Very Deep Convolutional Networks
for Large-Scale Image Recognition
用于大规模图像识别的非常深的卷积网络

During training, the input to our ConvNets is a fixed-size $224\times 224$ RGB image. The only pre-processing we do is subtracting the mean RGB value, computed on the training set, from each pixel. The image is passed through a stack of convolutional (conv.) layers, where we use filters with a very small receptive field: $3\times 3$ (which is the smallest size to capture the notion of left/right, up/down, center). In one of the configurations we also utilise $1\times 1$ convolution filters, which can be seen as a linear transformation of the input channels (followed by non-linearity). The convolution stride is fixed to $1$ pixel; the spatial padding of conv. layer input is such that the spatial resolution is preserved after convolution, i.e. the padding is $1$ pixel for $3\times 3$ conv. layers. Spatial pooling is carried out by five max-pooling layers, which follow some of the conv. layers (not all the conv. layers are followed by max-pooling). Max-pooling is performed over a $2\times 2$ pixel window, with stride $2$.
在训练过程中，我们卷积神经网络的输入是一张固定大小的 $224\times 224$ RGB 图像。我们进行的唯一预处理是从每个像素中减去在训练集上计算得到的 RGB 均值。图像通过一个卷积层堆栈，我们使用具有非常小感受野的滤波器： $3\times 3$ （这是捕获左右、上下、中心概念的最小尺寸）。在一种配置中，我们还利用了 $1\times 1$ 卷积滤波器，这可以看作是对输入通道的线性变换（随后接非线性）。卷积步长固定为 $1$ 像素；卷积层输入的空间填充方式是保持卷积后的空间分辨率，即对于 $3\times 3$ 卷积层，填充为 $1$ 像素。空间池化由五个最大池化层执行，这些池化层跟在一些卷积层后面（并非所有卷积层都跟有最大池化）。最大池化是在一个 $2\times 2$ 像素窗口上执行的，步长为 $2$ 。

A stack of convolutional layers (which has a different depth in different architectures) is followed by three Fully-Connected (FC) layers: the first two have 4096 channels each, the third performs 1000-way ILSVRC classification and thus contains 1000 channels (one for each class). The final layer is the soft-max layer. The configuration of the fully connected layers is the same in all networks.
一个卷积层堆栈（在不同架构中深度不同）后面跟着三个全连接（FC）层：前两层每个有 4096 个通道，第三层执行 1000 种 ILSVRC 分类，因此包含 1000 个通道（每个类别一个）。最后一层是 softmax 层。所有网络的全连接层配置相同。

All hidden layers are equipped with the rectification (ReLU (Krizhevsky et al., 2012)) non-linearity. We note that none of our networks (except for one) contain Local Response Normalisation (LRN) normalisation (Krizhevsky et al., 2012): as will be shown in Sect. 4, such normalisation does not improve the performance on the ILSVRC dataset, but leads to increased memory consumption and computation time. Where applicable, the parameters for the LRN layer are those of (Krizhevsky et al., 2012).
所有隐藏层都配备了整流（ReLU (Krizhevsky 等人，2012)）非线性。我们注意到我们的网络（除一个外）都不包含局部响应归一化（LRN）归一化（Krizhevsky 等人，2012）：正如第 4 节所示，这种归一化不会提高 ILSVRC 数据集的性能，但会导致内存消耗和计算时间增加。在适用的情况下，LRN 层的参数是(Krizhevsky 等人，2012)的参数。

The ConvNet configurations, evaluated in this paper, are outlined in Table 1, one per column. In the following we will refer to the nets by their names (A–E). All configurations follow the generic design presented in Sect. 2.1, and differ only in the depth: from 11 weight layers in the network A (8 conv. and 3 FC layers) to 19 weight layers in the network E (16 conv. and 3 FC layers). The width of conv. layers (the number of channels) is rather small, starting from $64$ in the first layer and then increasing by a factor of $2$ after each max-pooling layer, until it reaches $512$.
本文评估的卷积神经网络配置在表 1 中列出，每列一个。在以下内容中，我们将用它们的名称（A-E）来指代这些网络。所有配置都遵循第 2.1 节中介绍的一般设计，它们之间的区别仅在于深度：从网络 A 中的 11 个权重层（8 个卷积层和 3 个全连接层）到网络 E 中的 19 个权重层（16 个卷积层和 3 个全连接层）。卷积层的宽度（通道数）相对较小，从第一层的 $64$ 开始，然后在每次最大池化层后以 $2$ 的倍数增加，直到达到 $512$ 。

