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I read about analysis on local properties of neural networks. Some of them study the impact of "infinitesimal" change to an input. Like in Percy Liang's paper Understanding Black-box Predictions via Influence Functions. They have neat formalization with these properties but I don't really get the point.

My questions are:

  1. How local is local? Or maybe how small is "infinitesimal". I get the idea that this is related to some derivative and stuff. But these results are applied to some changes that are not that small from my perspective, like a change of 0.1 in pixel value ranging from 0 to 1. Why is this considered correct?

  2. From my understanding, if the derivative does not change much, it can be applied to a larger area. But is this the case in neural networks? It would be great if I can get some references on this topic.

  3. Since I don't really have much background in ml theory and serious math stuff. Other pointers that may help me understand these topics are also appreciated.

Thank you for the help in advance.

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