原名：ImageNet Classification with Deep Convolutional Neural Networks

作者：Alex Krizhevsky，Ilya Sutskever，Geoffrey E. Hinton

摘要

We trained a large, deep convolutional neural network to classify the 1.2 million high-resolution images in the ImageNet ILSVRC-2010 contest into the 1000 different classes.

我们训练了一个大型深度卷积神经网络，将 ImageNet ILSVRC-2010 竞赛中的 120 万张高分辨率图像分为 1000 个不同的类别。

On the test data, we achieved top-1 and top-5 error rates of 37.5% and 17.0% which is considerably better than the previous state-of-the-art. The neural network, which has 60 million parameters and 650,000 neurons, consists of five convolutional layers, some of which are followed by max-pooling layers, and three fully-connected layers with a final 1000-way softmax.

在测试数据上，我们实现了 37.5% 和 17.0% 的 top-1 和 top-5 错误率，这比之前的最新技术要好得多。该神经网络有 6000 万个参数和 650,000 个神经元，由五个卷积层组成，其中一些卷积层后面是最大池化层，以及三个全连接层，最终是 1000 维度的softmax层。

To make training faster, we used non-saturating neurons and a very efficient GPU implementation of the convolution operation

为了使训练更快，我们使用了非饱和神经元和卷积运算的非常有效的 GPU 实现

To reduce overfitting in the fully-connected layers we employed a recently-developed regularization method called “dropout” that proved to be very effective.

为了减少全连接层的过度拟合，我们采用了最近开发的一种称为“dropout”的正则化方法，事实证明这种方法非常有效。

We also entered a variant of this model in the ILSVRC-2012 competition and achieved a winning top-5 test error rate of 15.3%, compared to 26.2% achieved by the second-best entry.

我们还在 ILSVRC-2012 竞赛中输入了该模型的一个变体，并获得了 15.3% 的前 5 名测试错误率，而第二名的测试错误率为 26.2%。

1、背景

Current approaches to object recognition make essential use of machine learning methods. To improve their performance, we can collect larger datasets, learn more powerful models, and use better techniques for preventing overfitting.

当前的对象识别方法必须使用机器学习方法。为了提高它们的性能，我们可以收集更大的数据集，学习更强大的模型，并使用更好的技术来防止过度拟合。

Until recently, datasets of labeled images were relatively small — on the order of tens of thousands of images (e.g., NORB [16], Caltech-101/256, and CIFAR-10/100). Simple recognition tasks can be solved quite well with datasets of this size, especially if they are augmented with label-preserving transformations. For example, the current- best error rate on the MNIST digit recognition task (<0.3%) approaches human performance.

直到最近，标记图像的数据集还相对较小——大约有数万张图像（例如 NORB [16]、Caltech-101/256 和 CIFAR-10/100）。使用这种大小的数据集可以很好地解决简单的识别任务，特别是如果它们通过保留标签的转换进行增强。例如，MNIST 数字识别任务的当前最佳错误率 (<0.3%) 接近人类表现。

But objects in realistic settings exhibit considerable variability, so to learn to recognize them it is necessary to use much larger training sets. And indeed, the shortcomings of small image datasets have been widely recognized (e.g., Pinto et al. ), but it has only recently become possible to collect labeled datasets with millions of images。

但是现实环境中的物体表现出相当大的可变性，因此要学会识别它们，就必须使用更大的训练集。事实上，小图像数据集的缺点已经得到广泛认可（例如 Pinto 等人），但直到最近才可能收集具有数百万张图像的标记数据集。

The new larger datasets include LabelMe , which consists of hundreds of thousands of fully-segmented images, and ImageNet , which consists of over 15 million labeled high-resolution images in over 22,000 categories.

新的更大的数据集包括由数十万张完全分割的图像组成的 LabelMe 和由超过 22,000 个类别的超过 1500 万张标记的高分辨率图像组成的 ImageNet。

To learn about thousands of objects from millions of images, we need a model with a large learning capacity. However, the immense complexity of the object recognition task means that this prob- lem cannot be specified even by a dataset as large as ImageNet, so our model should also have lots of prior knowledge to compensate for all the data we don’t have.

要从数百万张图像中了解数千个对象，我们需要一个具有大学习能力的模型。然而，对象识别任务的巨大复杂性意味着即使像 ImageNet 这样大的数据集也无法指定这个问题，所以我们的模型也应该有很多先验知识来补偿我们没有的所有数据。

Convolutional neural networks (CNNs) constitute one such class of models. Their capacity can be con- trolled by varying their depth and breadth, and they also make strong and mostly correct assumptions about the nature of images (namely, stationarity of statistics and locality of pixel dependencies). Thus, compared to standard feedforward neural networks with similarly-sized layers, CNNs have much fewer connections and parameters and so they are easier to train, while their theoretically-best performance is likely to be only slightly worse.

卷积神经网络 (CNN) 构成了这样一类模型。它们的容量可以通过改变它们的深度和广度来控制，并且它们还对图像的性质（即统计数据的平稳性和像素依赖性的局部性）做出了强有力且基本正确的假设。因此，与具有类似大小层的标准前馈神经网络相比，CNN 具有更少的连接和参数，因此它们更容易训练，而它们理论上的最佳性能可能只是稍微差一点。

Despite the attractive qualities of CNNs, and despite the relative efficiency of their local architecture, they have still been prohibitively expensive to apply in large scale to high-resolution images. Luckily, current GPUs, paired with a highly-optimized implementation of 2D convolution, are powerful enough to facilitate the training of interestingly-large CNNs, and recent datasets such as ImageNet contain enough labeled examples to train such models without severe overfitting.