In Table 2 we report the number of parameters for each configuration. In spite of a large depth, the number of weights in our nets is not greater than the number of weights in a more shallow net with larger conv. layer widths and receptive fields (144M weights in (Sermanet et al., 2014)).
表 2 报告了每种配置的参数数量。尽管深度很大，但我们网络中的权重数量并不大于一个更浅的网络中的权重数量，该浅网络的卷积层宽度和感受野更大（(Sermanet et al., 2014)中的 1.44 亿权重）。

Our ConvNet configurations are quite different from the ones used in the top-performing entries of the ILSVRC-2012 (Krizhevsky et al., 2012) and ILSVRC-2013 competitions (Zeiler & Fergus, 2013; Sermanet et al., 2014). Rather than using relatively large receptive fields in the first conv. layers (e.g. $11\times 11$ with stride $4$ in (Krizhevsky et al., 2012), or $7\times 7$ with stride $2$ in (Zeiler & Fergus, 2013; Sermanet et al., 2014)), we use very small $3\times 3$ receptive fields throughout the whole net, which are convolved with the input at every pixel (with stride $1$). It is easy to see that a stack of two $3\times 3$ conv. layers (without spatial pooling in between) has an effective receptive field of $5\times 5$; three such layers have a $7\times 7$ effective receptive field. So what have we gained by using, for instance, a stack of three $3\times 3$ conv. layers instead of a single $7\times 7$ layer? First, we incorporate three non-linear rectification layers instead of a single one, which makes the decision function more discriminative. Second, we decrease the number of parameters: assuming that both the input and the output of a three-layer $3\times 3$ convolution stack has $C$ channels, the stack is parametrised by $3\left(3^{2}C^{2}\right)=27C^{2}$ weights; at the same time, a single $7\times 7$ conv. layer would require $7^{2}C^{2}=49C^{2}$ parameters, i.e. $81\%$ more. This can be seen as imposing a regularisation on the $7\times 7$ conv. filters, forcing them to have a decomposition through the $3\times 3$ filters (with non-linearity injected in between).
我们的卷积神经网络配置与 ILSVRC-2012（Krizhevsky 等人，2012 年）和 ILSVRC-2013 竞赛（Zeiler & Fergus，2013 年；Sermanet 等人，2014 年）中表现最佳的配置差异很大。我们没有在第一层卷积层中使用相对较大的感受野（例如在 Krizhevsky 等人，2012 年中使用步长为 1 的 $11\times 11$ ，或在 Zeiler & Fergus，2013 年；Sermanet 等人，2014 年中使用步长为 3 的 $7\times 7$ ），而是在整个网络中使用非常小的 $3\times 3$ 感受野，这些感受野在每一个像素点都与输入进行卷积（步长为 $1$ ）。很容易看出，两个 $3\times 3$ 卷积层堆叠（中间没有空间池化）的有效感受野为 $5\times 5$ ；三个这样的层具有 $7\times 7$ 的有效感受野。那么，我们通过使用三个 $3\times 3$ 卷积层堆叠而不是单个 $7\times 7$ 层获得了什么？首先，我们使用了三个非线性整流层而不是单个层，这使得决策函数更具判别性。 其次，我们减少参数数量：假设三层 $3\times 3$ 卷积堆的输入和输出都有 $C$ 个通道，该堆由 $3\left(3^{2}C^{2}\right)=27C^{2}$ 个权重参数化；同时，一个单独的 $7\times 7$ 卷积层需要 $7^{2}C^{2}=49C^{2}$ 个参数，即 $81\%$ 个更多。这可以看作是对 $7\times 7$ 卷积滤波器施加正则化，迫使它们通过 $3\times 3$ 滤波器进行分解（在中间注入非线性）。

