连续深度学习架构能够学习用于预测建模的灵活概率模型作为神经常规差分方程（ODES），以及作为连续标准化流动的生成建模。在这项工作中，我们通过修剪网络架构来设计一个框架来破译这些连续深度模型的内部动态。我们的经验结果表明，修剪改善了生成建模中神经杂物的概括。我们经验表明，改进是因为修剪有助于避免模式塌陷并压平损耗表面。此外，修剪发现高效的神经颂歌表示，与原始网络相比，参数减少了98％，而不会损失准确性。我们希望我们的结果将使进一步研究现代持续深度模型的性能尺寸折衷。

正常化流动在过去几年中已经变得更加流行;然而，他们继续计算得昂贵，使得它们难以被接受到更广泛的机器学习界中。在本文中，我们介绍了一个简单的一维一层网络，其封闭形式的Lipschitz常数;使用此，我们介绍了一种新的精确嘴唇流（ELF），这些流量（ELF）结合了剩余流量的易于采样，并具有自回归流的强烈性能。此外，我们表明，与多个其他流相比，ELF被证明是通用密度近似器，更新和参数有效，并且在多个大规模数据集上实现最先进的性能。

Normalizing Flows are generative models which produce tractable distributions where both sampling and density evaluation can be efficient and exact. The goal of this survey article is to give a coherent and comprehensive review of the literature around the construction and use of Normalizing Flows for distribution learning. We aim to provide context and explanation of the models, review current state-of-the-art literature, and identify open questions and promising future directions.

Normalizing flows provide a general mechanism for defining expressive probability distributions, only requiring the specification of a (usually simple) base distribution and a series of bijective transformations. There has been much recent work on normalizing flows, ranging from improving their expressive power to expanding their application. We believe the field has now matured and is in need of a unified perspective. In this review, we attempt to provide such a perspective by describing flows through the lens of probabilistic modeling and inference. We place special emphasis on the fundamental principles of flow design, and discuss foundational topics such as expressive power and computational trade-offs. We also broaden the conceptual framing of flows by relating them to more general probability transformations. Lastly, we summarize the use of flows for tasks such as generative modeling, approximate inference, and supervised learning.

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。

We propose a simultaneous learning and pruning algorithm capable of identifying and eliminating irrelevant structures in a neural network during the early stages of training. Thus, the computational cost of subsequent training iterations, besides that of inference, is considerably reduced. Our method, based on variational inference principles using Gaussian scale mixture priors on neural network weights, learns the variational posterior distribution of Bernoulli random variables multiplying the units/filters similarly to adaptive dropout. Our algorithm, ensures that the Bernoulli parameters practically converge to either 0 or 1, establishing a deterministic final network. We analytically derive a novel hyper-prior distribution over the prior parameters that is crucial for their optimal selection and leads to consistent pruning levels and prediction accuracy regardless of weight initialization or the size of the starting network. We prove the convergence properties of our algorithm establishing theoretical and practical pruning conditions. We evaluate the proposed algorithm on the MNIST and CIFAR-10 data sets and the commonly used fully connected and convolutional LeNet and VGG16 architectures. The simulations show that our method achieves pruning levels on par with state-of the-art methods for structured pruning, while maintaining better test-accuracy and more importantly in a manner robust with respect to network initialization and initial size.

修剪是稀疏深神经网络的任务，最近受到了越来越多的关注。尽管最先进的修剪方法提取了高度稀疏的模型，但它们忽略了两个主要挑战：（1）寻找这些稀疏模型的过程通常非常昂贵； （2）非结构化的修剪在GPU记忆，训练时间或碳排放方面没有提供好处。我们提出了通过梯度流量保存（早期CROP）提出的早期压缩，该压缩在训练挑战（1）的培训（1）中有效提取最先进的稀疏模型，并且可以以结构化的方式应用来应对挑战（2）。这使我们能够在商品GPU上训练稀疏的网络，该商品GPU的密集版本太大，从而节省了成本并减少了硬件要求。我们从经验上表明，早期杂交的表现优于许多任务（包括分类，回归）和域（包括计算机视觉，自然语言处理和增强学习）的丰富基线。早期杂交导致准确性与密集训练相当，同时超过修剪基线。