尽管 CNN 具有吸引人的品质，并且尽管它们的局部架构相对高效，但将它们大规模应用于高分辨率图像的成本仍然高得令人望而却步。幸运的是，当前的 GPU 与高度优化的 2D 卷积实现相结合，足以促进有趣的大型 CNN 的训练，并且最近的数据集（例如 ImageNet）包含足够的标记示例来训练此类模型而不会出现严重的过度拟合。

The specific contributions of this paper are as follows: we trained one of the largest convolutional neural networks to date on the subsets of ImageNet used in the ILSVRC-2010 and ILSVRC-2012 competitions and achieved by far the best results ever reported on these datasets. We wrote a highly-optimized GPU implementation of 2D convolution and all the other operations inherent in training convolutional neural networks, which we make available publicly.

本文的具体贡献如下：我们在 ILSVRC-2010 和 ILSVRC-2012 比赛中使用的 ImageNet 子集上训练了迄今为止最大的卷积神经网络之一，并取得了迄今为止在这些数据集上报告的最佳结果。我们编写了一个高度优化的 GPU 实现，用于 2D 卷积和训练卷积神经网络中固有的所有其他操作，我们将其公开。

Our network contains a number of new and unusual features which improve its performance and reduce its training time, which are detailed in Section 3. The size of our network made overfitting a significant problem, even with 1.2 million labeled training examples, so we used several effective techniques for preventing overfitting, which are described in Section 4. Our final network contains five convolutional and three fully-connected layers, and this depth seems to be important: we found that removing any convolutional layer (each of which contains no more than 1% of the model’s parameters) resulted in inferior performance.

我们的网络包含许多新的和不寻常的特征，可以提高其性能并减少其训练时间，详见第 3 节。我们网络的规模使得过度拟合成为一个严重问题，即使有 120 万个标记的训练示例，因此我们使用了几个防止过拟合的有效技术，在第 4 节中描述。我们的最终网络包含五个卷积层和三个全连接层，这个深度似乎很重要：我们发现去除任何卷积层（每个卷积层包含不超过 1 % 的模型参数）导致性能较差。

In the end, the network’s size is limited mainly by the amount of memory available on current GPUs and by the amount of training time that we are willing to tolerate. Our network takes between five and six days to train on two GTX 580 3GB GPUs. All of our experiments suggest that our results can be improved simply by waiting for faster GPUs and bigger datasets to become available.

最后，网络的大小主要受限于当前 GPU 上可用的内存量以及我们愿意容忍的训练时间量。我们的网络需要五到六天的时间在两个 GTX 580 3GB GPU 上进行训练。我们所有的实验都表明，只需等待更快的 GPU 和更大的数据集可用，就可以改善我们的结果。

2、数据集

ImageNet is a dataset of over 15 million labeled high-resolution images belonging to roughly 22,000 categories. The images were collected from the web and labeled by human labelers using Ama- zon’s Mechanical Turk crowd-sourcing tool. Starting in 2010, as part of the Pascal Visual Object Challenge, an annual competition called the ImageNet Large-Scale Visual Recognition Challenge (ILSVRC) has been held. ILSVRC uses a subset of ImageNet with roughly 1000 images in each of 1000 categories. In all, there are roughly 1.2 million training images, 50,000 validation images, and 150,000 testing images.

ImageNet 是一个包含超过 1500 万张标记的高分辨率图像的数据集，属于大约 22,000 个类别。这些图像是从网络上收集的，并使用亚马逊的 Mechanical Turk 众包工具由人工标记器进行标记。从 2010 年开始，作为 Pascal 视觉对象挑战赛的一部分，一年一度的竞赛称为 ImageNet 大规模视觉识别挑战赛 (ILSVRC)。 ILSVRC 使用 ImageNet 的一个子集，在 1000 个类别中的每个类别中包含大约 1000 个图像。总共有大约 120 万张训练图像、50,000 张验证图像和 150,000 张测试图像。

ILSVRC-2010 is the only version of ILSVRC for which the test set labels are available, so this is the version on which we performed most of our experiments. Since we also entered our model in the ILSVRC-2012 competition, in Section 6 we report our results on this version of the dataset as well, for which test set labels are unavailable. On ImageNet, it is customary to report two error rates: top-1 and top-5, where the top-5 error rate is the fraction of test images for which the correct label is not among the five labels considered most probable by the model.

ILSVRC-2010 是唯一提供测试集标签的 ILSVRC 版本，因此这是我们执行大部分实验的版本。由于我们也在 ILSVRC-2012 竞赛中输入了我们的模型，因此在第 6 节中，我们也报告了此版本数据集的结果，其中测试集标签不可用。在 ImageNet 上，通常会报告两个错误率：top-1 和 top-5，其中 top-5 错误率是测试图像中正确标签不在模型认为最可能的五个标签中的部分 .

ImageNet consists of variable-resolution images, while our system requires a constant input dimen- sionality. Therefore, we down-sampled the images to a fixed resolution of 256 × 256. Given a rectangular image, we first rescaled the image such that the shorter side was of length 256, and then cropped out the central 256×256 patch from the resulting image. We did not pre-process the images in any other way, except for subtracting the mean activity over the training set from each pixel. So we trained our network on the (centered) raw RGB values of the pixels.

ImageNet 由可变分辨率的图像组成，而我们的系统需要恒定的输入维度。因此，我们将图像下采样到 256 × 256 的固定分辨率。给定一个矩形图像，我们首先重新缩放图像，使短边的长度为 256，然后从结果中裁剪出中心的 256 × 256 块图片。除了从每个像素中减去训练集上的平均活动外，我们没有以任何其他方式对图像进行预处理。因此，我们在像素的（居中）原始 RGB 值上训练了我们的网络。

3、网络结构

The architecture of our network is summarized in Figure 2. It contains eight learned layers — five convolutional and three fully-connected. Below, we describe some of the novel or unusual features of our network’s architecture. Sections 3.1-3.4 are sorted according to our estimation of their importance, with the most important first.

