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**Goal**: identifying training points most responsible for a given prediction. Given training points $z_1, \dots, z_n$, let loss function be $\frac{1}{n}\sum_{i=1}^nL(z_i, \theta)$ A function called influence function let us compute the parameter change if $z$ were upweighted by some small $\epsilon$. $$\hat{\theta}_{\epsilon, z} := \arg \min_{\theta \in \Theta} \frac{1}{n}\sum_{i=1}^n L(z_i, \theta) + \epsilon L(z, \theta)$$ $$\mathcal{I}_{\text{up, params}}(z) := \frac{d\hat{\theta}_{\epsilon, z}}{d\epsilon} = H_{\hat{\theta}}^{1} \nabla_\theta L(z, \hat{\theta})$$ $\mathcal{I}_{\text{up, params}}(z)$ shows how uplifting one point $z$ affect the estimate of the parameters $\theta$. Furthermore, we could determine how uplifting $z$ affect the loss estimate of a test point through chain rule. $$\mathcal{I}_{\text{up, loss}}(z, z_{\text{test}}) = \nabla_\theta L(z_{\text{test}}, \hat{\theta})^\top \mathcal{I}_{\text{up, params}}(z)$$ Apart from lifting one training point, change of the parameters with the change of a training point could also be estimated. $$\frac{d\hat{\theta}_{\epsilon, z_\delta, z}}{d\epsilon} = \mathcal{I}_{\text{up, params}}(z_\delta)  \mathcal{I}_{\text{up, params}}(z)$$ This measures how purturbation $\delta$ to training point $z$ affect the parameter estimation $\theta$. Section 3 describes some practicals about efficient implementing. This set of tool could be used for some interpretable machine learning tasks.
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