We investigate the use of information from all second order derivatives of the error function to perfonn network pruning (i.e., removing unimportant weights from a trained network) in order to improve generalization, simplify networks, reduce hardware or storage requirements, increase the speed of further training, and in some cases enable rule extraction. Our method, Optimal Brain Surgeon (OBS), is Significantly better than magnitude-based methods and Optimal Brain Damage [Le Cun, Denker and Sol1a, 1990], which often remove the wrong weights. OBS permits the pruning of more weights than other methods (for the same error on the training set), and thus yields better generalization on test data. Crucial to OBS is a recursion relation for calculating the inverse Hessian matrix H-I from training data and structural information of the net. OBS permits a 90%, a 76%, and a 62% reduction in weights over backpropagation with weighL decay on three benchmark MONK's problems [Thrun et aI., 1991]. Of OBS, Optimal Brain Damage, and magnitude-based methods, only OBS deletes the correct weights from a trained XOR network in every case. Finally, whereas Sejnowski and Rosenberg [1987J used 18,000 weights in their NETtalk network, we used OBS to prune a network to just 1560 weights, yielding better generalization.

We have used information-theoretic ideas to derive a class of practical and nearly optimal schemes for adapting the size of a neural network. By removing unimportant weights from a network, several improvements can be expected: better generalization, fewer training examples required, and improved speed of learning and/or classification. The basic idea is to use second-derivative information to make a tradeoff between network complexity and training set error. Experiments confirm the usefulness of the methods on a real-world application.

We present a new algorithm for nding low complexity neural networks with high generalization capability. The algorithm searches for a \ at" minimum of the error function. A at minimum is a large connected region in weight-space where the error remains approximately constant. An MDL-based, Bayesian argument suggests that at minima correspond to \simple" networks and low expected over tting. The argument is based on a Gibbs algorithm variant and a novel way of splitting generalization error into under tting and over tting error. Unlike many previous approaches, ours does not require Gaussian assumptions and does not depend on a \good" weight prior { instead we have a prior over input/output functions, thus taking into account net architecture and training set. Although our algorithm requires the computation of second order derivatives, it has backprop's order of complexity. Automatically, it e ectively prunes units, weights, and input lines. Various experiments with feedforward and recurrent nets are described. In an application to stock market prediction, at minimum search outperforms (1) conventional backprop, (2) weight decay, (3) \optimal brain surgeon" / \optimal brain damage". We also provide pseudo code of the algorithm (omitted from the NC-version).The appendix presents a detailed theoretical justi cation of our approach. Using a variant of the Gibbs algorithm, appendix A.1 de nes generalization, under tting and over tting error in a novel way. By de ning an appropriate prior over input-output functions, we postulate that the most probable network is a \ at" one. Appendix A.2 formally justi es the error function minimized by our algorithm. Appendix A.3 describes an e cient implementation of the algorithm. Appendix A.4 nally presents pseudo code of the algorithm. TASK / ARCHITECTURE / BOXESGeneralization task. The task is to approximate an unknown function f X Y mapping a nite set of possible inputs X R N to a nite set of possible outputs Y R K . A data set D is obtained from f (see appendix A.1). All training information is given by a nite set D 0 D. D 0 is called the training set. The pth element of D 0 is denoted by an input/target pair (x p ; y p ).Architecture/ Net functions. For simplicity, we will focus on a standard feedforward net (but in the experiments, we will use recurrent nets as well). The net has N input units, K output units, L weights, and di erentiable activation functions. It maps input vectors x 2 R N to output vectors o(w; x) 2 R K , where w is the L-dimensional weight vector, and the weight on the connection from unit j to i is denoted w ij . The net function induced by w is denoted net(w): for x 2 R N , net(w)(x) = o(w; x) = o 1 (w; x); o 2 (w; x); : : : ; o K 1 (w; x); o K (w; x) , where o i (w; x) denotes the i-th component of o(w; x), corresponding to output unit i. Training error. We use squared error E(net(w); D 0 ) := P (xp;yp)2D0 k y p o(w; x p ) k 2 , where k : k denotes the Euclidean norm.Tolerable error. To

