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.
Concurrent to our work, Cho and Lee @cite_33 propose to optimize over Grassmann manifold, aiming to improve the performance of neural networks equipped with batch normalization @cite_17 . The differences between their work and our work are in two aspects: (1) they only use the traditional Riemannian optimization method ( Riemannian gradient + exponential maps' @cite_15 ) to solve the constraint optimization problem, which introduce non-trivial commutation cost; while we consider both Riemannian optimization method ( Riemannian gradient+ retraction' @cite_2 ) and further proposed a more general and efficient projection based weight normalization framework, which introduces negligible extra computation cost; (2) @cite_33 requires gradient clipping technique @cite_1 to make optimization stable and also needs tailored revision for SGD with momentum. On the contrary, our method is more general without requiring any extra tailored revision, and it can also collaborate well with other techniques of training neural networks.
@cite_33 @cite_1 @cite_2 @cite_15 @cite_17
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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. Parallel implementations of stochastic gradient descent (SGD) have received significant research attention, thanks to its excellent scalability properties. A fundamental barrier when parallelizing SGD is the high bandwidth cost of communicating gradient updates between nodes; consequently, several lossy compresion heuristics have been proposed, by which nodes only communicate quantized gradients. Although effective in practice, these heuristics do not always guarantee convergence, and it is not clear whether they can be improved. In this paper, we propose Quantized SGD (QSGD), a family of compression schemes for gradient updates which provides convergence guarantees. QSGD allows the user to smoothly trade off and : nodes can adjust the number of bits sent per iteration, at the cost of possibly higher variance. We show that this trade-off is inherent, in the sense that improving it past some threshold would violate information-theoretic lower bounds. QSGD guarantees convergence for convex and non-convex objectives, under asynchrony, and can be extended to stochastic variance-reduced techniques. When applied to training deep neural networks for image classification and automated speech recognition, QSGD leads to significant reductions in end-to-end training time. For example, on 16GPUs, we can train the ResNet152 network to full accuracy on ImageNet 1.8x faster than the full-precision variant. We address the numerical problem of recovering large matrices of low rank when most of the entries are unknown. We exploit the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on a single Grassmann manifold. We then apply second-order Riemannian trust-region methods (RTRMC 2) and Riemannian conjugate gradient methods (RCGMC) to solve it. A preconditioner for the Hessian is introduced that helps control the conditioning of the problem and we detail preconditioned versions of Riemannian optimization algorithms. The cost of each iteration is linear in the number of known entries. The proposed methods are competitive with state-of-the-art algorithms on a wide range of problem instances. In particular, they perform well on rectangular matrices. We further note that second-order and preconditioned methods are well suited to solve badly conditioned matrix completion tasks. We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of pixel images in about 0.4 s using a single graphics processing unit (GPU). Despite superior training outcomes, adaptive optimization methods such as Adam, Adagrad or RMSprop have been found to generalize poorly compared to Stochastic gradient descent (SGD). These methods tend to perform well in the initial portion of training but are outperformed by SGD at later stages of training. We investigate a hybrid strategy that begins training with an adaptive method and switches to SGD when appropriate. Concretely, we propose SWATS, a simple strategy which switches from Adam to SGD when a triggering condition is satisfied. The condition we propose relates to the projection of Adam steps on the gradient subspace. By design, the monitoring process for this condition adds very little overhead and does not increase the number of hyperparameters in the optimizer. We report experiments on several standard benchmarks such as: ResNet, SENet, DenseNet and PyramidNet for the CIFAR-10 and CIFAR-100 data sets, ResNet on the tiny-ImageNet data set and language modeling with recurrent networks on the PTB and WT2 data sets. The results show that our strategy is capable of closing the generalization gap between SGD and Adam on a majority of the tasks. Nesterov's accelerated gradient descent (AGD), an instance of the general family of "momentum methods", provably achieves faster convergence rate than gradient descent (GD) in the convex setting. However, whether these methods are superior to GD in the nonconvex setting remains open. This paper studies a simple variant of AGD, and shows that it escapes saddle points and finds a second-order stationary point in @math iterations, faster than the @math iterations required by GD. To the best of our knowledge, this is the first Hessian-free algorithm to find a second-order stationary point faster than GD, and also the first single-loop algorithm with a faster rate than GD even in the setting of finding a first-order stationary point. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases per step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Nesterov's accelerated gradient descent (AGD), an instance of the general family of momentum methods,'' provably achieves faster convergence rate than gradient descent (GD) in the convex setting. While these methods are widely used in modern nonconvex applications, including training of deep neural networks, whether they are provably superior to GD in the nonconvex setting remains open. This paper studies a simple variant of Nesterov's AGD, and shows that it escapes saddle points and finds a second-order stationary point in ( O (1 ^ 7 4 ) ) iterations, matching the best known convergence rate, which is faster than the ( O (1 ^ 2 ) ) iterations required by GD. To the best of our knowledge, this is the first direct acceleration (single-loop) algorithm that is provably faster than GD in general nonconvex setting---all previous nonconvex accelerated algorithms rely on more complex mechanisms such as nested loops and proximal terms. Our analysis is based on two key ideas: (1) the use of a simple Hamiltonian function, inspired by a continuous-time perspective, which AGD monotonically decreases on each step even for nonconvex functions, and (2) a novel framework called improve or localize, which is useful for tracking the long-term behavior of gradient-based optimization algorithms. We believe that these techniques may deepen our understanding of both acceleration algorithms and nonconvex optimization. Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide significant improvements over stochastic gradient descent (SGD). In general, “fast gradient” methods have provable improvements over gradient descent only for the deterministic case, where the gradients are exact. In the stochastic case, the popular explanations for their wide applicability is that when these fast gradient methods are applied in the stochastic case, they partially mimic their exact gradient counterparts, resulting in some practical gain. This work provides a counterpoint to this belief by proving that there are simple problem instances where these methods cannot outperform SGD despite the best setting of its parameters. These negative problem instances are, in an informal sense, generic; they do not look like carefully constructed pathological instances. These results suggest (along with empirical evidence) that HB or NAG's practical performance gains are a by-product of minibatching. Furthermore, this work provides a viable (and provable) alternative, which, on the same set of problem instances, significantly improves over HB, NAG, and SGD's performance. This algorithm, denoted as ASGD, is a simple to implement stochastic algorithm, based on a relatively less popular version of Nesterov's AGD. Extensive empirical results in this paper show that ASGD has performance gains over HB, NAG, and SGD. Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches. In recent years, stochastic gradient descent (SGD) based techniques has become the standard tools for training neural networks. However, formal theoretical understanding of why SGD can train neural networks in practice is largely missing. In this paper, we make progress on understanding this mystery by providing a convergence analysis for SGD on a rich subset of two-layer feedforward networks with ReLU activations. This subset is characterized by a special structure called "identity mapping". We prove that, if input follows from Gaussian distribution, with standard @math initialization of the weights, SGD converges to the global minimum in polynomial number of steps. Unlike normal vanilla networks, the "identity mapping" makes our network asymmetric and thus the global minimum is unique. To complement our theory, we are also able to show experimentally that multi-layer networks with this mapping have better performance compared with normal vanilla networks. Our convergence theorem differs from traditional non-convex optimization techniques. We show that SGD converges to optimal in "two phases": In phase I, the gradient points to the wrong direction, however, a potential function @math gradually decreases. Then in phase II, SGD enters a nice one point convex region and converges. We also show that the identity mapping is necessary for convergence, as it moves the initial point to a better place for optimization. Experiment verifies our claims. Several recent trends in machine learning theory and practice, from the design of state-of-the-art Gaussian Process to the convergence analysis of deep neural nets (DNNs) under stochastic gradient descent (SGD), have found it fruitful to study wide random neural networks. Central to these