图 2 总结了我们网络的架构。它包含八个学习层——五个卷积层和三个全连接层。下面，我们将描述我们网络架构的一些新颖或不寻常的功能。第 3.1-3.4 节根据我们对其重要性的估计进行排序，最重要的在前。

3.1 ReLU 非线性

The standard way to model a neuron’s output f as a function of its input x is with f(x) = tanh(x) or f(x) = (1 + e−x)−1. In terms of training time with gradient descent, these saturating nonlinearities are much slower than the non-saturating nonlinearity f(x) = max(0, x). Following Nair and Hinton [20], we refer to neurons with this nonlinearity as Rectified Linear Units (ReLUs). Deep convolutional neural net- works with ReLUs train several times faster than their equivalents with tanh units. This is demonstrated in Figure 1, which shows the number of iterations re- quired to reach 25% training error on the CIFAR-10 dataset for a particular four-layer convolutional net- work. This plot shows that we would not have been able to experiment with such large neural networks for this work if we had used traditional saturating neuron models.

将神经元的输出 f 建模为其输入 x 的函数的标准方法是 f(x) = tanh(x) 或 f(x) = (1 + e的−x方)的−1次方。就梯度下降的训练时间而言，这些饱和非线性比非饱和非线性 f(x) = max(0, x) 慢得多。继 Nair 和 Hinton [20] 之后，我们将具有这种非线性的神经元称为整流线性单元 (ReLU)。 使用 ReLU 的深度卷积神经网络的训练速度比使用 tanh 单元的等价物快几倍。这在图 1 中得到了证明，它显示了在 CIFAR-10 数据集上针对特定的四层卷积网络达到 25% 训练误差所需的迭代次数。该图表明，如果我们使用传统的饱和神经元模型，我们将无法在这项工作中使用如此大的神经网络进行实验。

We are not the first to consider alternatives to tradi- tional neuron models in CNNs. For example, Jarrett et al. claim that the nonlinearity f(x) = |tanh(x)| works particularly well with their type of contrast nor- malization followed by local average pooling on the Caltech-101 dataset. However, on this dataset the pri- mary concern is preventing overfitting, so the effect they are observing is different from the accelerated ability to fit the training set which we report when us- ing ReLUs. Faster learning has a great influence on the performance of large models trained on large datasets.

我们并不是第一个考虑替代 CNN 中传统神经元模型的人。例如，Jarrett 等人。声称非线性 f(x) = |tanh(x)| 在 Caltech-101 数据集上进行局部平均池化后的对比度归一化类型特别有效。然而，在这个数据集上，主要关注的是防止过度拟合，因此他们观察到的效果不同于我们在使用 ReLU 时报告的加速拟合训练集的能力。更快的学习对在大型数据集上训练的大型模型的性能有很大影响。

Figure 1: A four-layer convolutional neural network with ReLUs (solid line) reaches a 25% training error rate on CIFAR-10 six times faster than an equivalent network with tanh neurons (dashed line). The learning rates for each net- work were chosen independently to make train- ing as fast as possible. No regularization of any kind was employed. The magnitude of the effect demonstrated here varies with network architecture, but networks with ReLUs consis- tently learn several times faster than equivalents with saturating neurons.

图 1：具有 ReLU 的四层卷积神经网络（实线）在 CIFAR-10 上达到 25% 的训练错误率，比具有 tanh 神经元的等效网络（虚线）快六倍。每个网络的学习率都是独立选择的，以使训练尽可能快。没有采用任何形式的正则化。这里展示的效果的大小因网络架构而异，但使用 ReLU 的网络始终比使用饱和神经元的网络学习速度快几倍。

3.2 在多 GPU 上训练

A single GTX 580 GPU has only 3GB of memory, which limits the maximum size of the networks that can be trained on it. It turns out that 1.2 million training examples are enough to train networks which are too big to fit on one GPU. Therefore we spread the net across two GPUs.

单个 GTX 580 GPU 只有 3GB 的内存，这限制了可以在其上训练的网络的最大大小。事实证明，120 万个训练示例足以训练太大而无法在一个 GPU 上安装的网络。因此，我们将网络分布在两个 GPU 上。

Current GPUs are particularly well-suited to cross-GPU parallelization, as they are able to read from and write to one another’s memory directly, without going through host machine memory. The parallelization scheme that we employ essentially puts half of the kernels (or neurons) on each GPU, with one additional trick: the GPUs communicate only in certain layers.

当前的 GPU 特别适合跨 GPU 并行化，因为它们能够直接读取和写入彼此的内存，而无需通过主机内存。我们采用的并行化方案本质上是将一半的内核（或神经元）放在每个 GPU 上，还有一个技巧：GPU 仅在某些层中进行通信。

This means that, for example, the kernels of layer 3 take input from all kernel maps in layer 2. However, kernels in layer 4 take input only from those kernel maps in layer 3 which reside on the same GPU. Choosing the pattern of connectivity is a problem for cross-validation, but this allows us to precisely tune the amount of communication until it is an acceptable fraction of the amount of computation.

这意味着，例如，第 3 层的内核从第 2 层中的所有内核映射中获取输入。然而，第 4 层中的内核仅从位于同一 GPU 上的第 3 层中的那些内核映射中获取输入。选择连接模式是交叉验证的一个问题，但这使我们能够精确地调整通信量，直到它是计算量的一个可接受的部分。

The resultant architecture is somewhat similar to that of the “columnar” CNN employed by Ciresan et al. , except that our columns are not independent (see Figure 2). This scheme reduces our top-1 and top-5 error rates by 1.7% and 1.2%, respectively, as compared with a net with half as many kernels in each convolutional layer trained on one GPU. The two-GPU net takes slightly less time to train than the one-GPU net.

