Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works have been focusing on why we can train neural networks when there is only one hidden layer. The theory of multi-layer networks remains unsettled. In this work, we prove simple algorithms such as stochastic gradient descent (SGD) can find global minima on the training objective of DNNs in polynomial time. We only make two assumptions: the inputs do not degenerate and the network is over-parameterized. The latter means the number of hidden neurons is sufficiently large: polynomial in L, the number of DNN layers and in n, the number of training samples. As concrete examples, starting from randomly initialized weights, we show that SGD attains 100% training accuracy in classification tasks, or minimizes regression loss in linear convergence speed ε ∝ e −Ω(T ) , with running time polynomial in n and L. Our theory applies to the widely-used but non-smooth ReLU activation, and to any smooth and possibly non-convex loss functions. In terms of network architectures, our theory at least applies to fully-connected neural networks, convolutional neural networks (CNN), and residual neural networks (ResNet).* Equal contribution . Full version and future updates are available at https://arxiv.org/abs/1811.03962.This paper is a follow up to the recurrent neural network (RNN) paper (Allen-Zhu et al., 2018b) by the same set of authors. Most of the techniques used in this paper were already discovered in the RNN paper, and this paper can be viewed as a simplification (or to some extent a special case) of the RNN setting in order to reach out to a wider audience. We compare the difference and mention our additional contribution in Section 1.2.

The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesn't the trained network overfit when it is overparameterized?In this work, we prove that overparameterized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network.On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic approximation of the neural network (that can be viewed as a second-order variant of NTK), and connect it to the SGD theory of escaping saddle points.

Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep overparameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. We further extend our analysis to deep residual convolutional neural networks and obtain a similar convergence result.

Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter characterization of training speed than recent papers, an explanation for why training a neural net with random labels leads to slower training, as originally observed in [Zhang et al. ICLR'17]. (ii) Generalization bound independent of network size, using a data-dependent complexity measure. Our measure distinguishes clearly between random labels and true labels on MNIST and CIFAR, as shown by experiments. Moreover, recent papers require sample complexity to increase (slowly) with the size, while our sample complexity is completely independent of the network size. (iii) Learnability of a broad class of smooth functions by 2-layer ReLU nets trained via gradient descent.The key idea is to track dynamics of training and generalization via properties of a related kernel.

尽管使用对抗性训练捍卫深度学习模型免受对抗性扰动的经验成功，但到目前为止，仍然不清楚对抗性扰动的存在背后的原则是什么，而对抗性培训对神经网络进行了什么来消除它们。在本文中，我们提出了一个称为特征纯化的原则，在其中，我们表明存在对抗性示例的原因之一是在神经网络的训练过程中，在隐藏的重量中积累了某些小型密集混合物；更重要的是，对抗训练的目标之一是去除此类混合物以净化隐藏的重量。我们介绍了CIFAR-10数据集上的两个实验，以说明这一原理，并且一个理论上的结果证明，对于某些自然分类任务，使用随机初始初始化的梯度下降训练具有RELU激活的两层神经网络确实满足了这一原理。从技术上讲，我们给出了我们最大程度的了解，第一个结果证明，以下两个可以同时保持使用RELU激活的神经网络。 （1）对原始数据的训练确实对某些半径的小对抗扰动确实不舒适。 （2）即使使用经验性扰动算法（例如FGM），实际上也可以证明对对抗相同半径的任何扰动也可以证明具有强大的良好性。最后，我们还证明了复杂性的下限，表明该网络的低复杂性模型，例如线性分类器，低度多项式或什至是神经切线核，无论使用哪种算法，都无法防御相同半径的扰动训练他们。

在本文中，我们研究了学习最适合培训数据集的浅层人工神经网络的问题。我们在过度参数化的制度中研究了这个问题，在该制度中，观测值的数量少于模型中的参数数量。我们表明，通过二次激活，训练的优化景观这种浅神经网络具有某些有利的特征，可以使用各种局部搜索启发式方法有效地找到全球最佳模型。该结果适用于输入/输出对的任意培训数据。对于可区分的激活函数，我们还表明，适当初始化的梯度下降以线性速率收敛到全球最佳模型。该结果着重于选择输入的可实现模型。根据高斯分布和标签是根据种植的重量系数生成的。

