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In Understanding Black-box Predictions via Influence Functions paper Appendix A, the authors provide a standard derivation for influence functions, however, I could not understand one of the steps. They mentioned that from the step $(12)$

$$\Delta_{\epsilon} \approx -[\nabla^2R(\hat{\theta}) + \epsilon\nabla^2L(z, \hat{\theta})]^{-1}[\nabla R(\hat{\theta}) + \epsilon\nabla L(z, \hat{\theta})]$$

to step $(13)$

$$ \Delta_\epsilon \approx -\nabla^2R(\hat{\theta})\nabla L(z, \hat{\theta})\epsilon $$

by dropping $o(\epsilon)$ terms. Also, we have that $\nabla R(\hat{\theta}) = 0$. Therefore, I could not get what it meant when they mentioned dropping $o(\epsilon)$ terms.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Oct 11, 2023 at 16:44

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