由此产生的架构有点类似于 Ciresan 等人使用的“柱状”CNN 的架构。，除了我们的列不是独立的（见图 2）。与在一个 GPU 上训练的每个卷积层中内核数量减半的网络相比，该方案分别将我们的 top-1 和 top-5 错误率降低了 1.7% 和 1.2%。与单 GPU 网络相比，双 GPU 网络的训练时间略少。

3.3 局部响应归一化

ReLUs have the desirable property that they do not require input normalization to prevent them from saturating. If at least some training examples produce a positive input to a ReLU, learning will happen in that neuron. However, we still find that the following local normalization scheme aids generalization. Denoting by ai x,y the activity of a neuron computed by applying kernel i at position(x, y) and then applying the ReLU nonlinearity, the response-normalized activity bix,y is given by the expression

ReLU 具有理想的特性，即它们不需要输入归一化来防止它们饱和。如果至少有一些训练样例对 ReLU 产生了正输入，那么学习就会发生在那个神经元中。但是，我们仍然发现以下局部归一化方案有助于泛化。用 ai x,y 表示通过在位置 (x, y) 处应用内核 i 然后应用 ReLU 非线性计算出的神经元活动，响应归一化激活表达式 bix,y 由下给出

where the sum runs over n “adjacent” kernel maps at the same spatial position, and N is the total number of kernels in the layer. The ordering of the kernel maps is of course arbitrary and determined before training begins. This sort of response normalization implements a form of lateral inhibition inspired by the type found in real neurons, creating competition for big activities amongst neuron outputs computed using different kernels.

其中总和在相同空间位置的 n 个“相邻”内核映射上运行，N 是层中内核的总数。内核映射的排序当然是任意的，并且在训练开始之前就已确定。这种响应归一化实现了一种受真实神经元类型启发的侧向抑制形式，在使用不同内核计算的神经元输出之间创建大型活动的竞争。

The constants k, n, α, and β are hyper-parameters whose values are determined using a validation set; we used k = 2, n = 5, α = 10−4, and β = 0.75. We applied this normalization after applying the ReLU nonlinearity in certain layers (see Section 3.5).

常数 k、n、α 和 β 是超参数，其值由验证集确定；我们使用 k = 2、n = 5、α = 10−4 和 β = 0.75。我们在某些层中应用了 ReLU 非线性之后应用了这种归一化（参见第 3.5 节）。

This scheme bears some resemblance to the local contrast normalization scheme of Jarrett et al. but ours would be more correctly termed “brightness normalization”, since we do not subtract the mean activity. Response normalization reduces our top-1 and top-5 error rates by 1.4% and 1.2%, respectively. We also verified the effectiveness of this scheme on the CIFAR-10 dataset: a four-layer CNN achieved a 13% test error rate without normalization and 11% with normalization.

该方案与 Jarrett 等人的局部对比度归一化方案有些相似。但我们将其称为“亮度归一化”更正确，因为我们没有减去平均活动。响应归一化将我们的 top-1 和 top-5 错误率分别降低了 1.4% 和 1.2%。我们还在 CIFAR-10 数据集上验证了该方案的有效性：一个四层 CNN 在没有归一化的情况下实现了 13% 的测试错误率，在归一化的情况下达到了 11%。

3.4 重叠池

Pooling layers in CNNs summarize the outputs of neighboring groups of neurons in the same kernel map. Traditionally, the neighborhoods summarized by adjacent pooling units do not overlap (e.g., [17, 11, 4]). To be more precise, a pooling layer can be thought of as consisting of a grid of pooling units spaced s pixels apart, each summarizing a neighborhood of size z × z centered at the location of the pooling unit. If we set s = z, we obtain traditional local pooling as commonly employed in CNNs.

CNN 中的池化层汇总了同一内核映射中相邻神经元组的输出。传统上，相邻池化单元汇总的邻域不重叠（例如，[17, 11, 4]）。更准确地说，一个池化层可以被认为是由一个间隔 s 个像素的池化单元网格组成，每个网格总结了一个大小为 z × z 的邻域，以池化单元的位置为中心。如果我们设置 s = z，我们将获得 CNN 中常用的传统局部池化。

If we set s < z, we obtain overlapping pooling. This is what we use throughout our network, with s = 2 and z = 3. This scheme reduces the top-1 and top-5 error rates by 0.4% and 0.3%, respectively, as compared with the non-overlapping scheme s = 2, z = 2, which produces output of equivalent dimensions. We generally observe during training that models with overlapping pooling find it slightly more difficult to overfit.

如果我们设置 s < z，我们将获得重叠池化。这是我们在整个网络中使用的，s = 2 和 z = 3。与产生等效维度输出的非重叠方案 s = 2, z = 2 相比，该方案分别将 top-1 和 top-5 错误率降低了 0.4% 和 0.3%。我们通常在训练期间观察到具有重叠池化的模型发现过拟合稍微困难一些。

3.5 总体架构

Now we are ready to describe the overall architecture of our CNN. As depicted in Figure 2, the net contains eight layers with weights; the first five are convolutional and the remaining three are fully- connected. The output of the last fully-connected layer is fed to a 1000-way softmax which produces a distribution over the 1000 class labels. Our network maximizes the multinomial logistic regression objective, which is equivalent to maximizing the average across training cases of the log-probability of the correct label under the prediction distribution.

现在我们准备描述我们的 CNN 的整体架构。如图 2 所示，网络包含八层权重；前五个是卷积的，其余三个是全连接的。最后一个全连接层的输出被馈送到 1000 路 softmax，它产生 1000 个类别标签的分布。我们的网络最大化多项逻辑回归目标，这相当于最大化预测分布下正确标签的对数概率的训练案例的平均值。

The kernels of the second, fourth, and fifth convolutional layers are connected only to those kernel maps in the previous layer which reside on the same GPU (see Figure 2). The kernels of the third convolutional layer are connected to all kernel maps in the second layer. The neurons in the fully- connected layers are connected to all neurons in the previous layer. Response-normalization layers follow the first and second convolutional layers. Max-pooling layers, of the kind described in Section 3.4, follow both response-normalization layers as well as the fifth convolutional layer. The ReLU non-linearity is applied to the output of every convolutional and fully-connected layer.