The incorporation of $1\times 1$ conv. layers (configuration C, Table 1) is a way to increase the non-linearity of the decision function without affecting the receptive fields of the conv. layers. Even though in our case the $1\times 1$ convolution is essentially a linear projection onto the space of the same dimensionality (the number of input and output channels is the same), an additional non-linearity is introduced by the rectification function. It should be noted that $1\times 1$ conv. layers have recently been utilised in the “Network in Network” architecture of Lin et al. (2014).
将 $1\times 1$ 卷积层（配置 C，表 1）的加入是一种在不影响卷积层感受野的情况下增加决策函数非线性的方法。尽管在我们的情况下 $1\times 1$ 卷积本质上是对同一维度的线性投影（输入和输出通道数相同），但通过整流函数引入了额外的非线性。需要注意的是 $1\times 1$ 卷积层最近已被用于 Lin 等人（2014）提出的“网络中的网络”架构。

Small-size convolution filters have been previously used by Ciresan et al. (2011), but their nets are significantly less deep than ours, and they did not evaluate on the large-scale ILSVRC dataset. Goodfellow et al. (2014) applied deep ConvNets ($11$ weight layers) to the task of street number recognition, and showed that the increased depth led to better performance. GoogLeNet (Szegedy et al., 2014), a top-performing entry of the ILSVRC-2014 classification task, was developed independently of our work, but is similar in that it is based on very deep ConvNets (22 weight layers) and small convolution filters (apart from $3\times 3$, they also use $1\times 1$ and $5\times 5$ convolutions). Their network topology is, however, more complex than ours, and the spatial resolution of the feature maps is reduced more aggressively in the first layers to decrease the amount of computation. As will be shown in Sect. 4.5, our model is outperforming that of Szegedy et al. (2014) in terms of the single-network classification accuracy.
小尺寸卷积滤波器之前已被 Ciresan 等人（2011）使用，但他们的网络深度远小于我们，并且他们没有在大型 ILSVRC 数据集上进行评估。Goodfellow 等人（2014）将深度卷积网络（ $11$ 权重层）应用于街道号码识别任务，并表明增加深度可以提高性能。GoogLeNet（Szegedy 等人，2014），作为 ILSVRC-2014 分类任务中的顶级表现者，独立于我们的工作开发，但在基于非常深的卷积网络（22 权重层）和小卷积滤波器（除了 $3\times 3$ ，他们还使用 $1\times 1$ 和 $5\times 5$ 卷积）方面相似。然而，他们的网络拓扑比我们更复杂，并且在第一层中更激进地降低了特征图的空间分辨率以减少计算量。第 4.5 节将表明，我们的模型在单网络分类精度方面优于 Szegedy 等人（2014）的模型。

The ConvNet training procedure generally follows Krizhevsky et al. (2012) (except for sampling the input crops from multi-scale training images, as explained later). Namely, the training is carried out by optimising the multinomial logistic regression objective using mini-batch gradient descent (based on back-propagation (LeCun et al., 1989)) with momentum. The batch size was set to $256$, momentum to $0.9$. The training was regularised by weight decay (the $L_{2}$ penalty multiplier set to $5\cdot 10^{-4}$) and dropout regularisation for the first two fully-connected layers (dropout ratio set to $0.5$). The learning rate was initially set to $10^{-2}$, and then decreased by a factor of $10$ when the validation set accuracy stopped improving. In total, the learning rate was decreased 3 times, and the learning was stopped after $370$K iterations (74 epochs). We conjecture that in spite of the larger number of parameters and the greater depth of our nets compared to (Krizhevsky et al., 2012), the nets required less epochs to converge due to (a) implicit regularisation imposed by greater depth and smaller conv. filter sizes; (b) pre-initialisation of certain layers.
卷积神经网络（ConvNet）的训练过程通常遵循 Krizhevsky 等人（2012 年）的方法（除了从多尺度训练图像中采样输入裁剪，如后文所述）。具体来说，通过使用基于反向传播（LeCun 等人，1989 年）的动量小批量梯度下降来优化多项式逻辑回归目标。批大小设置为 $256$ ，动量设置为 $0.9$ 。训练通过权重衰减（ $L_{2}$ 惩罚乘数设置为 $5\cdot 10^{-4}$ ）和前两层全连接层的 dropout 正则化进行正则化（dropout 比率设置为 $0.5$ ）。初始学习率设置为 $10^{-2}$ ，当验证集准确率停止提升时，学习率以因子 $10$ 减少。总共学习率减少了 3 次，并在 $370$ K 次迭代（74 个 epoch）后停止训练。我们推测，尽管我们的网络参数数量更多、深度更大，但与（Krizhevsky 等人，2012 年）相比，由于（a）深度增加和卷积滤波器尺寸减小带来的隐式正则化；（b）某些层的预初始化，我们的网络收敛所需的 epoch 数更少。