A quantitative and practical Bayesian framework is described for learning of mappings in feedforward networks. The framework makes possible (1) objective comparisons between solutions using alternative network architectures, (2) objective stopping rules for network pruning or growing procedures, (3) objective choice of magnitude and type of weight decay terms or additive regularizers (for penalizing large weights, etc.), (4) a measure of the effective number of well-determined parameters in a model, (5) quantified estimates of the error bars on network parameters and on network output, and (6) objective comparisons with alternative learning and interpolation models such as splines and radial basis functions. The Bayesian "evidence" automatically embodies "Occam's razor,'' penalizing overflexible and overcomplex models.The Bayesian approach helps detect poor underlying assumptions in learning models. For learning models well matched to a problem, a good correlation between generalization ability and the Bayesian evidence is obtained.This paper makes use of the Bayesian framework for regularization and model comparison described in the companion paper "Bayesian Interpolation" (MacKay 1992a). This framework is due to Gull and Skilling (Gull 1989). The Gaps in BackpropThere are many knobs on the black box of "backprop" [learning by backpropagation of errors (Rumelhart et al. 198611. Generally these knobs are set by rules of thumb, trial and error, and the use of reserved test data to assess generalization ability (or more sophisticated cross-validation). The knobs fall into two classes: (1) parameters that change the effective learning model, for example, number of hidden units, and weight decay

网络压缩对于使深网的效率更高，更快且可推广到低端硬件至关重要。当前的网络压缩方法有两个开放问题：首先，缺乏理论框架来估计最大压缩率；其次，有些层可能会过多地进行，从而导致网络性能大幅下降。为了解决这两个问题，这项研究提出了一种基于梯度矩阵分析方法，以估计最大网络冗余。在最大速率的指导下，开发了一种新颖而有效的层次网络修剪算法，以最大程度地凝结神经元网络结构而无需牺牲网络性能。进行实质性实验以证明新方法修剪几个高级卷积神经网络（CNN）体系结构的功效。与现有的修剪方法相比，拟议的修剪算法实现了最先进的性能。与其他方法相比，在相同或相似的压缩比下，新方法提供了最高的网络预测准确性。

We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases is completely non-sparse. It is derived by approximating various large blocks of the Fisher (corresponding to entire layers) as being the Kronecker product of two much smaller matrices. While only several times more expensive to compute than the plain stochastic gradient, the updates produced by K-FAC make much more progress optimizing the objective, which results in an algorithm that can be much faster than stochastic gradient descent with momentum in practice. And unlike some previously proposed approximate natural-gradient/Newton methods which use high-quality non-diagonal curvature matrices (such as Hessian-free optimization), K-FAC works very well in highly stochastic optimization regimes. This is because the cost of storing and inverting K-FAC's approximation to the curvature matrix does not depend on the amount of data used to estimate it, which is a feature typically associated only with diagonal or low-rank approximations to the curvature matrix.

Variational methods have been previously explored as a tractable approximation to Bayesian inference for neural networks. However the approaches proposed so far have only been applicable to a few simple network architectures. This paper introduces an easy-to-implement stochastic variational method (or equivalently, minimum description length loss function) that can be applied to most neural networks. Along the way it revisits several common regularisers from a variational perspective. It also provides a simple pruning heuristic that can both drastically reduce the number of network weights and lead to improved generalisation. Experimental results are provided for a hierarchical multidimensional recurrent neural network applied to the TIMIT speech corpus.

