目录

• 概
• 主要内容
  • 符号
  • MART

Wang Y, Zou D, Yi J, et al. Improving Adversarial Robustness Requires Revisiting Misclassified Examples[C]. international conference on learning representations, 2020.

@article{wang2020improving,

title={Improving Adversarial Robustness Requires Revisiting Misclassified Examples},

author={Wang, Yisen and Zou, Difan and Yi, Jinfeng and Bailey, James and Ma, Xingjun and Gu, Quanquan},

year={2020}}

作者认为, 错分样本对于提高网络的鲁棒性是很重要的, 为此提出了一个启发于此的新的损失函数.

符号

\(h_{\theta}\): 参数为\(\theta\)的神经网络;

\((x,y) \in \mathbb{R}^d \times \{1,\ldots, K\}\): 类别及其标签;

\[\tag{2}
h_{\boldsymbol{\theta}}\left(\mathbf{x}_{i}\right)=\underset{k=1, \ldots, K}{\arg \max } \mathbf{p}_{k}\left(\mathbf{x}_{i}, \boldsymbol{\theta}\right), \quad \mathbf{p}_{k}\left(\mathbf{x}_{i}, \boldsymbol{\theta}\right)=\exp \left(\mathbf{z}_{k}\left(\mathbf{x}_{i}, \boldsymbol{\theta}\right)\right) / \sum_{k^{\prime}=1}^{K} \exp \left(\mathbf{z}_{k^{\prime}}\left(\mathbf{x}_{i}, \boldsymbol{\theta}\right)\right)
\]

定义正分类样本和误分类样本

\[\mathcal{S}_{h_{\theta}}^+ = \{i : i \in [n], h_{\theta} (x_i)=y_i \} \quad \mathrm{and} \quad \mathcal{S}_{h_{\theta}}^- = \{i : i \in [n], h_{\theta} (x_i) \not =y_i \}.
\]

MART

在所有样本上的鲁棒分类误差:

\[\tag{3}
\mathcal{R}(h_{\theta}) = \frac{1}{n} \sum_{i=1}^n \max_{x_i' \in \mathcal{B}_{\epsilon}(x_i)} \mathbb{1}(h_{\theta}(x_i') \not= y_i),
\]

并定义在错分样本上的鲁棒分类误差

\[\tag{4}
\mathcal{R}^- (h_{\theta}, x_i):= \mathbb{1} (h_{\theta}(\hat{x}_i') \not=y_i) + \mathbb{1}(h_{\theta}(x_i) \not= h_{\theta} (\hat{x}_i'))
\]

其中

\[\tag{5}
\hat{x}_i'=\arg \max_{x_i' \in \mathcal{B}_{\epsilon} (x_i)} \mathbb{1} (h_{\theta} (x_i') \not = y_i).
\]

以及正分样本上的鲁棒分类误差:

\[\tag{6}
\mathcal{R}^+(h_{\theta}, x_i):=\mathbb{1}(h_{\theta}(\hat{x}_i') \not = y_i).
\]

最后, 我们要最小化的是二者的混合误差:

\[\tag{7}
\begin{aligned}
\min _{\boldsymbol{\theta}} \mathcal{R}_{\text {misc }}\left(h_{\boldsymbol{\theta}}\right): &=\frac{1}{n}\left(\sum_{i \in \mathcal{S}_{h}^{+}} \mathcal{R}^{+}\left(h_{\boldsymbol{\theta}}, \mathbf{x}_{i}\right)+\sum_{i \in \mathcal{S}_{\boldsymbol{h}_{\boldsymbol{\theta}}}^{-}} \mathcal{R}^{-}\left(h_{\boldsymbol{\theta}}, \mathbf{x}_{i}\right)\right) \\
&=\frac{1}{n} \sum_{i=1}^{n}\left\{\mathbb{1}\left(h_{\boldsymbol{\theta}}\left(\hat{\mathbf{x}}_{i}^{\prime}\right) \neq y_{i}\right)+\mathbb{1}\left(h_{\boldsymbol{\theta}}\left(\mathbf{x}_{i}\right) \neq h_{\boldsymbol{\theta}}\left(\hat{\mathbf{x}}_{i}^{\prime}\right)\right) \cdot \mathbb{1}\left(h_{\boldsymbol{\theta}}\left(\mathbf{x}_{i}\right) \neq y_{i}\right)\right\}
\end{aligned}.
\]

为了能够传递梯度, 需要利用一些替代函数"软化"上面的损失函数, 对于\(\mathbb{1}(h_{\theta}(\hat{x}_i')\not = y_i)\)利用BCE损失函数替代

\[\tag{8}
\mathrm{BCE} (p(\hat{x}_i, \theta),y_i)= -\log (p_{y_i} (\hat{x}_i',\theta))- \log (1-\max_{k\not=y_i} p_k(\hat{x}_i',\theta)),
\]

第一项为普通的交叉熵损失, 第二项用于提高分类边界.

对于第二项\(\mathbb{1}(h_{\theta}(x_i)\not=h_{\theta}(\hat{x}_i'))\), 用KL散度作为替代

\[\tag{9}
\mathrm{KL} (p(x_i, \theta)\| p(\hat{x}_i', \theta))=\sum_{k=1}^K p_k(x_i, \theta)\log \frac{p_k(x_i,\theta)}{p_k(\hat{x}_i',\theta)}.
\]

最后一项\(\mathbb{1}(h_{\theta}(x_i) \not =y_i)\)则可用 \(1-p_{y_i}(x_i,\theta)\)来代替.

于是最后的损失函数便是

\[\tag{11}
\mathcal{L}^{\mathrm{MART}}(\theta)= \frac{1}{n} \sum_{i=1}^n \ell(x_i, y_i, \theta),
\]

其中

\[\ell (x_i,y_i,\theta):=\mathrm{BCE}(p(\hat{x}_i', \theta),y_i)+\lambda \cdot \mathrm{KL} (p(x_i,\theta) \|p(\hat{x}_i,\theta)) \cdot (1-p_{y_i}(x_i, \theta)).
\]