The initialisation of the network weights is important, since bad initialisation can stall learning due to the instability of gradient in deep nets. To circumvent this problem, we began with training the configuration A (Table 1), shallow enough to be trained with random initialisation. Then, when training deeper architectures, we initialised the first four convolutional layers and the last three fully-connected layers with the layers of net A (the intermediate layers were initialised randomly). We did not decrease the learning rate for the pre-initialised layers, allowing them to change during learning. For random initialisation (where applicable), we sampled the weights from a normal distribution with the zero mean and $10^{-2}$ variance. The biases were initialised with zero. It is worth noting that after the paper submission we found that it is possible to initialise the weights without pre-training by using the random initialisation procedure of Glorot & Bengio (2010).
网络权重的初始化很重要，因为不良的初始化会导致深度网络中的梯度不稳定，从而停滞学习。为了解决这个问题，我们首先从训练配置 A（表 1）开始，这个配置足够浅，可以用随机初始化来训练。然后，在训练更深层次的网络时，我们用网络 A 的层来初始化前四个卷积层和最后三个全连接层（中间层随机初始化）。我们没有降低预初始化层的初始学习率，允许它们在学习过程中发生变化。对于随机初始化（适用时），我们从均值为零、方差为 $10^{-2}$ 的正态分布中采样权重。偏置项被初始化为零。值得注意的是，在论文提交后我们发现，可以通过使用 Glorot & Bengio（2010）的随机初始化方法来初始化权重，而无需预训练。

To obtain the fixed-size $224\times 224$ ConvNet input images, they were randomly cropped from rescaled training images (one crop per image per SGD iteration). To further augment the training set, the crops underwent random horizontal flipping and random RGB colour shift (Krizhevsky et al., 2012). Training image rescaling is explained below.
为了获得固定大小的 $224\times 224$ 卷积网络输入图像，它们被随机裁剪自重新缩放的训练图像（每张图像每个 SGD 迭代裁剪一次）。为了进一步扩充训练集，这些裁剪图像会进行随机水平翻转和随机 RGB 颜色偏移（Krizhevsky 等人，2012 年）。训练图像的重新缩放将在下文解释。

At test time, given a trained ConvNet and an input image, it is classified in the following way. First, it is isotropically rescaled to a pre-defined smallest image side, denoted as $Q$ (we also refer to it as the test scale). We note that $Q$ is not necessarily equal to the training scale $S$ (as we will show in Sect. 4, using several values of $Q$ for each $S$ leads to improved performance). Then, the network is applied densely over the rescaled test image in a way similar to (Sermanet et al., 2014). Namely, the fully-connected layers are first converted to convolutional layers (the first FC layer to a $7\times 7$ conv. layer, the last two FC layers to $1\times 1$ conv. layers). The resulting fully-convolutional net is then applied to the whole (uncropped) image. The result is a class score map with the number of channels equal to the number of classes, and a variable spatial resolution, dependent on the input image size. Finally, to obtain a fixed-size vector of class scores for the image, the class score map is spatially averaged (sum-pooled). We also augment the test set by horizontal flipping of the images; the soft-max class posteriors of the original and flipped images are averaged to obtain the final scores for the image.
在测试时，给定一个训练好的卷积神经网络和一张输入图像，它的分类方式如下。首先，图像被等比例缩放到一个预定义的最小图像边长，记为 $Q$ （我们也将它称为测试尺度）。我们注意到 $Q$ 不一定等于训练尺度 $S$ （如我们在第 4 节将展示的，对每个 $S$ 使用多个 $Q$ 的值可以提升性能）。然后，网络以类似于(Sermanet 等人，2014)的方式密集地应用于缩放后的测试图像。具体来说，首先将全连接层转换为卷积层（第一个全连接层转换为 $7\times 7$ 卷积层，最后两个全连接层转换为 $1\times 1$ 卷积层）。然后，将生成的全卷积网络应用于整张（未裁剪的）图像。结果是一个具有与类别数量相同通道数的类别分数图，以及一个取决于输入图像大小的可变空间分辨率。最后，为了获得图像的固定大小类别分数向量，对类别分数图进行空间平均（求和池化）。 我们还通过图像的水平翻转来增强测试集；原始图像和翻转图像的 softmax 类后验概率被平均，以获得图像的最终得分。

