problem. Optimizing deep neural networks (DNNs) often suffers from the ill-conditioned problem. We observe that the scaling-based weight space symmetry property in rectified nonlinear network will cause this negative effect. Therefore, we propose to constrain the incoming weights of each neuron to be unit-norm, which is formulated as an optimization problem over Oblique manifold. A simple yet efficient method referred to as projection based weight normalization (PBWN) is also developed to solve this problem. PBWN executes standard gradient updates, followed by projecting the updated weight back to Oblique manifold. This proposed method has the property of regularization and collaborates well with the commonly used batch normalization technique. We conduct comprehensive experiments on several widely-used image datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet for supervised learning over the state-of-the-art convolutional neural networks, such as Inception, VGG and residual networks. The results show that our method is able to improve the performance of DNNs with different architectures consistently. We also apply our method to Ladder network for semi-supervised learning on permutation invariant MNIST dataset, and our method outperforms the state-of-the-art methods: we obtain test errors as 2.52 , 1.06 , and 0.91 with only 20, 50, and 100 labeled samples, respectively. Learning long term dependencies in recurrent networks is difficult due to vanishing and exploding gradients. To overcome this difficulty, researchers have developed sophisticated optimization techniques and network architectures. In this paper, we propose a simpler solution that use recurrent neural networks composed of rectified linear units. Key to our solution is the use of the identity matrix or its scaled version to initialize the recurrent weight matrix. We find that our solution is comparable to LSTM on our four benchmarks: two toy problems involving long-range temporal structures, a large language modeling problem and a benchmark speech recognition problem. In recent years, state-of-the-art methods in computer vision have utilized increasingly deep convolutional neural network architectures (CNNs), with some of the most successful models employing hundreds or even thousands of layers. A variety of pathologies such as vanishing exploding gradients make training such deep networks challenging. While residual connections and batch normalization do enable training at these depths, it has remained unclear whether such specialized architecture designs are truly necessary to train deep CNNs. In this work, we demonstrate that it is possible to train vanilla CNNs with ten thousand layers or more simply by using an appropriate initialization scheme. We derive this initialization scheme theoretically by developing a mean field theory for signal propagation and by characterizing the conditions for dynamical isometry, the equilibration of singular values of the input-output Jacobian matrix. These conditions require that the convolution operator be an orthogonal transformation in the sense that it is norm-preserving. We present an algorithm for generating such random initial orthogonal convolution kernels and demonstrate empirically that they enable efficient training of extremely deep architectures. We propose a deep recurrent belief network with distributed time delays for learning multivariate Gaussians. Learning long time delays in deep belief networks is difficult due to the problem of vanishing or exploding gradients with increase in delay. To mitigate this problem and improve the transparency of learning time-delays, we introduce the use of Gaussian networks with time-delays to initialize the weights of each hidden neuron. From our knowledge of time delays, it is possible to learn the long delays from short delays in a hierarchical manner. In contrast to previous works, here dynamic Gaussian Bayesian networks over training samples are evolved using Markov Chain Monte Carlo to determine the initial weights of each hidden layer of neurons. In this way, the time-delayed network motifs of increasing Markov order across layers can be modeled hierarchically using a deep model. To validate the proposed Variable-order Belief Network (VBN) framework, it is applied for modeling word dependencies in text. To explore the generality of VBN, it is further considered for a real-world scenario where the dynamic movements of basketball players are modeled. Experimental results obtained showed that the proposed VBN could achieve over 30 improvement in accuracy on real-world scenarios compared to the state-of-the-art baselines. Recent advances in deep neural networks (DNNs) owe their success to training algorithms that use backpropagation and gradient-descent. Backpropagation, while highly effective on von Neumann architectures, becomes inefficient when scaling to large networks. Commonly referred to as the weight transport problem, each neuron's dependence on the weights and errors located deeper in the network require exhaustive data movement which presents a key problem in enhancing the performance and energy-efficiency of machine-learning hardware. In this work, we propose a bio-plausible alternative to backpropagation drawing from advances in feedback alignment algorithms in which the error computation at a single synapse reduces to the product of three scalar values, satisfying the three factor rule. Using a sparse feedback matrix, we show that a neuron needs only a fraction of the information