第二、第四和第五卷积层的内核仅连接到前一层中位于同一 GPU 上的内核映射（见图 2）。第三个卷积层的内核连接到第二层的所有内核映射。全连接层中的神经元连接到前一层中的所有神经元。响应归一化层跟在第一和第二个卷积层之后。 3.4 节中描述的那种最大池化层，跟在响应归一化层和第五个卷积层之后。 ReLU 非线性应用于每个卷积层和全连接层的输出。

The first convolutional layer filters the 224×224×3 input image with 96 kernels of size 11×11×3 with a stride of 4 pixels (this is the distance between the receptive field centers of neighboring neurons in a kernel map). The second convolutional layer takes as input the (response-normalized and pooled) output of the first convolutional layer and filters it with 256 kernels of size 5 × 5 × 48. The third, fourth, and fifth convolutional layers are connected to one another without any intervening pooling or normalization layers. The third convolutional layer has 384 kernels of size 3 × 3 × 256 connected to the (normalized, pooled) outputs of the second convolutional layer. The fourth convolutional layer has 384 kernels of size 3 × 3 × 192 , and the fifth convolutional layer has 256 kernels of size 3 × 3 × 192. The fully-connected layers have 4096 neurons each.

第一个卷积层用 96 个大小为 11×11×3 的内核以 4 个像素（这是内核图中相邻神经元的感受野中心之间的距离）过滤 224×224×3 输入图像。第二个卷积层将第一个卷积层的（响应归一化和池化）输出作为输入，并用 256 个大小为 5 × 5 × 48 的内核对其进行过滤。第三、第四和第五卷积层相互连接，中间没有任何池化或归一化层。第三个卷积层有 384 个大小为 3 × 3 × 256 的内核，连接到第二个卷积层的（归一化、池化）输出。第四个卷积层有 384 个大小为 3 × 3 × 192 的内核，第五个卷积层有 256 个大小为 3 × 3 × 192 的内核。每个全连接层有 4096 个神经元。

Figure 2: An illustration of the architecture of our CNN, explicitly showing the delineation of responsibilities between the two GPUs. One GPU runs the layer-parts at the top of the figure while the other runs the layer-parts at the bottom. The GPUs communicate only at certain layers. The network’s input is 150,528-dimensional, and the number of neurons in the network’s remaining layers is given by 253,440–186,624–64,896–64,896–43,264– 4096–4096–1000

图 2：我们的 CNN 架构图，明确显示了两个 GPU 之间的职责划分。一个 GPU 运行图形顶部的层部分，而另一个 GPU 运行底部的层部分。 GPU 仅在某些层进行通信。网络的输入为 150,528 维，网络其余层的神经元数量为 253,440-186,624-64,896-64,896-43,264-4096-4096-1000

4 、减少过拟合

Our neural network architecture has 60 million parameters. Although the 1000 classes of ILSVRC make each training example impose 10 bits of constraint on the mapping from image to label, this turns out to be insufficient to learn so many parameters without considerable overfitting. Below, we describe the two primary ways in which we combat overfitting.

我们的神经网络架构有 6000 万个参数。尽管 ILSVRC 的 1000 个类别使每个训练示例对从图像到标签的映射施加了 10 位的约束，但事实证明，这不足以学习如此多的参数而不会出现严重的过拟合。下面，我们将描述我们对抗过度拟合的两种主要方式。

4.1 数据增强

The easiest and most common method to reduce overfitting on image data is to artificially enlarge the dataset using label-preserving transformations (e.g., [25, 4, 5]). We employ two distinct forms of data augmentation, both of which allow transformed images to be produced from the original images with very little computation, so the transformed images do not need to be stored on disk. In our implementation, the transformed images are generated in Python code on the CPU while the GPU is training on the previous batch of images. So these data augmentation schemes are, in effect, computationally free.

减少图像数据过度拟合的最简单和最常见的方法是使用标签保留转换（例如 [25, 4, 5]）人为地扩大数据集。我们采用了两种不同形式的数据增强，这两种形式都允许以很少的计算从原始图像生成转换后的图像，因此转换后的图像不需要存储在磁盘上。在我们的实现中，转换后的图像是在 CPU 上用 Python 代码生成的，而 GPU 正在对上一批图像进行训练。因此，这些数据增强方案实际上在计算上是免费的。

The first form of data augmentation consists of generating image translations and horizontal reflec- tions. We do this by extracting random 224×224 patches (and their horizontal reflections) from the 256×256 images and training our network on these extracted patches4. This increases the size of our training set by a factor of 2048, though the resulting training examples are, of course, highly inter- dependent. Without this scheme, our network suffers from substantial overfitting, which would have forced us to use much smaller networks. At test time, the network makes a prediction by extracting five 224 × 224 patches (the four corner patches and the center patch) as well as their horizontal reflections (hence ten patches in all), and averaging the predictions made by the network’s softmax layer on the ten patches.

数据增强的第一种形式包括生成图像平移和水平反射。我们通过从 256×256 图像中提取随机 224×224 块（及其水平反射）并在这些提取的块上训练我们的网络来做到这一点。这将我们的训练集的大小增加了 2048 倍，尽管由此产生的训练示例当然是高度相互依赖的。如果没有这个方案，我们的网络就会遭受严重的过度拟合，这将迫使我们使用更小的网络。在测试时，网络通过提取 5 个 224 × 224 块（四个角块和中心块）及其水平反射（因此总共有 10 个块）并在十个补丁上取网络的 softmax 层所做的预测的平均来进行预测。

The second form of data augmentation consists of altering the intensities of the RGB channels in training images. Specifically, we perform PCA on the set of RGB pixel values throughout the ImageNet training set. To each training image, we add multiples of the found principal components.