近期在应用于培训深度神经网络和数据分析中的其他优化问题中的非凸优化的优化算法的兴趣增加，我们概述了最近对非凸优化优化算法的全球性能保证的理论结果。我们从古典参数开始，显示一般非凸面问题无法在合理的时间内有效地解决。然后，我们提供了一个问题列表，可以通过利用问题的结构来有效地找到全球最小化器，因为可能的问题。处理非凸性的另一种方法是放宽目标，从找到全局最小，以找到静止点或局部最小值。对于该设置，我们首先为确定性一阶方法的收敛速率提出了已知结果，然后是最佳随机和随机梯度方案的一般理论分析，以及随机第一阶方法的概述。之后，我们讨论了非常一般的非凸面问题，例如最小化$ \ alpha $ -weakly-are-convex功能和满足Polyak-lojasiewicz条件的功能，这仍然允许获得一阶的理论融合保证方法。然后，我们考虑更高阶和零序/衍生物的方法及其收敛速率，以获得非凸优化问题。

探讨了第一层神经网络中的参数和输入数据的乘法结构，以在丢失功能的景观与模型函数的景观与输入数据的景观之间建立连接。通过这种连接，示出了平坦的最小值规范了模型功能的梯度，这解释了扁平最小值的良好泛化性能。然后，我们超越平坦度并考虑梯度噪声的高阶矩，并且表明随机梯度下降（SGD）倾向于通过全球最小值的SGD的线性稳定性分析对这些瞬间施加约束。我们与乘法结构一起，我们识别SGD的SoboLev正则化效果，即SGD对输入数据的模型函数的SoboLev Semininorms进行了规范。最后，提供了在数据分布的假设下由SGD发现的解决方案的泛化误差和逆势鲁棒性的界限。

神经网络在许多领域取得了巨大的经验成功。已经观察到，通过一阶方法训练的随机初始化的神经网络能够实现接近零的训练损失，尽管其损失景观是非凸的并且不平滑的。这种现象很少有理论解释。最近，通过分析过参数化制度中的梯度下降〜（GD）和重球方法〜（HB）的梯度来弥合实践和理论之间的这种差距。在这项工作中，通过考虑Nesterov的加速梯度方法〜（nag），我们通过恒定的动量参数进行进一步进展。我们通过Relu激活分析其用于过度参数化的双层完全连接神经网络的收敛性。具体而言，我们证明了NAG的训练误差以非渐近线性收敛率$（1- \θ（1 / \ sqrt {\ kappa}））收敛到零（1 / \ sqrt {\ kappa}）^ t $ the $ t $迭代，其中$ \ Kappa> 1 $由神经网络的初始化和架构决定。此外，我们在NAG和GD和HB的现有收敛结果之间提供了比较。我们的理论结果表明，NAG实现了GD的加速度，其会聚率与HB相当。此外，数值实验验证了我们理论分析的正确性。

Consider the multivariate nonparametric regression model. It is shown that estimators based on sparsely connected deep neural networks with ReLU activation function and properly chosen network architecture achieve the minimax rates of convergence (up to log nfactors) under a general composition assumption on the regression function. The framework includes many well-studied structural constraints such as (generalized) additive models. While there is a lot of flexibility in the network architecture, the tuning parameter is the sparsity of the network. Specifically, we consider large networks with number of potential network parameters exceeding the sample size. The analysis gives some insights into why multilayer feedforward neural networks perform well in practice. Interestingly, for ReLU activation function the depth (number of layers) of the neural network architectures plays an important role and our theory suggests that for nonparametric regression, scaling the network depth with the sample size is natural. It is also shown that under the composition assumption wavelet estimators can only achieve suboptimal rates.