previously used by the feedback alignment algorithms to yield results which are competitive with backpropagation. Consequently, memory and compute can be partitioned and distributed whichever way produces the most efficient forward pass so long as a single error can be delivered to each neuron. We evaluate our algorithm using standard data sets, including ImageNet, to address the concern of scaling to challenging problems. Our results show orders of magnitude improvement in data movement and @math improvement in multiply-and-accumulate operations over backpropagation. All the code and results are available under https: github.com bcrafton ssdfa. Deep neural networks such as Convolutional Networks (ConvNets) and Deep Belief Networks (DBNs) represent the state-of-the-art for many machine learning and computer vision classification problems. To overcome the large computational cost of deep networks, spiking deep networks have recently been proposed, given the specialized hardware now available for spiking neural networks (SNNs). However, this has come at the cost of performance losses due to the conversion from analog neural networks (ANNs) without a notion of time, to sparsely firing, event-driven SNNs. Here we analyze the effects of converting deep ANNs into SNNs with respect to the choice of parameters for spiking neurons such as firing rates and thresholds. We present a set of optimization techniques to minimize performance loss in the conversion process for ConvNets and fully connected deep networks. These techniques yield networks that outperform all previous SNNs on the MNIST database to date, and many networks here are close to maximum performance after only 20 ms of simulated time. The techniques include using rectified linear units (ReLUs) with zero bias during training, and using a new weight normalization method to help regulate firing rates. Our method for converting an ANN into an SNN enables low-latency classification with high accuracies already after the first output spike, and compared with previous SNN approaches it yields improved performance without increased training time. The presented analysis and optimization techniques boost the value of spiking deep networks as an attractive framework for neuromorphic computing platforms aimed at fast and efficient pattern recognition. Weight modifications in traditional neural nets are computed by hard-wired algorithms. Without exception, all previous weight change algorithms have many specific limitations. Is it (in principle) possible to overcome limitations of hard-wired algorithms by allowing neural nets to run and improve their own weight change algorithms? This paper constructively demonstrates that the answer (in principle) is ‘yes’. I derive an initial gradientbased sequence learning algorithm for a ‘self-referential’ recurrent network that can ‘speak’ about its own weight matrix in terms of activations. It uses some of its input and output units for observing its own errors and for explicitly analyzing and modifying its own weight matrix, including those parts of the weight matrix responsible for analyzing and modifying the weight matrix. The result is the first ‘introspective’ neural net with explicit potential control over all of its own adaptive parameters. A disadvantage of the algorithm is its high computational complexity per time step which is independent of the sequence length and equals O(nconnlognconn), where riconn is the number of connections. Another disadvantage is the high number of local minima of the unusually complex error surface. The purpose of this paper, however, is not to come up with the most efficient ‘introspective’ or ‘self-referential’ weight change algorithm, but to show that such algorithms are possible at all. Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. Deep neural network is difficult to train and this predicament becomes worse as the depth increases. The essence of this problem exists in the magnitude of backpropagated errors that will result in gradient vanishing or exploding phenomenon. We show that a variant of regularizer which utilizes orthonormality among different filter banks can alleviate this problem. Moreover, we design a backward error modulation mechanism based on the quasi-isometry assumption between two consecutive parametric layers. Equipped with these two ingredients, we propose several novel optimization solutions that can be utilized for training a specific-structured (repetitively triple modules of Conv-BNReLU) extremely deep convolutional neural network (CNN) WITHOUT any shortcuts identity mappings from scratch. Experiments show that our proposed solutions can achieve distinct improvements for a 44-layer and a 110-layer plain networks on both the CIFAR-10 and ImageNet datasets. Moreover, we can successfully train plain CNNs to match the performance of the residual counterparts. Besides, we propose new principles for designing network structure from the insights evoked by orthonormality. Combined with residual structure, we achieve comparative performance on the ImageNet dataset. 