Since the fully-convolutional network is applied over the whole image, there is no need to sample multiple crops at test time (Krizhevsky et al., 2012), which is less efficient as it requires network re-computation for each crop. At the same time, using a large set of crops, as done by Szegedy et al. (2014), can lead to improved accuracy, as it results in a finer sampling of the input image compared to the fully-convolutional net. Also, multi-crop evaluation is complementary to dense evaluation due to different convolution boundary conditions: when applying a ConvNet to a crop, the convolved feature maps are padded with zeros, while in the case of dense evaluation the padding for the same crop naturally comes from the neighbouring parts of an image (due to both the convolutions and spatial pooling), which substantially increases the overall network receptive field, so more context is captured. While we believe that in practice the increased computation time of multiple crops does not justify the potential gains in accuracy, for reference we also evaluate our networks using $50$ crops per scale ($5\times 5$ regular grid with $2$ flips), for a total of $150$ crops over $3$ scales, which is comparable to $144$ crops over $4$ scales used by Szegedy et al. (2014).
由于全卷积网络应用于整张图像，因此在测试时无需采样多个裁剪区域（Krizhevsky 等人，2012 年），这种方式效率较低，因为它需要对每个裁剪区域进行网络重新计算。同时，使用大量裁剪区域（如 Szegedy 等人，2014 年所做的那样）可以提高准确率，因为它导致对输入图像的采样比全卷积网络更精细。此外，多裁剪评估与密集评估互补，因为它们具有不同的卷积边界条件：当将卷积神经网络应用于裁剪区域时，卷积特征图会用零填充，而在密集评估的情况下，相同裁剪区域的填充自然来自图像的相邻部分（由于卷积和空间池化），这显著增加了整个网络的感受野，从而捕获了更多上下文信息。 虽然我们相信在实践中，多裁剪带来的计算时间增加并不能证明其可能带来的准确率提升，但为了参考，我们也使用每个尺度 $50$ 个裁剪（ $5\times 5$ 个规则的网格， $2$ 次翻转）来评估我们的网络，总共在 $3$ 个尺度上进行 $150$ 个裁剪，这相当于 Szegedy 等人（2014 年）使用的在 $4$ 个尺度上进行 $144$ 个裁剪。

Our implementation is derived from the publicly available C++ Caffe toolbox (Jia, 2013) (branched out in December 2013), but contains a number of significant modifications, allowing us to perform training and evaluation on multiple GPUs installed in a single system, as well as train and evaluate on full-size (uncropped) images at multiple scales (as described above). Multi-GPU training exploits data parallelism, and is carried out by splitting each batch of training images into several GPU batches, processed in parallel on each GPU. After the GPU batch gradients are computed, they are averaged to obtain the gradient of the full batch. Gradient computation is synchronous across the GPUs, so the result is exactly the same as when training on a single GPU.
我们的实现基于公开可用的 C++ Caffe 工具箱（Jia，2013）（于 2013 年 12 月分支出来），但包含许多重大修改，使我们能够在单个系统中的多个 GPU 上进行训练和评估，以及在多种尺度（如上所述）的全尺寸（未裁剪）图像上进行训练和评估。多 GPU 训练利用数据并行性，通过将每个训练图像批次分割成多个 GPU 批次，并在每个 GPU 上并行处理这些批次来进行。在计算 GPU 批次的梯度后，将它们平均以获得整个批次的梯度。梯度计算在 GPU 之间是同步的，因此结果与在单个 GPU 上训练时完全相同。