Network pruning is widely used for reducing the heavy inference cost of deep models in low-resource settings. A typical pruning algorithm is a three-stage pipeline, i.e., training (a large model), pruning and fine-tuning. During pruning, according to a certain criterion, redundant weights are pruned and important weights are kept to best preserve the accuracy. In this work, we make several surprising observations which contradict common beliefs. For all state-of-the-art structured pruning algorithms we examined, fine-tuning a pruned model only gives comparable or worse performance than training that model with randomly initialized weights. For pruning algorithms which assume a predefined target network architecture, one can get rid of the full pipeline and directly train the target network from scratch. Our observations are consistent for multiple network architectures, datasets, and tasks, which imply that: 1) training a large, over-parameterized model is often not necessary to obtain an efficient final model, 2) learned "important" weights of the large model are typically not useful for the small pruned model, 3) the pruned architecture itself, rather than a set of inherited "important" weights, is more crucial to the efficiency in the final model, which suggests that in some cases pruning can be useful as an architecture search paradigm. Our results suggest the need for more careful baseline evaluations in future research on structured pruning methods. We also compare with the "Lottery Ticket Hypothesis" (Frankle & Carbin, 2019), and find that with optimal learning rate, the "winning ticket" initialization as used in Frankle & Carbin (2019) does not bring improvement over random initialization. * Equal contribution. † Work done while visiting UC Berkeley.

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.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a blackbox differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

训练有素的神经网络的性能至关重要。加上深度学习模型的不断增长的规模，这种观察激发了对学习稀疏模型的广泛研究。在这项工作中，我们专注于控制稀疏学习时的稀疏水平的任务。基于稀疏性惩罚的现有方法涉及对罚款因素的昂贵反复试验调整，因此缺乏直接控制所得模型的稀疏性。作为响应，我们采用了一个约束的公式：使用Louizos等人提出的栅极机制。 （2018年），我们制定了一个受约束的优化问题，其中稀疏以训练目标和所需的稀疏目标以端到端的方式指导。使用WIDERESNET和RESNET {18，50}模型进行了CIFAR-10/100，Tinyimagenet和ImageNet的实验验证了我们的提案的有效性，并证明我们可以可靠地实现预定的稀疏目标，而不会损害预测性能。

由于稀疏神经网络通常包含许多零权重，因此可以在不降低网络性能的情况下潜在地消除这些不必要的网络连接。因此，设计良好的稀疏神经网络具有显着降低拖鞋和计算资源的潜力。在这项工作中，我们提出了一种新的自动修剪方法 - 稀疏连接学习（SCL）。具体地，重量被重新参数化为可培训权重变量和二进制掩模的元素方向乘法。因此，由二进制掩模完全描述网络连接，其由单位步进函数调制。理论上，从理论上证明了使用直通估计器（STE）进行网络修剪的基本原理。这一原则是STE的代理梯度应该是积极的，确保掩模变量在其最小值处收敛。在找到泄漏的Relu后，SoftPlus和Identity Stes可以满足这个原理，我们建议采用SCL的身份STE以进行离散面膜松弛。我们发现不同特征的面具梯度非常不平衡，因此，我们建议将每个特征的掩模梯度标准化以优化掩码变量训练。为了自动训练稀疏掩码，我们将网络连接总数作为我们的客观函数中的正则化术语。由于SCL不需要由网络层设计人员定义的修剪标准或超级参数，因此在更大的假设空间中探讨了网络，以实现最佳性能的优化稀疏连接。 SCL克服了现有自动修剪方法的局限性。实验结果表明，SCL可以自动学习并选择各种基线网络结构的重要网络连接。 SCL培训的深度学习模型以稀疏性，精度和减少脚波特的SOTA人类设计和自动修剪方法训练。

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。

A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.

The rectified linear unit (ReLU) is a highly successful activation function in neural networks as it allows networks to easily obtain sparse representations, which reduces overfitting in overparameterized networks. However, in network pruning, we find that the sparsity introduced by ReLU, which we quantify by a term called dynamic dead neuron rate (DNR), is not beneficial for the pruned network. Interestingly, the more the network is pruned, the smaller the dynamic DNR becomes during optimization. This motivates us to propose a method to explicitly reduce the dynamic DNR for the pruned network, i.e., de-sparsify the network. We refer to our method as Activating-while-Pruning (AP). We note that AP does not function as a stand-alone method, as it does not evaluate the importance of weights. Instead, it works in tandem with existing pruning methods and aims to improve their performance by selective activation of nodes to reduce the dynamic DNR. We conduct extensive experiments using popular networks (e.g., ResNet, VGG) via two classical and three state-of-the-art pruning methods. The experimental results on public datasets (e.g., CIFAR-10/100) suggest that AP works well with existing pruning methods and improves the performance by 3% - 4%. For larger scale datasets (e.g., ImageNet) and state-of-the-art networks (e.g., vision transformer), we observe an improvement of 2% - 3% with AP as opposed to without. Lastly, we conduct an ablation study to examine the effectiveness of the components comprising AP.