approaches are certain scaling limits of such networks. We unify these results by introducing a notion of a straightline that can express most neural network computations, and we characterize its scaling limit when its tensors are large and randomized. From our framework follows (1) the convergence of random neural networks to Gaussian processes for architectures such as recurrent neural networks, convolutional neural networks, residual networks, attention, and any combination thereof, with or without batch normalization; (2) conditions under which the -- that weights in backpropagation can be assumed to be independent from weights in the forward pass -- leads to correct computation of gradient dynamics, and corrections when it does not; (3) the convergence of the Neural Tangent Kernel, a recently proposed kernel used to predict training dynamics of neural networks under gradient descent, at initialization for all architectures in (1) without batch normalization. Mathematically, our framework is general enough to rederive classical random matrix results such as the semicircle and the Marchenko-Pastur laws, as well as recent results in neural network Jacobian singular values. We hope our work opens a way toward design of even stronger Gaussian Processes, initialization schemes to avoid gradient explosion vanishing, and deeper understanding of SGD dynamics in modern architectures. In this paper, we propose a second order optimization method to learn models where both the dimensionality of the parameter space and the number of training samples is high. In our method, we construct on each iteration a Krylov subspace formed by the gradient and an approximation to the Hessian matrix, and then use a subset of the training data samples to optimize over this subspace. As with the Hessian Free (HF) method of Martens (2010), the Hessian matrix is never explicitly constructed, and is computed using a subset of data. In practice, as in HF, we typically use a positive definite substitute for the Hessian matrix such as the Gauss-Newton matrix. We investigate the effectiveness of our proposed method on deep neural networks, and compare its performance to widely used methods such as stochastic gradient descent, conjugate gradient descent and L-BFGS, and also to HF. Our method leads to faster convergence than either L-BFGS or HF, and generally performs better than either of them in cross-validation accuracy. It is also simpler and more general than HF, as it does not require a positive semidefinite approximation of the Hessian matrix to work well nor the setting of a damping parameter. The chief drawback versus HF is the need for memory to store a basis for the Krylov subspace. Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on lightweight estimates of sample uncertainty in stochastic gradient descent (SGD): the variance in predicted probability of the correct class across iterations of mini-batch SGD, and the proximity of the correct class probability to the decision threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains on top of other popular training techniques, such as residual learning, momentum, ADAM, batch normalization, dropout, and distillation. Self-paced learning and hard example mining re-weight training instances to improve learning accuracy. This paper presents two improved alternatives based on lightweight estimates of sample uncertainty in stochastic gradient descent (SGD): the variance in predicted probability of the correct class across iterations of mini-batch SGD, and the proximity of the correct class probability to the decision threshold. Extensive experimental results on six datasets show that our methods reliably improve accuracy in various network architectures, including additional gains on top of other popular training techniques, such as residual learning, momentum, ADAM, batch normalization, dropout, and distillation. We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent (SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent (Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Given a strongly convex optimization and for a total number of @math iterations, Epro-SGD requires only @math projections, and meanwhile attains an optimal convergence rate of @math , both in expectation and with a high probability. To exploit the structure of the optimization problem, we propose a proximal variant of Epro-SGD, namely Epro-ORDA, based on the optimal regularized dual averaging method. We apply the proposed methods on real-world applications; the empirical results demonstrate the effectiveness of our methods. Maximization of mutual information of voxel intensities has been demonstrated to be a very powerful criterion for three-dimensional medical image registration, allowing robust and accurate fully automated affine registration of multimodal images in a variety of applications, without the need for segmentation or other preprocessing of the images. In this paper, we investigate the performance of various optimization methods and multiresolution strategies for maximization of mutual information, aiming at increasing registration speed when