这项正在进行的工作旨在为统计学习提供统一的介绍，从诸如GMM和HMM等经典模型到现代神经网络（如VAE和扩散模型）缓慢地构建。如今，有许多互联网资源可以孤立地解释这一点或新的机器学习算法，但是它们并没有（也不能在如此简短的空间中）将这些算法彼此连接起来，或者与统计模型的经典文献相连现代算法出现了。同样明显缺乏的是一个单一的符号系统，尽管对那些已经熟悉材料的人（如这些帖子的作者）不满意，但对新手的入境造成了重大障碍。同样，我的目的是将各种模型（尽可能）吸收到一个用于推理和学习的框架上，表明（以及为什么）如何以最小的变化将一个模型更改为另一个模型（其中一些是新颖的，另一些是文献中的）。某些背景当然是必要的。我以为读者熟悉基本的多变量计算，概率和统计以及线性代数。这本书的目标当然不是​​完整性，而是从基本知识到过去十年中极强大的新模型的直线路径或多或少。然后，目标是补充而不是替换，诸如Bishop的\ emph {模式识别和机器学习}之类的综合文本，该文本现在已经15岁了。

有效地近似损失函数的局部曲率信息是用于深神经网络的优化和压缩的关键工具。然而，大多数现有方法近似二阶信息具有高计算或存储成本，这可以限制其实用性。在这项工作中，我们调查矩阵，用于估计逆象征的矢量产品（IHVPS）的矩阵线性时间方法，因为当Hessian可以近似为乘语 - 一个矩阵的总和时，如Hessian的经典近似由经验丰富的Fisher矩阵。我们提出了两个新的算法作为称为M-FAC的框架的一部分：第一个算法朝着网络压缩量身定制，如果Hessian给出了M $等级的总和，则可以计算Dimension $ D $的IHVP。 ，使用$ O（DM ^ 2）$预压制，$ O（DM）$代价计算IHVP，并查询逆Hessian的任何单个元素的费用$ O（m）$。第二算法针对优化设置，我们希望在反向Hessian之间计算产品，估计在优化步骤的滑动窗口和给定梯度方向上，根据预先说明的SGD所需的梯度方向。我们为计算IHVP和OHVP和O（DM + M ^ 3）$ of $ o（dm + m ^ 2）$提供算法，以便从滑动窗口添加或删除任何渐变。这两种算法产生最先进的结果，用于网络修剪和相对于现有二阶方法的计算开销的优化。在[9]和[17]可用实现。

Pruning large neural networks while maintaining their performance is often desirable due to the reduced space and time complexity. In existing methods, pruning is done within an iterative optimization procedure with either heuristically designed pruning schedules or additional hyperparameters, undermining their utility. In this work, we present a new approach that prunes a given network once at initialization prior to training. To achieve this, we introduce a saliency criterion based on connection sensitivity that identifies structurally important connections in the network for the given task. This eliminates the need for both pretraining and the complex pruning schedule while making it robust to architecture variations. After pruning, the sparse network is trained in the standard way. Our method obtains extremely sparse networks with virtually the same accuracy as the reference network on the MNIST, CIFAR-10, and Tiny-ImageNet classification tasks and is broadly applicable to various architectures including convolutional, residual and recurrent networks. Unlike existing methods, our approach enables us to demonstrate that the retained connections are indeed relevant to the given task.

Deep Learning optimization involves minimizing a high-dimensional loss function in the weight space which is often perceived as difficult due to its inherent difficulties such as saddle points, local minima, ill-conditioning of the Hessian and limited compute resources. In this paper, we provide a comprehensive review of 12 standard optimization methods successfully used in deep learning research and a theoretical assessment of the difficulties in numerical optimization from the optimization literature.

Neural networks are both computationally intensive and memory intensive, making them difficult to deploy on embedded systems. Also, conventional networks fix the architecture before training starts; as a result, training cannot improve the architecture. To address these limitations, we describe a method to reduce the storage and computation required by neural networks by an order of magnitude without affecting their accuracy by learning only the important connections. Our method prunes redundant connections using a three-step method. First, we train the network to learn which connections are important. Next, we prune the unimportant connections. Finally, we retrain the network to fine tune the weights of the remaining connections. On the ImageNet dataset, our method reduced the number of parameters of AlexNet by a factor of 9×, from 61 million to 6.7 million, without incurring accuracy loss. Similar experiments with VGG-16 found that the total number of parameters can be reduced by 13×, from 138 million to 10.3 million, again with no loss of accuracy.

