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Robust Graph Convolutional Networks Against Adversarial Attacks

时间:2022-10-07 11:11:42浏览次数:98  
标签:Convolutional odot Attacks bm Graph mu tilde mathcal sigma

目录

Zhu D., Zhang Z., Cui P. and Zhu W. Robust graph convolutional networks against adversarial attacks. In ACM International Conference on Knowledge Discovery and Data Mining (KDD), 2019.

RGCN 设计了一种 Gaussian-based 的模块, 利用 Variance 来设计 attention 用于提高网络的鲁棒性. 从我的理解来看, 它提高对抗鲁棒性的能力堪忧.

符号说明

  • \(G = (V, E)\), 图;
  • \(V = \{v_1, \ldots, v_N\}\), nodes;
  • \(A\), 邻接矩阵;
  • \(\tilde{A}\), add self-loop;
  • \(\tilde{D}_{ii} = \sum_{j} \tilde{A}_{ij}\);
  • \(\bm{h}_i^{(l)}\), node \(v_i\) 在第 \(l\) 层的特征表示;
  • \(N(i) = \{v: (v, v_i) \in E\} \cup \{v_i\}\);
  • \(\odot\), element-wise 乘法;

算法

  • 初始化均值和方差特征:

    \[M^{(0)} = [\bm{\mu}_1^{(0)}, \bm{\mu}_2^{(0)}, \ldots, \bm{\mu}_N^{(0)}]^T, \\ \Sigma^{(0)} = [\bm{\sigma}_1^{(0)}, \bm{\sigma}_2^{(0)}, \ldots, \bm{\sigma}_N^{(0)}]^T, \\ \]

    假设

    \[\bm{h}_i^{(0)} \sim \mathcal{N}(\bm{\mu}_i^{(0)}, \bm{\sigma}_i^{(0)}); \]

  • 第 \(l\) 层进行如下操作:

    1. 首先计算 attention:

      \[\bm{\alpha}_j^{(l)} = \exp(-\gamma \bm{\sigma}_j^{(l - 1)}), \]

      可以看出, 方差越大对应的权重越小 (因为作者认为方差越大的越容易是噪声);

    2. 计算如下该层的均值和方差特征:

      \[\bm{\mu}_i^{(l)} = \rho(\sum_{j \in N(i)} \frac{\bm{\mu}_j^{(l-1)} \odot \bm{\alpha}_j^{(l)}}{\sqrt{\tilde{D}_{ii} \tilde{D}_{jj}}} W_{\mu}^{(l)}), \\ \bm{\sigma}_i^{(l)} = \rho(\sum_{j \in N(i)} \frac{\bm{\sigma}_j^{(l-1)} \odot \bm{\alpha}_j^{(l)} \odot \bm{\alpha}_j^{(l)}}{\sqrt{\tilde{D}_{ii} \tilde{D}_{jj}}}W_{\sigma}^{(l)}); \]

      此时

      \[\bm{h}_i^{(l)} \sim \mathcal{N}(\bm{\mu}_i^{(l)}, \bm{\sigma}_i^{(l)}); \]

    3. 用矩阵表示为如下结果:

      \[M^{(l)} = \rho(\tilde{D}^{-1/2} \tilde{A} \tilde{D}^{-1/2} (M^{(l-1)} \odot \mathcal{A}^{(l)}) W_{\mu}^{(l)}), \\ \Sigma^{(l)} = \rho(\tilde{D}^{-1/2} \tilde{A} \tilde{D}^{-1/2} (\Sigma^{(l-1)} \odot \mathcal{A}^{(l)} \odot \mathcal{A}^{(l)}) W_{\sigma}^{(l)}); \]

  • 最后一层, 我们得到最后的输出:

    \[\bm{z}_i = \bm{\mu}_i^{(L)} + \bm{\epsilon} \odot \sqrt{\bm{\sigma}_i^{(L)}}, \: \bm{\epsilon} \sim \mathcal{N}(\bm{0}, I); \]

  • 最后通过如下损失进行训练:

    \[\mathcal{L} = \mathcal{L}_{cls} + \beta_1 \mathcal{L}_{reg1} + \beta_2 \mathcal{L}_{reg2}, \]

    其中 \(\mathcal{L}_{cls}\) 就是普通的分类损失 (基于 \(\bm{z}_i\)), 然后

    \[\mathcal{L}_{reg1} = \sum_{i=1}^N \text{KL}(\mathcal{N}(\bm{\mu}_i^{(1)}, \bm{\sigma_i}^{(1)}) \| \mathcal{N}(\bm{0}, I)) \]

    确保第一层的输出是正态分布?

    \[\mathcal{L}_{reg2} = \|W_{\mu}^{(0)}\|_2^2 + \|W_{\sigma}^{(0)}\|_2^2 \]

    为普通的对第一层的 L2 正则.

标签:Convolutional,odot,Attacks,bm,Graph,mu,tilde,mathcal,sigma
From: https://www.cnblogs.com/MTandHJ/p/16759276.html

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