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 constraining the transition matrix to be unitary is a special case of an orthogonal constraint. Then we present a new parametrisation of the transition matrix which allows efficient training of an RNN while ensuring that the matrix is always orthogonal. Our results show that the orthogonal constraint on the transition matrix applied through our parametrisation gives similar benefits to the unitary constraint, without the time complexity limitations. The problem of quantizing the activations of a deep neural network is considered. An examination of the popular binary quantization approach shows that this consists of approximating a classical non-linearity, the hyperbolic tangent, by two functions: a piecewise constant sign function, which is used in feedforward network computations, and a piecewise linear hard tanh function, used in the backpropagation step during network learning. The problem of approximating the ReLU non-linearity, widely used in the recent deep learning literature, is then considered. An half-wave Gaussian quantizer (HWGQ) is proposed for forward approximation and shown to have efficient implementation, by exploiting the statistics of of network activations and batch normalization operations commonly used in the literature. To overcome the problem of gradient mismatch, due to the use of different forward and backward approximations, several piece-wise backward approximators are then investigated. The implementation of the resulting quantized network, denoted as HWGQ-Net, is shown to achieve much closer performance to full precision networks, such as AlexNet, ResNet, GoogLeNet and VGG-Net, than previously available low-precision networks, with 1-bit binary weights and 2-bit quantized activations. Deep convolutional neural networks (CNNs) have shown appealing performance on various computer vision tasks in recent years. This motivates people to deploy CNNs to realworld applications. However, most of state-of-art CNNs require large memory and computational resources, which hinders the deployment on mobile devices. Recent studies show that low-bit weight representation can reduce much storage and memory demand, and also can achieve efficient network inference. To achieve this goal, we propose a novel approach named BWNH to train Binary Weight Networks via Hashing. In this paper, we first reveal the strong connection between inner-product preserving hashing and binary weight networks, and show that training binary weight networks can be intrinsically regarded as a hashing problem. Based on this perspective, we propose an alternating optimization method to learn the hash codes instead of directly learning binary weights. Extensive experiments on CIFAR10, CIFAR100 and ImageNet demonstrate that our proposed BWNH outperforms current state-of-art by a large margin. The problem of quantizing the activations of a deep neural network is considered. An examination of the popular binary quantization approach shows that this consists of approximating a classical non-linearity, the hyperbolic tangent, by two functions: a piecewise constant sign function, which is used in feedforward network computations, and a piecewise linear hard tanh function, used in the backpropagation step during network learning. The problem of approximating the widely used ReLU non-linearity is then considered. An half-wave Gaussian quantizer (HWGQ) is proposed for forward approximation and shown to have efficient implementation, by exploiting the statistics of of network activations and batch normalization operations. To overcome the problem of gradient mismatch, due to the use of different forward and backward approximations, several piece-wise backward approximators are then investigated. The implementation of the resulting quantized network, denoted as HWGQ-Net, is shown to achieve much closer performance to full precision networks, such as AlexNet, ResNet, GoogLeNet and VGG-Net, than previously available low-precision networks, with 1-bit binary weights and 2-bit quantized activations. Neural networks with their inherent learning ability have been widely applied to solve the robot manipulator inverse kinematics problems. However, there are still two open problems: (1) without knowing inverse kinematic expressions, these solutions have the difficulty of how to collect training sets, and (2) the gradient-based learning algorithms can cause a very slow training process, especially for a complex configuration, or a large set of training data. Unlike these traditional implementations, the proposed metho trains neural network in joint subspace which can be easily calculated with electromagnetism-like method. The kinematics equation and its inverse are one-to-one mapping within the subspace. Thus the constrained training sets can be easily collected by forward kinematics relations. For issue 2, this paper uses a novel learning algorithm called extreme learning machine (ELM) which randomly choose the input weights and analytically determines the output weights of the single hidden layer feedforward neural networks (SLFNs). In theory, this algorithm tends to provide the best generalization performance at extremely fast learning speed. The results show that the proposed approach has not only greatly reduced the computation time but also improved the precision. There are two widely known issues with properly training recurrent neural networks, the vanishing and the exploding gradient problems detailed in (1994). In this paper we attempt to improve the understanding of the underlying issues by exploring these problems from an analytical, a geometric and a dynamical systems perspective. Our analysis is used to justify a simple yet effective solution. We propose a gradient norm clipping strategy to deal with exploding gradients and a soft constraint for the vanishing gradients problem. We validate empirically our hypothesis and proposed solutions in the experimental section.