Multilayer Neural Networks trained with the backpropagation algorithm constitute the best example of a successful Gradient-Based Learning technique. Given an appropriate network architecture, Gradient-Based Learning algorithms can be used to synthesize a complex decision surface that can classify high-dimensional patterns such as handwritten characters, with minimal preprocessing. This paper reviews various methods applied to handwritten character recognition and compares them on a standard handwritten digit recognition task. Convolutional Neural Networks, that are specifically designed to deal with the variability of 2D shapes, are shown to outperform all other techniques.Real-life document recognition systems are composed of multiple modules including eld extraction, segmentation, recognition, and language modeling. A new learning paradigm, called Graph Transformer Networks (GTN), allows such multi-module systems to be trained globally using Gradient-Based methods so as to minimize an overall performance measure.Two systems for on-line handwriting recognition are described. Experiments demonstrate the advantage of global training, and the exibility of Graph Transformer Networks.A Graph Transformer Network for reading bank check is also described. It uses Convolutional Neural Network character recognizers combined with global training techniques to provides record accuracy on business and personal checks. It is deployed commercially and reads several million checks per day.

Neural networks are both computationally intensive and memory intensive, making them difficult to deploy on embedded systems. Also, conventional networks fix the architecture before training starts; as a result, training cannot improve the architecture. To address these limitations, we describe a method to reduce the storage and computation required by neural networks by an order of magnitude without affecting their accuracy by learning only the important connections. Our method prunes redundant connections using a three-step method. First, we train the network to learn which connections are important. Next, we prune the unimportant connections. Finally, we retrain the network to fine tune the weights of the remaining connections. On the ImageNet dataset, our method reduced the number of parameters of AlexNet by a factor of 9×, from 61 million to 6.7 million, without incurring accuracy loss. Similar experiments with VGG-16 found that the total number of parameters can be reduced by 13×, from 138 million to 10.3 million, again with no loss of accuracy.

We propose a new formulation for pruning convolutional kernels in neural networks to enable efficient inference. We interleave greedy criteria-based pruning with finetuning by backpropagation-a computationally efficient procedure that maintains good generalization in the pruned network. We propose a new criterion based on Taylor expansion that approximates the change in the cost function induced by pruning network parameters. We focus on transfer learning, where large pretrained networks are adapted to specialized tasks. The proposed criterion demonstrates superior performance compared to other criteria, e.g. the norm of kernel weights or feature map activation, for pruning large CNNs after adaptation to fine-grained classification tasks (Birds-200 and Flowers-102) relaying only on the first order gradient information. We also show that pruning can lead to more than 10× theoretical reduction in adapted 3D-convolutional filters with a small drop in accuracy in a recurrent gesture classifier. Finally, we show results for the largescale ImageNet dataset to emphasize the flexibility of our approach.

如果神经网络更大，即使所产生的模型过度参数化，神经网络也倾向于通过训练获得更好的准确性。然而，在训练之前，期间或之后，仔细删除此类多余参数也可能会产生具有相似精度甚至提高的模型。在许多情况下，可以通过启发式方法奇怪地实现，就像去除具有最小绝对价值的权重一样 - 即使幅度并不是重量相关性的完美代理。以这样的前提是，从修剪中获得更好的性能取决于删除多个权重的综合效果的考虑，我们重新审视了基于影响的基于撞击的经典方法之一：最佳脑外科医生（obs）。我们提出了一种可拖动的启发式方法，用于求解OBS的组合扩展，其中我们选择了同时删除的权重，以及剩余权重的系统更新。我们的选择方法在高稀疏性下的其他方法优于其他方法，即使与其他方法结合使用，重量更新也是有利的。