How well does a classic deep net architecture like AlexNet or VGG19 classify on a standard dataset such as CIFAR-10 when its "width"-namely, number of channels in convolutional layers, and number of nodes in fully-connected internal layers -is allowed to increase to infinity? Such questions have come to the forefront in the quest to theoretically understand deep learning and its mysteries about optimization and generalization. They also connect deep learning to notions such as Gaussian processes and kernels. A recent paper [Jacot et al., 2018] introduced the Neural Tangent Kernel (NTK) which captures the behavior of fully-connected deep nets in the infinite width limit trained by gradient descent; this object was implicit in some other recent papers. An attraction of such ideas is that a pure kernel-based method is used to capture the power of a fully-trained deep net of infinite width. The current paper gives the first efficient exact algorithm for computing the extension of NTK to convolutional neural nets, which we call Convolutional NTK (CNTK), as well as an efficient GPU implementation of this algorithm. This results in a significant new benchmark for performance of a pure kernel-based method on CIFAR-10, being 10% higher than the methods reported in [Novak et al., 2019], and only 6% lower than the performance of the corresponding finite deep net architecture (once batch normalization etc. are turned off). Theoretically, we also give the first non-asymptotic proof showing that a fully-trained sufficiently wide net is indeed equivalent to the kernel regression predictor using NTK.

过度参数化神经网络（NN）的损失表面具有许多全球最小值，却零训练误差。我们解释了标准NN训练程序的常见变体如何改变获得的最小化器。首先，我们明确说明了强烈参数化的NN初始化的大小如何影响最小化器，并可能恶化其最终的测试性能。我们提出了限制这种效果的策略。然后，我们证明，对于自适应优化（例如Adagrad），所获得的最小化器通常与梯度下降（GD）最小化器不同。随机迷你批次训练，即使在非自适应情况下，GD和随机GD基本相同的最小化器，这种自适应最小化器也会进一步改变。最后，我们解释说，这些效果仍然与较少参数化的NN相关。尽管过度参数具有其好处，但我们的工作强调，它会导致参数化模型缺乏错误来源。

This paper shows that a perturbed form of gradient descent converges to a second-order stationary point in a number iterations which depends only poly-logarithmically on dimension (i.e., it is almost "dimension-free"). The convergence rate of this procedure matches the wellknown convergence rate of gradient descent to first-order stationary points, up to log factors. When all saddle points are non-degenerate, all second-order stationary points are local minima, and our result thus shows that perturbed gradient descent can escape saddle points almost for free.Our results can be directly applied to many machine learning applications, including deep learning. As a particular concrete example of such an application, we show that our results can be used directly to establish sharp global convergence rates for matrix factorization. Our results rely on a novel characterization of the geometry around saddle points, which may be of independent interest to the non-convex optimization community.

我们考虑培训多层过参数化神经网络的问题，以最大限度地减少损失函数引起的经验风险。在过度参数化的典型设置中，网络宽度$ M $远大于数据维度$ D $和培训数量$ N $（$ m = \ mathrm {poly}（n，d）$），其中诱导禁止的大量矩阵$ w \ in \ mathbb {r} ^ {m \ times m} $每层。天真地，一个人必须支付$ O（m ^ 2）$时间读取权重矩阵并评估前向和后向计算中的神经网络功能。在这项工作中，我们展示了如何降低每个迭代的培训成本，具体而言，我们提出了一个仅在初始化阶段使用M ^ 2美元的框架，并且在$ M $的情况下实现了每次迭代的真正子种化成本。 ，$ m ^ {2- \ oomga（1）} $次迭代。为了获得此结果，我们利用各种技术，包括偏移的基于Relu的稀释器，懒惰的低级维护数据结构，快速矩阵矩阵乘法，张量的草图技术和预处理。

鉴于密集的浅色神经网络，我们专注于迭代创建，培训和组合随机选择的子网（代理函数），以训练完整模型。通过仔细分析$ i）$ Subnetworks的神经切线内核，II美元）$代理职能'梯度，以及$ iii）$我们如何对替代品函数进行采样并结合训练错误的线性收敛速度 - 内部一个错误区域 - 对于带有回归任务的Relu激活的过度参数化单隐藏层Perceptron。我们的结果意味着，对于固定的神经元选择概率，当我们增加代理模型的数量时，误差项会减少，并且随着我们增加每个所选子网的本地训练步骤的数量而增加。考虑的框架概括并提供了关于辍学培训，多样化辍学培训以及独立的子网培训的新见解;对于每种情况，我们提供相应的收敛结果，作为我们主要定理的冠状动脉。

We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) and [Nguyen 21, Section 3.2] (training both layers with Le Cun's initialization). We also report several simple experiments with synthetic data. They strongly suggest that, at least in our model, the convergence phenomenon extends well beyond the ``NTK regime''.