我们提出了特征神经常规差分方程（C节点），该框架用于扩展神经常规微分方程（节点）之外的缺点。虽然节点模型将潜在状态的演变为对颂歌的解决方案，但是所提出的C节点模拟了潜在的潜在的演变作为其特征的一阶准线性部分微分方程（PDE）的解决方案，定义为PDE减少到ODES的曲线。反过来，还原允许应用标准框架，以解决PDE设置的杂散。另外，所提出的框架可以作为现有节点架构的扩展来投用，从而允许使用现有的黑盒颂歌求解器。我们证明了C节点框架通过展示不能由节点表示的功能来扩展经典节点，而是由C节点表示。我们通过在许多合成和实际数据场景中展示其性能，进一步研究了C节点框架的功效。经验结果展示了CIFAR-10，SVHN和MNIST数据集的提出方法提供的改进，如类似的计算预算作为现有节点方法。

Many applications require sparse neural networks due to space or inference time restrictions. There is a large body of work on training dense networks to yield sparse networks for inference, but this limits the size of the largest trainable sparse model to that of the largest trainable dense model. In this paper we introduce a method to train sparse neural networks with a fixed parameter count and a fixed computational cost throughout training, without sacrificing accuracy relative to existing dense-tosparse training methods. Our method updates the topology of the sparse network during training by using parameter magnitudes and infrequent gradient calculations. We show that this approach requires fewer floating-point operations (FLOPs) to achieve a given level of accuracy compared to prior techniques. We demonstrate state-of-the-art sparse training results on a variety of networks and datasets, including ResNet-50, MobileNets on Imagenet-2012, and RNNs on WikiText-103. Finally, we provide some insights into why allowing the topology to change during the optimization can overcome local minima encountered when the topology remains static * .

Neural network pruning techniques can reduce the parameter counts of trained networks by over 90%, decreasing storage requirements and improving computational performance of inference without compromising accuracy. However, contemporary experience is that the sparse architectures produced by pruning are difficult to train from the start, which would similarly improve training performance.We find that a standard pruning technique naturally uncovers subnetworks whose initializations made them capable of training effectively. Based on these results, we articulate the lottery ticket hypothesis: dense, randomly-initialized, feed-forward networks contain subnetworks (winning tickets) that-when trained in isolationreach test accuracy comparable to the original network in a similar number of iterations. The winning tickets we find have won the initialization lottery: their connections have initial weights that make training particularly effective.We present an algorithm to identify winning tickets and a series of experiments that support the lottery ticket hypothesis and the importance of these fortuitous initializations. We consistently find winning tickets that are less than 10-20% of the size of several fully-connected and convolutional feed-forward architectures for MNIST and CIFAR10. Above this size, the winning tickets that we find learn faster than the original network and reach higher test accuracy.

Sparse neural networks attract increasing interest as they exhibit comparable performance to their dense counterparts while being computationally efficient. Pruning the dense neural networks is among the most widely used methods to obtain a sparse neural network. Driven by the high training cost of such methods that can be unaffordable for a low-resource device, training sparse neural networks sparsely from scratch has recently gained attention. However, existing sparse training algorithms suffer from various issues, including poor performance in high sparsity scenarios, computing dense gradient information during training, or pure random topology search. In this paper, inspired by the evolution of the biological brain and the Hebbian learning theory, we present a new sparse training approach that evolves sparse neural networks according to the behavior of neurons in the network. Concretely, by exploiting the cosine similarity metric to measure the importance of the connections, our proposed method, Cosine similarity-based and Random Topology Exploration (CTRE), evolves the topology of sparse neural networks by adding the most important connections to the network without calculating dense gradient in the backward. We carried out different experiments on eight datasets, including tabular, image, and text datasets, and demonstrate that our proposed method outperforms several state-of-the-art sparse training algorithms in extremely sparse neural networks by a large gap. The implementation code is available on https://github.com/zahraatashgahi/CTRE

彩票假设引发了通过识别大型随机初始化神经网络的稀疏子网来实现结构学习的修剪算法的快速发展。这些“胜利门票”的存在理论上已被证明，但在次优稀疏水平。当代修剪算法还在努力确定复杂的学习任务的稀疏彩票票。这个次优稀疏仅仅是存在证明和算法的文物还是修剪方法的一般限制？并且，如果存在非常稀疏的罚单，则当前算法是能够找到它们的当前算法，或者是实现有效网络压缩所需的进一步改进吗？为了系统地回答这些问题，我们推导了一个框架来植物并隐藏大型随机初始化的神经网络中的目标架构。对于机器学习中的三个共同挑战，我们手工制作极其稀疏的网络拓扑，将它们植入大型神经网络，并评估最先进的彩票修剪方法。我们发现，修剪算法的当前局限性识别极其稀疏的票证是算法的，而不是基本的性质，并且预期我们的种植框架将促进有效修剪算法的未来发展，因为我们已经解决了所提出的领域缺失基线的问题Frankle等人。