matching large high-resolution images. We show that mutual information is a continuous function of the affine registration parameters when appropriate interpolation is used and we derive analytic expressions of its derivatives that allow numerically exact evaluation of its gradient. Various multiresolution gradient- and non-gradient-based optimization strategies, such as Powell, simplex, steepest-descent, conjugate-gradient, quasi-Newton and Levenberg—Marquardt methods, are evaluated for registration of computed tomography (CT) and magnetic resonance images of the brain. Speed-ups of a factor of 3 on average compared to Powell's method at full resolution are achieved with similar precision and without a loss of robustness with the simplex, conjugate-gradient and Levenberg—Marquardt method using a two-level multiresolution scheme. Large data sets such as 2562 × 128 MR and 5122 × 48 CT images can be registered with subvoxel precision in < 5 min CPU time on current workstations. Synchronized stochastic gradient descent (SGD) optimizers with data parallelism are widely used in training large-scale deep neural networks. Although using larger mini-batch sizes can improve the system scalability by reducing the communication-to-computation ratio, it may hurt the generalization ability of the models. To this end, we build a highly scalable deep learning training system for dense GPU clusters with three main contributions: (1) We propose a mixed-precision training method that significantly improves the training throughput of a single GPU without losing accuracy. (2) We propose an optimization approach for extremely large mini-batch size (up to 64k) that can train CNN models on the ImageNet dataset without losing accuracy. (3) We propose highly optimized all-reduce algorithms that achieve up to 3x and 11x speedup on AlexNet and ResNet-50 respectively than NCCL-based training on a cluster with 1024 Tesla P40 GPUs. On training ResNet-50 with 90 epochs, the state-of-the-art GPU-based system with 1024 Tesla P100 GPUs spent 15 minutes and achieved 74.9 top-1 test accuracy, and another KNL-based system with 2048 Intel KNLs spent 20 minutes and achieved 75.4 accuracy. Our training system can achieve 75.8 top-1 test accuracy in only 6.6 minutes using 2048 Tesla P40 GPUs. When training AlexNet with 95 epochs, our system can achieve 58.7 top-1 test accuracy within 4 minutes, which also outperforms all other existing systems. With the fast development of deep learning, it has become common to learn big neural networks using massive training data. Asynchronous Stochastic Gradient Descent (ASGD) is widely adopted to fulfill this task for its efficiency, which is, however, known to suffer from the problem of delayed gradients. That is, when a local worker adds its gradient to the global model, the global model may have been updated by other workers and this gradient becomes "delayed". We propose a novel technology to compensate this delay, so as to make the optimization behavior of ASGD closer to that of sequential SGD. This is achieved by leveraging Taylor expansion of the gradient function and efficient approximation to the Hessian matrix of the loss function. We call the new algorithm Delay Compensated ASGD (DC-ASGD). We evaluated the proposed algorithm on CIFAR-10 and ImageNet datasets, and the experimental results demonstrate that DC-ASGD outperforms both synchronous SGD and asynchronous SGD, and nearly approaches the performance of sequential SGD. Second-order optimization methods such as natural gradient descent have the potential to speed up training of neural networks by correcting for the curvature of the loss function. Unfortunately, the exact natural gradient is impractical to compute for large models, and most approximations either require an expensive iterative procedure or make crude approximations to the curvature. We present Kronecker Factors for Convolution (KFC), a tractable approximation to the Fisher matrix for convolutional networks based on a structured probabilistic model for the distribution over backpropagated derivatives. Similarly to the recently proposed Kronecker-Factored Approximate Curvature (K-FAC), each block of the approximate Fisher matrix decomposes as the Kronecker product of small matrices, allowing for efficient inversion. KFC captures important curvature information while still yielding comparably efficient updates to stochastic gradient descent (SGD). We show that the updates are invariant to commonly used reparameterizations, such as centering of the activations. In our experiments, approximate natural gradient descent with KFC was able to train convolutional networks several times faster than carefully tuned SGD. Furthermore, it was able to train the networks in 10-20 times fewer iterations than SGD, suggesting its potential applicability in a distributed setting. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning @math (where @math is a bound on the smoothness and @math on the strong convexity) to a linear dependence on @math . Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios. Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate.