目录

• 概
• 主要内容
  • 符号说明
  • Error
  • Classification-calibrated surrogate loss
  • 引理2.1
  • 定理3.1
  • 定理3.2
  • 由此导出的TRADES算法
  • 实验概述
• 代码

Zhang H, Yu Y, Jiao J, et al. Theoretically Principled Trade-off between Robustness and Accuracy[J]. arXiv: Learning, 2019.

@article{zhang2019theoretically,

title={Theoretically Principled Trade-off between Robustness and Accuracy},

author={Zhang, Hongyang and Yu, Yaodong and Jiao, Jiantao and Xing, Eric P and Ghaoui, Laurent El and Jordan, Michael I},

journal={arXiv: Learning},

year={2019}}

从二分类问题入手, 拆分\(\mathcal{R}_{rob}\)为\(\mathcal{R}_{nat},\mathcal{R}_{bdy}\), 通过\(\mathcal{R}_{rob}-\mathcal{R}_{nat}^*\)的上界建立损失函数，并将这种思想推广到一般的多分类问题.

符号说明

\(X, Y\): 随机变量;

\(x\in \mathcal{X}, y\): 样本, 对应的标签(\(1, -1\));

\(f\): 分类器(如神经网络);

\(\mathbb{B}(x, \epsilon)\): \(\{x'\in \mathcal{X}:\|x'-x\| \le \epsilon\}\);

\(\mathbb{B}(DB(f),\epsilon)\): \(\{x \in \mathcal{X}: \exist x'\in \mathbb{B}(x,\epsilon), \mathrm{s.t.} \: f(x)f(x')\le0\}\) ;

\(\psi^*(u)\): \(\sup_u\{u^Tv-\psi(u)\}\), 共轭函数;

\(\phi\): surrogate loss.

Error

\[\tag{e.1}
\mathcal{R}_{rob}(f):= \mathbb{E}_{(X,Y)\sim \mathcal{D}}\mathbf{1}\{\exist X' \in \mathbb{B}(X, \epsilon), \mathrm{s.t.} \: f(X')Y \le 0\},
\]

其中\(\mathbf{1}(\cdot)\)表示指示函数, 显然\(\mathcal{R}_{rob}(f)\)是关于分类器\(f\)存在adversarial samples 的样本的点的测度.

\[\tag{e.2}
\mathcal{R}_{nat}(f) :=\mathbb{E}_{(X,Y)\sim \mathcal{D}}\mathbf{1}\{f(X)Y \le 0\},
\]

显然\(\mathcal{R}_{nat}(f)\)是\(f\)正确分类真实样本的概率, 并且\(\mathcal{R}_{rob} \ge \mathcal{R}_{nat}\).

\[\tag{e.3}
\mathcal{R}_{bdy}(f) :=\mathbb{E}_{(X,Y)\sim \mathcal{D}}\mathbf{1}\{X \in \mathbb{B}(DB(f), \epsilon), \:f(X)Y > 0\},
\]

显然

\[\tag{1}
\mathcal{R}_{rob}-\mathcal{R}_{nat}=\mathcal{R}_{bdy}.
\]

因为想要最优化\(0-1\)loss是很困难的, 我们往往用替代的loss \(\phi\), 定义:

\[\mathcal{R}_{\phi}(f):= \mathbb{E}_{(X, Y) \sim \mathcal{D}} \phi(f(X)Y), \\
\mathcal{R}^*_{\phi}(f):= \min_f \mathcal{R}_{\phi}(f).
\]

Classification-calibrated surrogate loss

这部分很重要, 但是篇幅很少, 我看懂, 等回看了引用的论文再讨论.

引理2.1

定理3.1

在假设1的条件下\(\phi(0)\ge1\), 任意的可测函数\(f:\mathcal{X} \rightarrow \mathbb{R}\), 任意的于\(\mathcal{X}\times \{\pm 1\}\)上的概率分布, 任意的\(\lambda > 0\), 有

\[\begin{array}{ll}
& \mathcal{R}_{rob}(f) - \mathcal{R}_{nat}^* \\
\le & \psi^{-1}(\mathcal{R}_{\phi}(f)-\mathcal{R}_{\phi}^*) + \mathbf{Pr}[X \in \mathbb{B}(DB(f), \epsilon), f(X)Y >0] \\
\le & \psi^{-1}(\mathcal{R}_{\phi}(f)-\mathcal{R}_{\phi}^*) + \mathbb{E} \quad \max _{X' \in \mathbb{B}(X, \epsilon)} \phi(f(X')f(X)/\lambda). \\
\end{array}
\]

最后一个不等式, 我知道是因为\(\phi(f(X')f(X)/\lambda) \ge1.\)

定理3.2

结合定理\(3.1, 3.2\)可知, 这个界是紧的.

由此导出的TRADES算法

二分类问题, 最优化上界, 即:

扩展到多分类问题, 只需:

算法如下:

实验概述

5.1: 衡量该算法下, 理论上界的大小差距;

5.2: MNIST, CIFAR10 上衡量\(\lambda\)的作用, \(\lambda\)越大\(\mathcal{R}_{nat}\)越小, \(\mathcal{R}_{rob}\)越大, CIFAR10上反映比较明显;

5.3: 在不同adversarial attacks 下不同算法的比较;

5.4: NIPS 2018 Adversarial Vision Challenge.

import torch
import torch.nn as nn

def quireone(func): #a decorator, for easy to define optimizer
    def wrapper1(*args, **kwargs):
        def wrapper2(arg):
            result = func(arg, *args, **kwargs)
            return result
        wrapper2.__doc__ = func.__doc__
        wrapper2.__name__ = func.__name__
        return wrapper2
    return wrapper1

class AdvTrain:

    def __init__(self, eta, k, lam,
                 net, lr = 0.01, **kwargs):
        """
        :param eta: step size for adversarial attacks
        :param lr: learning rate
        :param k: number of iterations K in inner optimization
        :param lam: lambda
        :param net: network
        :param kwargs: other configs for optim
        """
        kwargs.update({'lr':lr})
        self.net = net
        self.criterion = nn.CrossEntropyLoss()
        self.opti = self.optim(self.net.parameters(), **kwargs)
        self.eta = eta
        self.k = k
        self.lam = lam

    @quireone
    def optim(self, parameters, **kwargs):
        """
        quireone is decorator defined below
        :param parameters: net.parameteres()
        :param kwargs: other configs
        :return:
        """
        return torch.optim.SGD(parameters, **kwargs)

    def normal_perturb(self, x, sigma=1.):

        return x + sigma * torch.randn_like(x)

    @staticmethod
    def calc_jacobian(loss, inp):
        jacobian = torch.autograd.grad(loss, inp, retain_graph=True)[0]
        return jacobian

    @staticmethod
    def sgn(matrix):
        return torch.sign(matrix)

    def pgd(self, inp, y, perturb):
        boundary_low = inp - perturb
        boundary_up = inp + perturb
        inp.requires_grad_(True)
        out = self.net(inp)
        loss = self.criterion(out, y)
        delta = self.sgn(self.calc_jacobian(loss, inp)) * self.eta
        inp_new = inp.data
        for i in range(self.k):
            inp_new = torch.clamp(
                inp_new + delta,
                boundary_low,
                boundary_up
            )
        return inp_new

    def ipgd(self, inps, ys, perturb):
        N = len(inps)
        adversarial_samples = []
        for i in range(N):
            inp_new = self.pgd(
                inps[[i]], ys[[i]],
                perturb
            )
            adversarial_samples.append(inp_new)

        return torch.cat(adversarial_samples)

    def train(self, trainloader, epoches=50, perturb=1, normal=1):

        for epoch in range(epoches):
            running_loss = 0.
            for i, data in enumerate(trainloader, 1):
                inps, labels = data

                adv_inps = self.ipgd(self.normal_perturb(inps, normal),
                                     labels, perturb)

                out1 = self.net(inps)
                out2 = self.net(adv_inps)

                loss1 = self.criterion(out1, labels)
                loss2 = self.criterion(out2, labels)

                loss = loss1 + loss2

                self.opti.zero_grad()
                loss.backward()
                self.opti.step()

                running_loss += loss.item()

                if i % 10 is 0:
                    strings = "epoch {0:<3} part {1:<5} loss: {2:<.7f}\n".format(
                        epoch, i, running_loss
                    )
                    print(strings)
                    running_loss = 0.