PGD（projected gradient descent）算法源码解析

论文链接：https://arxiv.org/abs/1706.06083
源码出处：https://github.com/Harry24k/adversarial-attacks-pytorch/tree/master

---

PGDLinf源码

import torch
import torch.nn as nn

from ..attack import Attack


class PGD(Attack):
    r"""
    PGD in the paper 'Towards Deep Learning Models Resistant to Adversarial Attacks'
    [https://arxiv.org/abs/1706.06083]

    Distance Measure : Linf

    Arguments:
        model (nn.Module): model to attack.
        eps (float): maximum perturbation. (Default: 8/255)
        alpha (float): step size. (Default: 2/255)
        steps (int): number of steps. (Default: 10)
        random_start (bool): using random initialization of delta. (Default: True)

    Shape:
        - images: :math:`(N, C, H, W)` where `N = number of batches`, `C = number of channels`,        `H = height` and `W = width`. It must have a range [0, 1].
        - labels: :math:`(N)` where each value :math:`y_i` is :math:`0 \leq y_i \leq` `number of labels`.
        - output: :math:`(N, C, H, W)`.

    Examples::
        >>> attack = torchattacks.PGD(model, eps=8/255, alpha=1/255, steps=10, random_start=True)
        >>> adv_images = attack(images, labels)

    """
    def __init__(self, model, eps=8/255,
                 alpha=2/255, steps=10, random_start=True):
        super().__init__("PGD", model)
        self.eps = eps
        self.alpha = alpha
        self.steps = steps
        self.random_start = random_start
        self.supported_mode = ['default', 'targeted']

    def forward(self, images, labels):
        r"""
        Overridden.
        """
        self._check_inputs(images)

        images = images.clone().detach().to(self.device)
        labels = labels.clone().detach().to(self.device)

        if self.targeted:
            target_labels = self.get_target_label(images, labels)

        loss = nn.CrossEntropyLoss()

        adv_images = images.clone().detach()

        if self.random_start:
            # Starting at a uniformly random point
            adv_images = adv_images + torch.empty_like(adv_images).uniform_(-self.eps, self.eps)
            adv_images = torch.clamp(adv_images, min=0, max=1).detach()

        for _ in range(self.steps):
            adv_images.requires_grad = True
            outputs = self.get_logits(adv_images)

            # Calculate loss
            if self.targeted:
                cost = -loss(outputs, target_labels)
            else:
                cost = loss(outputs, labels)

            # Update adversarial images
            grad = torch.autograd.grad(cost, adv_images,
                                       retain_graph=False, create_graph=False)[0]

            adv_images = adv_images.detach() + self.alpha*grad.sign()
            delta = torch.clamp(adv_images - images, min=-self.eps, max=self.eps)
            adv_images = torch.clamp(images + delta, min=0, max=1).detach()

        return adv_images

解析

PGD算法（projected gradient descent）是在BIM算法的基础上的小改进，二者非常相近，BIM算法的源码解析在上一篇博客中，建议先看上一篇博客理解BIM算法的原理。

具体来说，在BIM算法开始迭代前，就先给图像加上扰动（在$ϵ$邻域内均匀分布）。换句话说，也就是图像开始迭代的起点随机，而不是像BIM算法一样从原始图像开始迭代。论文这么做的目的是为了研究从随机的起点开始迭代扰动，损失能够达到的不同的局部最大值的关系。

PGD算法的公式如下所示：$$X_0^{adv}=X+\eta ,X_{N+1}^{adv}=Cli{p}_{X,ϵ}\{X_N^{adv}+\alpha sign({▽}_{x}J(X_N^{adv},y_{true}))\}$$其中，$\eta$是一个随机扰动，在$ϵ$邻域内均匀分布。

eps：即$ϵ$，表示最大扰动。
alpha：即$\alpha$，表示每次迭代中扰动的增加量（或减少量）。
steps：表示迭代次数。
random_start：迭代的起点是否随机，也就是是否要加随机扰动$\eta$，若为False，则该算法就和BIM算法相同。
images = images.clone().detach().to(self.device)：clone()将图像克隆到一块新的内存区（pytorch默认同样的tensor共享一块内存区）；detach()是将克隆的新的tensor从当前计算图中分离下来，作为叶节点，从而可以计算其梯度；to()作用就是将其载入设备。
target_labels = self.get_target_label(images, labels)：若是有目标攻击的情况，获取目标标签。目标标签的选取有多种方式，例如可以选择与真实标签相差最大的标签，也可以随机选择除真实标签外的标签。
loss = nn.CrossEntropyLoss()：设置损失函数为交叉熵损失。

adv_images = adv_images + torch.empty_like(adv_images).uniform_(-self.eps, self.eps)
adv_images = torch.clamp(adv_images, min=0, max=1).detach()