第二种形式的数据增强包括改变训练图像中 RGB 通道的强度。具体来说，我们对整个 ImageNet 训练集的 RGB 像素值集执行 PCA。对于每个训练图像，我们添加找到的主成分的倍数。

with magnitudes proportional to the corresponding eigenvalues times a random variable drawn from a Gaussian with mean zero and standard deviation 0.1. Therefore to each RGB image pixel Ixy = [IRxy, IG xy, IBxy]T, we add the following quantity:[p1, p2, p3][α1λ1, α2λ2, α3λ3]T

幅度与相应的特征值乘以从均值为零和标准偏差为 0.1 的高斯分布的随机变量成正比。因此，对于每个 RGB 图像像素 Ixy = [IRxy, IG xy, IBxy]T，我们添加以下数量：[p1, p2, p3][α1λ1, α2λ2, α3λ3]T

where pi and λi are ith eigenvector and eigenvalue of the 3 × 3 covariance matrix of RGB pixel values, respectively, and αi is the aforementioned random variable. Each αi is drawn only once for all the pixels of a particular training image until that image is used for training again, at which point it is re-drawn. This scheme approximately captures an important property of natural images, namely, that object identity is invariant to changes in the intensity and color of the illumination. This scheme reduces the top-1 error rate by over 1%.

其中pi和λi分别是RGB像素值的3×3协方差矩阵的第i个特征向量和特征值，αi是上述随机变量。对于特定训练图像的所有像素，每个 αi 仅绘制一次，直到再次使用该图像进行训练，然后重新绘制。该方案近似地捕捉了自然图像的一个重要特性，即对象身份对于照明强度和颜色的变化是不变的。该方案将top-1错误率降低了1%以上。

4.2 Dropout

Combining the predictions of many different models is a very successful way to reduce test errors [1, 3], but it appears to be too expensive for big neural networks that already take several days to train. There is, however, a very efficient version of model combination that only costs about a factor of two during training. The recently-introduced technique, called “dropout” [10], consists of setting to zero the output of each hidden neuron with probability 0.5.

结合许多不同模型的预测是减少测试错误的一种非常成功的方法 [1, 3]，但对于已经需要几天时间训练的大型神经网络来说，这似乎太昂贵了。然而，有一个非常有效的模型组合版本，在训练期间只需花费大约两倍。最近引入的技术称为“dropout”[10]，包括将每个隐藏神经元的输出设置为零，概率为 0.5。

The neurons which are “dropped out” in this way do not contribute to the forward pass and do not participate in back- propagation. So every time an input is presented, the neural network samples a different architecture, but all these architectures share weights. This technique reduces complex co-adaptations of neurons, since a neuron cannot rely on the presence of particular other neurons.

以这种方式“失活”的神经元不参与前向传播，也不参与反向传播。因此，每次呈现输入时，神经网络都会对不同的架构进行采样，但所有这些架构共享权重。这种技术减少了神经元复杂的协同适应，因为神经元不能依赖于特定其他神经元的存在。

It is, therefore, forced to learn more robust features that are useful in conjunction with many different random subsets of the other neurons. At test time, we use all the neurons but multiply their outputs by 0.5, which is a reasonable approximation to taking the geometric mean of the predictive distributions produced by the exponentially-many dropout networks.

因此，它被迫学习与其他神经元的许多不同随机子集结合使用的更强大的特征。在测试时，我们使用所有的神经元，但将它们的输出乘以 0.5，这是对指数多 dropout 网络产生的预测分布的几何平均值的合理近似。

We use dropout in the first two fully-connected layers of Figure 2. Without dropout, our network exhibits substantial overfitting. Dropout roughly doubles the number of iterations required to converge.

我们在图 2 的前两个全连接层中使用了 dropout。当没有 dropout，我们的网络表现出严重的过度拟合。 Dropout 大约使收敛所需的迭代次数加倍。

5 、学习细节

We trained our models using stochastic gradient descent with a batch size of 128 examples, momentum of 0.9, and weight decay of 0.0005.

我们使用随机梯度下降训练我们的模型，批量大小为 128 个示例，动量为 0.9，权重衰减为 0.0005。

We found that this small amount of weight decay was important for the model to learn. In other words, weight decay here is not merely a regularizer : it reduces the model’s training error. The update rule for weight w was:

我们发现这种少量的权重衰减对于模型学习很重要。换句话说，这里的权重衰减不仅仅是一个正则化器：它减少了模型的训练误差。权重 w 的更新规则是：

where i is the iteration index, v is the momentum variable, ε is the learning rate, and is the average over the ith batch Di of the derivative of the objective with respect to w, evaluated at wi.

其中 i 是迭代索引，v 是动量变量，ε 是学习率，而是目标关于 w 的导数的第 i 批 Di 的平均值，在 wi 处评估。

Figure 3: 96 convolutional kernels of size 11×11×3 learned by the first convolutional layer on the 224×224×3 input images. The top 48 kernels were learned on GPU 1 while the bottom 48 kernels were learned on GPU 2. See Section 6.1 for details.

图 3：第一个卷积层在 224×224×3 输入图像上学习到的 96 个大小为 11×11×3 的卷积核。前 48 个内核是在 GPU 1 上学习的，而后 48 个内核是在 GPU 2 上学习的。有关详细信息，请参阅第 6.1 节。

We initialized the weights in each layer from a zero-mean Gaussian distribution with standard deviation 0.01. We initialized the neuron biases in the second, fourth, and fifth convolutional layers, as well as in the fully-connected hidden layers, with the constant 1. This initialization accelerates the early stages of learning by providing the ReLUs with positive inputs. We initialized the neuron biases in the remaining layers with the constant 0.

我们从标准差为 0.01 的零均值高斯分布初始化每一层的权重。我们将第二、第四和第五卷积层以及全连接隐藏层中的神经元偏置初始化为常数 1。此初始化通过为 ReLU 提供正输入来加速学习的早期阶段。我们将其余层中的神经元偏置初始化为常数 0。

We used an equal learning rate for all layers, which we adjusted manually throughout training. The heuristic which we followed was to divide the learning rate by 10 when the validation error rate stopped improving with the current learning rate. The learning rate was initialized at 0.01 and reduced three times prior to termination. We trained the network for roughly 90 cycles through the training set of 1.2 million images, which took five to six days on two NVIDIA GTX 580 3GB GPUs.