修剪神经网络可降低推理时间和记忆成本。在标准硬件上，如果修剪诸如特征地图之类的粗粒结构（例如特征地图），这些好处将特别突出。我们为二阶结构修剪（SOSP）设计了两种新型的基于显着性的方法，其中包括所有结构和层之间的相关性。我们的主要方法SOSP-H采用了创新的二阶近似，可以通过快速的Hessian-vector产品进行显着评估。 SOSP-H因此，尽管考虑到了完整的Hessian，但仍像一阶方法一样缩放。我们通过将SOSP-H与使用公认的Hessian近似值以及许多最先进方法进行比较来验证SOSP-H。尽管SOSP-H在准确性方面的表现或更好，但在可伸缩性和效率方面具有明显的优势。这使我们能够将SOSP-H扩展到大规模视觉任务，即使它捕获了网络所有层的相关性。为了强调我们修剪方法的全球性质，我们不仅通过删除预验证网络的结构，而且还通过检测建筑瓶颈来评估它们的性能。我们表明，我们的算法允许系统地揭示建筑瓶颈，然后将其删除以进一步提高网络的准确性。

Structural pruning of neural network parameters reduces computation, energy, and memory transfer costs during inference. We propose a novel method that estimates the contribution of a neuron (filter) to the final loss and iteratively removes those with smaller scores. We describe two variations of our method using the first and secondorder Taylor expansions to approximate a filter's contribution. Both methods scale consistently across any network layer without requiring per-layer sensitivity analysis and can be applied to any kind of layer, including skip connections. For modern networks trained on ImageNet, we measured experimentally a high (>93%) correlation between the contribution computed by our methods and a reliable estimate of the true importance. Pruning with the proposed methods leads to an improvement over state-ofthe-art in terms of accuracy, FLOPs, and parameter reduction. On ResNet-101, we achieve a 40% FLOPS reduction by removing 30% of the parameters, with a loss of 0.02% in the top-1 accuracy on ImageNet. Code is available at https://github.com/NVlabs/Taylor_pruning.

修剪深度神经网络的现有方法专注于去除训练有素的网络的不必要参数，然后微调模型，找到恢复训练模型的初始性能的良好解决方案。与其他作品不同，我们的方法特别注意通过修剪神经元的压缩模型和推理计算时间的解决方案的质量。通过探索Hessian的光谱半径，所提出的算法通过探索Hessian的光谱半径来指示压缩模型的参数，这导致了更好地推广了未经看涨的数据。此外，该方法不适用于预先训练的网络，并同时执行训练和修剪。我们的结果表明，它改善了神经元压缩的最先进的结果。该方法能够在不同神经网络模型上实现具有小精度下降的非常小的网络。

我们考虑在具有挑战性的训练后环境中，深度神经网络（DNN）的模型压缩问题，在该设置中，我们将获得精确的训练模型，并且必须仅基于少量校准输入数据而无需任何重新培训即可压缩它。鉴于新兴软件和硬件支持通过加速修剪和/或量化压缩的模型，并且已经针对两种压缩方法独立提出了良好的表现解决方案，因此该问题已变得流行。在本文中，我们引入了一个新的压缩框架，该框架涵盖了统一环境中的重量修剪和量化，时间和空间效率高，并且在现有的后训练方法的实际性能上大大改善。在技​​术层面上，我们的方法基于[Lecun，Denker和Solla，1990年]在现代DNN的规模上的经典最佳脑外科医生（OBS）框架的第一个精确实现，我们进一步扩展到覆盖范围。重量量化。这是通过一系列可能具有独立利益的算法开发来实现的。从实际的角度来看，我们的实验结果表明，它可以在现有后训练方法的压缩 - 准确性权衡方面显着改善，并且甚至可以在训练后进行修剪和量化的准确共同应用。