While more sophisticated methods of speeding up ConvNet training have been recently proposed (Krizhevsky, 2014), which employ model and data parallelism for different layers of the net, we have found that our conceptually much simpler scheme already provides a speedup of $3.75$ times on an off-the-shelf 4-GPU system, as compared to using a single GPU. On a system equipped with four NVIDIA Titan Black GPUs, training a single net took 2–3 weeks depending on the architecture.
虽然最近提出了更复杂的方法来加速 ConvNet 训练（Krizhevsky，2014），这些方法采用模型和数据并行性来处理网络的不同层，但我们发现我们概念上更简单的方案已经在 4-GPU 系统上提供了 $3.75$ 倍的加速，与使用单个 GPU 相比。在一台配备四块 NVIDIA Titan Black GPU 的系统上，根据网络架构的不同，训练单个网络需要 2-3 周的时间。

We begin with evaluating the performance of individual ConvNet models at a single scale with the layer configurations described in Sect. 2.2. The test image size was set as follows: $Q=S$ for fixed $S$, and $Q=0.5(S_{min}+S_{max})$ for jittered $S\in\left[S_{min},S_{max}\right]$. The results of are shown in Table 3.
我们首先使用第 2.2 节中描述的层配置，评估单个 ConvNet 模型在单尺度下的性能。测试图像大小设置如下： $Q=S$ 用于固定 $S$ ， $Q=0.5(S_{min}+S_{max})$ 用于抖动 $S\in\left[S_{min},S_{max}\right]$ 。结果如表 3 所示。

First, we note that using local response normalisation (A-LRN network) does not improve on the model A without any normalisation layers. We thus do not employ normalisation in the deeper architectures (B–E).
首先，我们注意到使用局部响应归一化（A-LRN 网络）并没有改善没有任何归一化层的模型 A 的性能。因此，我们在更深的架构（B-E）中不使用归一化。

Second, we observe that the classification error decreases with the increased ConvNet depth: from 11 layers in A to 19 layers in E. Notably, in spite of the same depth, the configuration C (which contains three $1\times 1$ conv. layers), performs worse than the configuration D, which uses $3\times 3$ conv. layers throughout the network. This indicates that while the additional non-linearity does help (C is better than B), it is also important to capture spatial context by using conv. filters with non-trivial receptive fields (D is better than C). The error rate of our architecture saturates when the depth reaches $19$ layers, but even deeper models might be beneficial for larger datasets. We also compared the net B with a shallow net with five $5\times 5$ conv. layers, which was derived from B by replacing each pair of $3\times 3$ conv. layers with a single $5\times 5$ conv. layer (which has the same receptive field as explained in Sect. 2.3). The top-1 error of the shallow net was measured to be $7\%$ higher than that of B (on a center crop), which confirms that a deep net with small filters outperforms a shallow net with larger filters.
其次，我们观察到分类错误随着卷积网络深度的增加而减少：从 A 中的 11 层到 E 中的 19 层。值得注意的是，尽管深度相同，包含三个 $1\times 1$ 卷积层的配置 C 表现不如整个网络使用 $3\times 3$ 卷积层的配置 D。这表明虽然额外的非线性有所帮助（C 比 B 好），但使用具有非平凡感受野的卷积滤波器来捕获空间上下文也同样重要（D 比 C 好）。当深度达到 $19$ 层时，我们架构的错误率达到饱和，但对于更大的数据集，更深层的模型可能仍然有益。我们还比较了网络 B 和一个具有五个 $5\times 5$ 卷积层的浅层网络，该网络由 B 通过将每对 $3\times 3$ 卷积层替换为单个具有与第 2.3 节所述相同感受野的 $5\times 5$ 卷积层得到。浅层网络的 Top-1 错误率（在中心裁剪上）测量值比 B 高 $7\%$ ，这证实了具有小滤波器的深层网络优于具有大滤波器的浅层网络。

