A. Helpful is a list of numbers, which are the IDs of the training data samples This packages offers two modes of computation to calculate the influence ICML 2017 best paperStanfordPang Wei KohCourseraStanfordNIPS 2019influence functionPercy Liang11Michael Jordan, , \hat{\theta}_{\epsilon, z} \stackrel{\text { def }}{=} \arg \min _{\theta \in \Theta} \frac{1}{n} \sum_{i=1}^{n} L\left(z_{i}, \theta\right)+\epsilon L(z, \theta), \left.\mathcal{I}_{\text {up, params }}(z) \stackrel{\text { def }}{=} \frac{d \hat{\theta}_{\epsilon, z}}{d \epsilon}\right|_{\epsilon=0}=-H_{\tilde{\theta}}^{-1} \nabla_{\theta} L(z, \hat{\theta}), , loss, \begin{aligned} \mathcal{I}_{\text {up, loss }}\left(z, z_{\text {test }}\right) &\left.\stackrel{\text { def }}{=} \frac{d L\left(z_{\text {test }}, \hat{\theta}_{\epsilon, z}\right)}{d \epsilon}\right|_{\epsilon=0} \\ &=\left.\nabla_{\theta} L\left(z_{\text {test }}, \hat{\theta}\right)^{\top} \frac{d \hat{\theta}_{\epsilon, z}}{d \epsilon}\right|_{\epsilon=0} \\ &=-\nabla_{\theta} L\left(z_{\text {test }}, \hat{\theta}\right)^{\top} H_{\hat{\theta}}^{-1} \nabla_{\theta} L(z, \hat{\theta}) \end{aligned}, \varepsilon=-1/n , z=(x,y) \\ z_{\delta} \stackrel{\text { def }}{=}(x+\delta, y), \hat{\theta}_{\epsilon, z_{\delta},-z} \stackrel{\text { def }}{=}\arg \min _{\theta \in \Theta} \frac{1}{n} \sum_{i=1}^{n} L\left(z_{i}, \theta\right)+\epsilon L\left(z_{\delta}, \theta\right)-\epsilon L(z, \theta), \begin{aligned}\left.\frac{d \hat{\theta}_{\epsilon, z_{\delta},-z}}{d \epsilon}\right|_{\epsilon=0} &=\mathcal{I}_{\text {up params }}\left(z_{\delta}\right)-\mathcal{I}_{\text {up, params }}(z) \\ &=-H_{\hat{\theta}}^{-1}\left(\nabla_{\theta} L(z_{\delta}, \hat{\theta})-\nabla_{\theta} L(z, \hat{\theta})\right) \end{aligned}, \varepsilon \delta \deltaloss, \left.\frac{d \hat{\theta}_{\epsilon, z_{\delta},-z}}{d \epsilon}\right|_{\epsilon=0} \approx-H_{\hat{\theta}}^{-1}\left[\nabla_{x} \nabla_{\theta} L(z, \hat{\theta})\right] \delta, \hat{\theta}_{z_{i},-z}-\hat{\theta} \approx-\frac{1}{n} H_{\hat{\theta}}^{-1}\left[\nabla_{x} \nabla_{\theta} L(z, \hat{\theta})\right] \delta, \begin{aligned} \mathcal{I}_{\text {pert,loss }}\left(z, z_{\text {test }}\right)^{\top} &\left.\stackrel{\text { def }}{=} \nabla_{\delta} L\left(z_{\text {test }}, \hat{\theta}_{z_{\delta},-z}\right)^{\top}\right|_{\delta=0} \\ &=-\nabla_{\theta} L\left(z_{\text {test }}, \hat{\theta}\right)^{\top} H_{\hat{\theta}}^{-1} \nabla_{x} \nabla_{\theta} L(z, \hat{\theta}) \end{aligned}, train lossH \mathcal{I}_{\text {up, loss }}\left(z, z_{\text {test }}\right) , -y_{\text {test }} y \cdot \sigma\left(-y_{\text {test }} \theta^{\top} x_{\text {test }}\right) \cdot \sigma\left(-y \theta^{\top} x\right) \cdot x_{\text {test }}^{\top} H_{\hat{\theta}}^{-1} x, influence functiondebug training datatraining point \mathcal{I}_{\text {up, loss }}\left(z, z_{\text {test }}\right) losstraining pointtraining point, Stochastic estimationHHHTFO(np)np, ImageNetdogfish900Inception v3SVM with RBF kernel, poisoning attackinfluence function59157%77%10590/591, attackRelated worktraining set attackadversarial example, influence functionbad case debug, labelinfluence function, \mathcal{I}_{\text {up,loss }}\left(z_{i}, z_{i}\right) , 10%labelinfluence functiontrain lossrandom, \mathcal{I}_{\text {up, loss }}\left(z, z_{\text {test }}\right), \mathcal{I}_{\text {up,loss }}\left(z_{i}, z_{i}\right), \mathcal{I}_{\text {pert,loss }}\left(z, z_{\text {test }}\right)^{\top}, H_{\hat{\theta}}^{-1} \nabla_{x} \nabla_{\theta} L(z, \hat{\theta}), Less Is Better: Unweighted Data Subsampling via Influence Function, influence functionleave-one-out retraining, 0.86H, SVMhinge loss0.95, straightforwardbest paper, influence functionloss.
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