我们对所有层使用相同的学习率，并在整个训练过程中手动调整。我们遵循的启发式方法是，当验证错误率随着当前学习率停止提高时，将学习率除以 10。学习率初始化为 0.01，并在终止前减少了 3 倍。我们通过 120 万张图像的训练集对网络进行了大约 90 个周期的训练，这在两个 NVIDIA GTX 580 3GB GPU 上耗时 5 到 6 天。

6 、结果

Our results on ILSVRC-2010 are summarized in Table 1. Our network achieves top-1 and top-5 test set error rates of 37.5% and 17.0%5. The best performance achieved during the ILSVRC- 2010 competition was 47.1% and 28.2% with an approach that averages the predictions produced from six sparse-coding models trained on different features [2], and since then the best pub- lished results are 45.7% and 25.7% with an approach that averages the predictions of two classi- fiers trained on Fisher Vectors (FVs) computed from two types of densely-sampled features [24].

我们在 ILSVRC-2010 上的结果总结在表 1 中。我们的网络实现了 37.5% 和 17.0%5 的 top-1 和 top-5 测试集错误率。在 ILSVRC-2010 竞赛期间获得的最佳性能分别为 47.1% 和 28.2%，其方法是对在不同特征上训练的六个稀疏编码模型产生的预测进行平均 [2]，此后发布的最佳结果为 45.7% 25.7% 的方法是平均两个分类器的预测，这些分类器在从两种密集采样特征[24] 计算的 Fisher 向量 (FV) 上训练。

We also entered our model in the ILSVRC-2012 competition and report our results in Table 2. Since the ILSVRC-2012 test set labels are not publicly available, we cannot report test error rates for all the models that we tried. In the remainder of this paragraph, we use validation and test error rates interchangeably because in our experience they do not differ by more than 0.1% (see Table 2). The CNN described in this paper achieves a top-5 error rate of 18.2%. Averaging the predictions of five similar CNNs gives an error rate of 16.4%.

我们还在 ILSVRC-2012 竞赛中输入了我们的模型，并在表 2 中报告了我们的结果。由于 ILSVRC-2012 测试集标签不是公开可用的，我们无法报告我们尝试过的所有模型的测试错误率。在本段的其余部分，我们交替使用验证和测试错误率，因为根据我们的经验，它们的差异不超过 0.1%（见表 2）。本文描述的 CNN 实现了 18.2% 的 top-5 错误率。对五个类似 CNN 的预测进行平均得出 16.4% 的错误率。

Table 1: Comparison of results on ILSVRC-2010 test set. In italics are best results achieved by others.

表 1：ILSVRC-2010 测试集上的结果比较。斜体是其他人取得的最好结果。

Training one CNN, with an extra sixth con- volutional layer over the last pooling layer, to classify the entire ImageNet Fall 2011 release (15M images, 22K categories), and then “fine-tuning” it on ILSVRC-2012 gives an error rate of 16.6%. Averaging the predictions of two CNNs that were pre-trained on the entire Fall 2011 re- lease with the aforementioned five CNNs gives an error rate of 15.3%. The second-best con- test entry achieved an error rate of 26.2% with an approach that averages the predictions of sev- eral classifiers trained on FVs computed from different types of densely-sampled features [7].

训练一个 CNN，在最后一个池化层上有一个额外的第六个卷积层，对整个 ImageNet 2011 年秋季发布（15M 图像，22K 类别）进行分类，然后在 ILSVRC-2012 上对其进行“微调”，给出错误率 16.6%。将在整个 2011 年秋季版本上预训练的两个 CNN 的预测与上述五个 CNN 的预测平均得出 15.3% 的错误率。第二好的竞赛条目的错误率达到了 26.2%，其方法是对在从不同类型的密集采样特征[7] 计算的 FV 上训练的几个分类器的预测进行平均。

Finally, we also report our error rates on the Fall 2009 version of ImageNet with 10,184 categories and 8.9 million images. On this dataset we follow the convention in the literature of using half of the images for training and half for testing. Since there is no established test set, our split necessarily differs from the splits used by previous authors, but this does not affect the results appreciably. Our top-1 and top-5 error rates on this dataset are 67.4% and 40.9%, attained by the net described above but with an additional, sixth convolutional layer over the last pooling layer. The best published results on this dataset are 78.1% and 60.9% [19].

最后，我们还报告了 2009 年秋季版 ImageNet 的错误率，该版本包含 10,184 个类别和 890 万张图像。在这个数据集上，我们遵循文献中的惯例，使用一半的图像进行训练，一半的图像进行测试。由于没有既定的测试集，我们的分割必然与之前作者使用的分割不同，但这不会明显影响结果。我们在这个数据集上的 top-1 和 top-5 错误率分别为 67.4% 和 40.9%，通过上述网络获得，但在最后一个池化层上增加了第六个卷积层。在该数据集上发布的最佳结果是 78.1% 和 60.9% [19]。

Table 2: Comparison of error rates on ILSVRC-2012 validation and test sets. In italics are best results achieved by others. Models with an asterisk* were “pre-trained” to classify the entire ImageNet 2011 Fall release. See Section 6 for details.

表 2：ILSVRC-2012 验证和测试集上的错误率比较。斜体是其他人取得的最好结果。带有星号 * 的模型经过“预训练”以对整个 ImageNet 2011 秋季版本进行分类。有关详细信息，请参阅第 6 节。

6.1 、定性评估

Figure 3 shows the convolutional kernels learned by the network’s two data-connected layers. The network has learned a variety of frequency- and orientation-selective kernels, as well as various col- ored blobs. Notice the specialization exhibited by the two GPUs, a result of the restricted connec- tivity described in Section 3.5. The kernels on GPU 1 are largely color-agnostic, while the kernels on on GPU 2 are largely color-specific. This kind of specialization occurs during every run and is independent of any particular random weight initialization (modulo a renumbering of the GPUs).