Finally, scale jittering at training time ($S\in[256;512]$) leads to significantly better results than training on images with fixed smallest side ($S=256$ or $S=384$), even though a single scale is used at test time. This confirms that training set augmentation by scale jittering is indeed helpful for capturing multi-scale image statistics.
最后，训练时的尺度抖动（ $S\in[256;512]$ ）比在具有固定最小边的图像上训练（ $S=256$ 或 $S=384$ ）的结果要好得多，即使测试时只使用单一尺度。这证实了通过尺度抖动对训练集进行增强确实有助于捕捉多尺度图像统计信息。

Having evaluated the ConvNet models at a single scale, we now assess the effect of scale jittering at test time. It consists of running a model over several rescaled versions of a test image (corresponding to different values of $Q$), followed by averaging the resulting class posteriors. Considering that a large discrepancy between training and testing scales leads to a drop in performance, the models trained with fixed $S$ were evaluated over three test image sizes, close to the training one: $Q=\{S-32,S,S+32\}$. At the same time, scale jittering at training time allows the network to be applied to a wider range of scales at test time, so the model trained with variable $S\in[S_{min};S_{max}]$ was evaluated over a larger range of sizes $Q=\{S_{min},0.5(S_{min}+S_{max}),S_{max}\}$.
在单一尺度下评估了 ConvNet 模型后，我们现在评估测试时尺度抖动的影响。它包括对测试图像的多个重缩放版本（对应不同的 $Q$ 值）运行模型，然后对得到的类后验进行平均。考虑到训练和测试尺度之间的较大差异会导致性能下降，使用固定 $S$ 训练的模型在接近训练尺度的三个测试图像尺寸 $Q=\{S-32,S,S+32\}$ 上进行了评估。同时，训练时的尺度抖动允许网络在测试时应用于更广泛的尺度范围，因此使用可变 $S\in[S_{min};S_{max}]$ 训练的模型在更大的尺寸范围 $Q=\{S_{min},0.5(S_{min}+S_{max}),S_{max}\}$ 上进行了评估。

The results, presented in Table 4, indicate that scale jittering at test time leads to better performance (as compared to evaluating the same model at a single scale, shown in Table 3). As before, the deepest configurations (D and E) perform the best, and scale jittering is better than training with a fixed smallest side $S$. Our best single-network performance on the validation set is $24.8\%/7.5\%$ top-1/top-5 error (highlighted in bold in Table 4). On the test set, the configuration E achieves $7.3\%$ top-5 error.
表 4 中的结果表明，在测试时进行尺度抖动可以获得更好的性能（与在单一尺度上评估相同模型相比，如表 3 所示）。如前所述，最深的配置（D 和 E）表现最佳，而尺度抖动比使用固定最小边长进行训练的效果更好。我们在验证集上的单网络最佳性能是 $24.8\%/7.5\%$ top-1/top-5 错误率（在表 4 中加粗显示）。在测试集上，配置 E 实现了 $7.3\%$ top-5 错误率。

In Table 5 we compare dense ConvNet evaluation with mult-crop evaluation (see Sect. 3.2 for details). We also assess the complementarity of the two evaluation techniques by averaging their soft-max outputs. As can be seen, using multiple crops performs slightly better than dense evaluation, and the two approaches are indeed complementary, as their combination outperforms each of them. As noted above, we hypothesize that this is due to a different treatment of convolution boundary conditions.
在表 5 中，我们比较了密集卷积网络评估与多裁剪评估（详见第 3.2 节）。我们还通过平均它们的 softmax 输出，评估了这两种评估技术的互补性。可以看出，使用多个裁剪的表现略好于密集评估，而且这两种方法确实互补，因为它们的组合表现优于各自单独使用。如前所述，我们假设这是由于对卷积边界条件的不同处理所致。