以上两行代码作用即为添加随机扰动，torch.empty_like(adv_images)会返回一个形状同adv_images的空的Tensor，uniform_(-self.eps, self.eps)将Tensor中的值在$[-ϵ,ϵ]$范围内的均匀分布中随机取值。torch.clamp(adv_images, min=0, max=1)会将图像中大于1的值设为1、小于0的值设为0，防止超出范围。
adv_images.requires_grad = True：将requires_grad 参数设置为True，torch就会在图像的计算过程中自动计算计算图，用于反向梯度计算。
outputs = self.get_logits(images)：获得图像的在模型中的输出值。
cost = -loss(outputs, target_labels)：有目标情况下计算损失。
cost = loss(outputs, labels)：无目标情况下计算损失。
grad = torch.autograd.grad(cost, images, retain_graph=False, create_graph=False)[0]：cost对images求导，得到梯度grad。
adv_images = images + self.alpha*grad.sign()：根据公式在图像上沿着梯度上升方向以步长为$\alpha$增加扰动。

delta = torch.clamp(adv_images - images, min=-self.eps, max=self.eps)  # 得到改变量
adv_images = torch.clamp(images + delta, min=0, max=1).detach()  # 防止图像超出有效范围

以上两行代码就是裁剪的过程，同BIM算法中的$Clip$过程，防止图像超出$[0,1]$范围。

---

PGDL2源码

import torch
import torch.nn as nn

from ..attack import Attack


class PGDL2(Attack):
    r"""
    PGD in the paper 'Towards Deep Learning Models Resistant to Adversarial Attacks'
    [https://arxiv.org/abs/1706.06083]

    Distance Measure : L2

    Arguments:
        model (nn.Module): model to attack.
        eps (float): maximum perturbation. (Default: 1.0)
        alpha (float): step size. (Default: 0.2)
        steps (int): number of steps. (Default: 10)
        random_start (bool): using random initialization of delta. (Default: True)

    Shape:
        - images: :math:`(N, C, H, W)` where `N = number of batches`, `C = number of channels`,        `H = height` and `W = width`. It must have a range [0, 1].
        - labels: :math:`(N)` where each value :math:`y_i` is :math:`0 \leq y_i \leq` `number of labels`.
        - output: :math:`(N, C, H, W)`.

    Examples::
        >>> attack = torchattacks.PGDL2(model, eps=1.0, alpha=0.2, steps=10, random_start=True)
        >>> adv_images = attack(images, labels)

    """

    def __init__(self, model, eps=1.0, alpha=0.2, steps=10,
                 random_start=True, eps_for_division=1e-10):
        super().__init__("PGDL2", model)
        self.eps = eps
        self.alpha = alpha
        self.steps = steps
        self.random_start = random_start
        self.eps_for_division = eps_for_division
        self.supported_mode = ['default', 'targeted']

    def forward(self, images, labels):
        r"""
        Overridden.
        """
        self._check_inputs(images)

        images = images.clone().detach().to(self.device)
        labels = labels.clone().detach().to(self.device)

        if self.targeted:
            target_labels = self.get_target_label(images, labels)

        loss = nn.CrossEntropyLoss()

        adv_images = images.clone().detach()
        batch_size = len(images)

        if self.random_start:
            # Starting at a uniformly random point
            delta = torch.empty_like(adv_images).normal_()
            d_flat = delta.view(adv_images.size(0), -1)  # 将图片矩阵展平，方便计算范数
            n = d_flat.norm(p=2, dim=1).view(adv_images.size(0), 1, 1, 1)  # 计算每个向量的模长
            r = torch.zeros_like(n).uniform_(0, 1)  # 随机[0,1]之间均匀分布
            delta *= r/n*self.eps  # 即将delta向量变为模长为[0,eps]之间的向量
            adv_images = torch.clamp(adv_images + delta, min=0, max=1).detach()

        for _ in range(self.steps):
            adv_images.requires_grad = True
            outputs = self.get_logits(adv_images)

            # Calculate loss
            if self.targeted:
                cost = -loss(outputs, target_labels)
            else:
                cost = loss(outputs, labels)

            # Update adversarial images
            grad = torch.autograd.grad(cost, adv_images,
                                       retain_graph=False, create_graph=False)[0]
            grad_norms = torch.norm(grad.view(batch_size, -1), p=2, dim=1) + self.eps_for_division  # 这边加上了self.eps_for_division是为了防止下面除0
            grad = grad / grad_norms.view(batch_size, 1, 1, 1)  # 使梯度变为单位向量
            adv_images = adv_images.detach() + self.alpha * grad
			
			# 下面是为了改变后的图像与原图像的L2距离不超过eps
            delta = adv_images - images
            delta_norms = torch.norm(delta.view(batch_size, -1), p=2, dim=1)  # 计算改变量的模长
            factor = self.eps / delta_norms
            # 如果eps/delta_norms小于1，则说明改变量的L2距离超过了eps
            # 那么就会在factor与delta相乘的过程中被替换为eps
            factor = torch.min(factor, torch.ones_like(delta_norms))
            delta = delta * factor.view(-1, 1, 1, 1)

            adv_images = torch.clamp(images + delta, min=0, max=1).detach()

        return adv_images

解析

PGDL2和PGDLinf的区别就在于度量样本之间的距离的范式不同，假设样本$X=(x_1,x_2,x_3,...,x_n)$，L2范数$||X|{|}_2=\sqrt{x_1^2+x_2^2+x_3^2+...+x_n^2}$，Linf范数$||X|{|}_{\infty }=\sqrt[n]{x_1^n+x_2^n+x_3^n+...+x_n^n}$，简单来说，L2范数可以理解为向量的模长，Linf范数可以理解为向量中最大元素的值。

二者在源码中的区别可以看我写在代码中的注释。