训练神经网络的一种常见方法是将所有权重初始化为独立的高斯向量。我们观察到，通过将权重初始化为独立对，每对由两个相同的高斯向量组成，我们可以显着改善收敛分析。虽然已经研究了类似的技术来进行随机输入[Daniely，Neurips 2020]，但尚未使用任意输入进行分析。使用此技术，我们展示了如何显着减少两层relu网络所需的神经元数量，均在逻辑损失的参数化设置不足的情况下，大约$ \ gamma^{ - 8} $ [Ji and telgarsky，ICLR， 2020]至$ \ gamma^{ - 2} $，其中$ \ gamma $表示带有神经切线内核的分离边距，以及在与平方损失的过度参数化设置中，从大约$ n^4 $ [song [song]和Yang，2019年]至$ n^2 $，隐含地改善了[Brand，Peng，Song和Weinstein，ITCS 2021]的近期运行时间。对于参数不足的设置，我们还证明了在先前工作时改善的新下限，并且在某些假设下是最好的。

最近的发现（例如ARXIV：2103.00065）表明，通过全批梯度下降训练的现代神经网络通常进入一个称为稳定边缘（EOS）的政权。在此制度中，清晰度（即最大的Hessian特征值）首先增加到值2/（步长尺寸）（渐进锐化阶段），然后在该值（EOS相）周围振荡。本文旨在分析沿优化轨迹的GD动力学和清晰度。我们的分析自然将GD轨迹分为四个阶段，具体取决于清晰度的变化。从经验上，我们将输出层重量的规范视为清晰动力学的有趣指标。基于这一经验观察，我们尝试从理论和经验上解释导致EOS每个阶段清晰度变化的各种关键量的动力学。此外，基于某些假设，我们提供了两层完全连接的线性神经网络中EOS制度的清晰度行为的理论证明。我们还讨论了其他一些经验发现以及我们的理论结果的局限性。

Neural networks have many successful applications, while much less theoretical understanding has been gained. Towards bridging this gap, we study the problem of learning a two-layer overparameterized ReLU neural network for multi-class classification via stochastic gradient descent (SGD) from random initialization. In the overparameterized setting, when the data comes from mixtures of wellseparated distributions, we prove that SGD learns a network with a small generalization error, albeit the network has enough capacity to fit arbitrary labels. Furthermore, the analysis provides interesting insights into several aspects of learning neural networks and can be verified based on empirical studies on synthetic data and on the MNIST dataset.

具有动量的随机梯度下降（SGD）被广泛用于训练现代深度学习体系结构。虽然可以很好地理解使用动量可以导致在各种环境中更快的收敛速率，但还观察到动量会产生更高的概括。先前的工作认为，动量在训练过程中稳定了SGD噪声，这会导致更高的概括。在本文中，我们采用了另一种观点，并首先在经验上表明，与梯度下降（GD）相比，具有动量（GD+M）的梯度下降在某些深度学习问题中显着改善了概括。从这个观察结果，我们正式研究了动量如何改善概括。我们设计了一个二进制分类设置，在该设置中，当两种算法都类似地初始化时，经过GD+M训练的单个隐藏层（过度参数化）卷积神经网络比使用GD训练的同一网络更好地概括了。我们分析中的关键见解是，动量在示例共享某些功能但边距不同的数据集中是有益的。与记住少量数据数据的GD相反，GD+M仍然通过其历史梯度来了解这些数据中的功能。最后，我们从经验上验证了我们的理论发现。