图 3 显示了网络的两个数据连接层学习的卷积核。该网络已经学习了各种频率和方向选择性内核，以及各种颜色的斑点。请注意两个 GPU 表现出的专业化，这是第 3.5 节中描述的受限连接的结果。 GPU 1 上的内核在很大程度上与颜色无关，而 GPU 2 上的内核在很大程度上与颜色相关。这种特殊化发生在每次运行期间，并且独立于任何特定的随机权重初始化（以 GPU 的重新编号为模）。

Figure 4: (Left) Eight ILSVRC-2010 test images and the five labels considered most probable by our model. The correct label is written under each image, and the probability assigned to the correct label is also shown with a red bar (if it happens to be in the top 5). (Right) Five ILSVRC-2010 test images in the first column. The remaining columns show the six training images that produce feature vectors in the last hidden layer with the smallest Euclidean distance from the feature vector for the test image.

图 4：（左）八个 ILSVRC-2010 测试图像和我们的模型认为最有可能的五个标签。正确的标签写在每张图像下，分配给正确标签的概率也用红色条显示（如果它恰好在前 5 个）。（右）第一列中的五张 ILSVRC-2010 测试图像。剩余的列显示了六个训练图像，它们在最后一个隐藏层中生成特征向量，与测试图像的特征向量的欧几里得距离最小。

In the left panel of Figure 4 we qualitatively assess what the network has learned by computing its top-5 predictions on eight test images. Notice that even off-center objects, such as the mite in the top-left, can be recognized by the net. Most of the top-5 labels appear reasonable. For example, only other types of cat are considered plausible labels for the leopard. In some cases (grille, cherry) there is genuine ambiguity about the intended focus of the photograph.

在图 4 的左侧面板中，我们通过计算对八张测试图像的前 5 名预测来定性评估网络所学到的东西。请注意，即使是偏离中心的物体，例如左上角的螨虫，也可以被网络识别。大多数前 5 个标签看起来都合理。例如，只有其他类型的猫才被认为是豹子的合理标签。在某些情况下（格栅、樱桃色），照片的预期焦点确实存在歧义。

Another way to probe the network’s visual knowledge is to consider the feature activations induced by an image at the last, 4096-dimensional hidden layer. If two images produce feature activation vectors with a small Euclidean separation, we can say that the higher levels of the neural network consider them to be similar. Figure 4 shows five images from the test set and the six images from the training set that are most similar to each of them according to this measure. Notice that at the pixel level, the retrieved training images are generally not close in L2 to the query images in the first column. For example, the retrieved dogs and elephants appear in a variety of poses. We present the results for many more test images in the supplementary material.

另一种探测网络视觉知识的方法是考虑图像在最后一个 4096 维隐藏层引起的特征激活。如果两幅图像产生具有小欧几里德分离的特征激活向量，我们可以说神经网络的更高级别认为它们是相似的。图 4 显示了来自测试集的五幅图像和来自训练集的六幅图像，根据该度量，它们与每幅图像最相似。请注意，在像素级别，检索到的训练图像在 L2 中通常与第一列中的查询图像不接近。例如，检索到的狗和大象以各种姿势出现。我们在补充材料中展示了更多测试图像的结果。

Computing similarity by using Euclidean distance between two 4096-dimensional, real-valued vec- tors is inefficient, but it could be made efficient by training an auto-encoder to compress these vectors to short binary codes. This should produce a much better image retrieval method than applying auto- encoders to the raw pixels [14], which does not make use of image labels and hence has a tendency to retrieve images with similar patterns of edges, whether or not they are semantically similar.

通过使用两个 4096 维实值向量之间的欧几里德距离计算相似性是低效的，但可以通过训练自动编码器将这些向量压缩为短二进制代码来提高效率。这应该产生比将自动编码器应用于原始像素 [14] 更好的图像检索方法，它不使用图像标签，因此倾向于检索具有相似边缘模式的图像，无论它们是否在语义上是相似的。

7、讨论

Our results show that a large, deep convolutional neural network is capable of achieving record- breaking results on a highly challenging dataset using purely supervised learning. It is notable that our network’s performance degrades if a single convolutional layer is removed. For example, removing any of the middle layers results in a loss of about 2% for the top-1 performance of the network. So the depth really is important for achieving our results.

我们的结果表明，使用纯监督学习，大型深度卷积神经网络能够在极具挑战性的数据集上取得破纪录的结果。值得注意的是，如果删除单个卷积层，我们的网络性能会下降。例如，移除任何中间层都会导致网络的 top-1 性能损失约 2%。所以深度对于实现我们的结果真的很重要。

To simplify our experiments, we did not use any unsupervised pre-training even though we expect that it will help, especially if we obtain enough computational power to significantly increase the size of the network without obtaining a corresponding increase in the amount of labeled data. Thus far, our results have improved as we have made our network larger and trained it longer but we still have many orders of magnitude to go in order to match the infero-temporal pathway of the human visual system. Ultimately we would like to use very large and deep convolutional nets on video sequences where the temporal structure provides very helpful information that is missing or far less obvious in static images.

为了简化我们的实验，我们没有使用任何无监督的预训练，尽管我们预计它会有所帮助，特别是如果我们获得足够的计算能力来显着增加网络的规模，而没有相应地增加标记数据量。到目前为止，我们的结果已经有所改善，因为我们已经扩大了我们的网络并对其进行了更长时间的训练，但为了匹配人类视觉系统的下时间路径，我们还有许多数量级的工作要做。最终，我们希望在视频序列上使用非常大且深的卷积网络，其中时间结构提供了非常有用的信息，而这些信息在静态图像中缺失或不那么明显。

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论文（二）：AlexNet

原名：ImageNet Classification with Deep Convolutional Neural Networks

摘要

1、背景

2、数据集