Up until now, we evaluated the performance of individual ConvNet models. In this part of the experiments, we combine the outputs of several models by averaging their soft-max class posteriors. This improves the performance due to complementarity of the models, and was used in the top ILSVRC submissions in 2012 (Krizhevsky et al., 2012) and 2013 (Zeiler & Fergus, 2013; Sermanet et al., 2014).
到目前为止，我们评估了单个卷积网络模型的性能。在这部分实验中，我们通过平均多个模型的 softmax 类别后验输出来组合它们的输出。由于模型的互补性，这种方法提高了性能，并在 2012 年（Krizhevsky 等人，2012 年）和 2013 年（Zeiler & Fergus，2013 年；Sermanet 等人，2014 年）的顶级 ILSVRC 提交中被使用。

The results are shown in Table 6. By the time of ILSVRC submission we had only trained the single-scale networks, as well as a multi-scale model D (by fine-tuning only the fully-connected layers rather than all layers). The resulting ensemble of 7 networks has $7.3\%$ ILSVRC test error. After the submission, we considered an ensemble of only two best-performing multi-scale models (configurations D and E), which reduced the test error to $7.0\%$ using dense evaluation and $6.8\%$ using combined dense and multi-crop evaluation. For reference, our best-performing single model achieves $7.1\%$ error (model E, Table 5).
结果如表 6 所示。在 ILSVRC 提交时，我们只训练了单尺度网络以及多尺度模型 D（通过仅微调全连接层而不是所有层）。由此产生的 7 个网络的集成具有 $7.3\%$ ILSVRC 测试错误。提交后，我们考虑了仅由两个性能最佳的多尺度模型（配置 D 和 E）组成的集成，使用密集评估将测试错误降低到 $7.0\%$ ，使用密集和多裁剪组合评估将测试错误降低到 $6.8\%$ 。作为参考，我们性能最佳的单一模型实现了 $7.1\%$ 错误（模型 E，表 5）。

Finally, we compare our results with the state of the art in Table 7. In the classification task of ILSVRC-2014 challenge (Russakovsky et al., 2014), our “VGG” team secured the 2nd place with $7.3\%$ test error using an ensemble of 7 models. After the submission, we decreased the error rate to $6.8\%$ using an ensemble of 2 models.
最后，我们在表 7 中将我们的结果与当前最优方法进行了比较。在 ILSVRC-2014 挑战赛（Russakovsky 等人，2014 年）的分类任务中，我们的“VGG”团队使用 7 个模型的集成获得了第 2 名，测试错误率为 $7.3\%$ 。提交后，我们使用 2 个模型的集成将错误率降低到 $6.8\%$ 。

As can be seen from Table 7, our very deep ConvNets significantly outperform the previous generation of models, which achieved the best results in the ILSVRC-2012 and ILSVRC-2013 competitions. Our result is also competitive with respect to the classification task winner (GoogLeNet with $6.7\%$ error) and substantially outperforms the ILSVRC-2013 winning submission Clarifai, which achieved $11.2\%$ with outside training data and $11.7\%$ without it. This is remarkable, considering that our best result is achieved by combining just two models – significantly less than used in most ILSVRC submissions. In terms of the single-net performance, our architecture achieves the best result ($7.0\%$ test error), outperforming a single GoogLeNet by $0.9\%$. Notably, we did not depart from the classical ConvNet architecture of LeCun et al. (1989), but improved it by substantially increasing the depth.
从表 7 可以看出，我们的非常深的卷积网络显著优于上一代模型，这些模型在 ILSVRC-2012 和 ILSVRC-2013 竞赛中取得了最佳结果。我们的结果在与分类任务冠军（GoogLeNet，错误率为 $6.7\%$ ）相比也具有竞争力，并且显著优于 ILSVRC-2013 冠军提交的 Clarifai，后者在外部训练数据下取得了 $11.2\%$ 的成绩，在无外部训练数据下取得了 $11.7\%$ 的成绩。考虑到我们的最佳结果是通过结合仅两个模型实现的——这远少于大多数 ILSVRC 提交中使用的模型数量。在单网络性能方面，我们的架构取得了最佳结果（ $7.0\%$ 测试错误率），优于单个 GoogLeNet $0.9\%$ 。值得注意的是，我们没有偏离 LeCun 等人（1989 年）提出的经典卷积网络架构，而是通过显著增加深度对其进行了改进。