Paper:
Multi-stage training and knowledge transfer from a large-scale pretraining task to various finetuning tasks have revolutionized natural language processing (NLP) and computer vision (CV), with state-of-the-art performances constantly being improved. In this paper, we develop a multi-stage influence function score to track predictions from a finetuned model all the way back to the pretraining data. With this score, we can identify the pretraining examples in the pretraining task that contribute most to a prediction in the finetuning task. The proposed multistage influence function generalizes the original influence function for a single model in , thereby enabling influence computation through both pretrained and finetuned models. We study two different scenarios with the pretrained embeddings fixed or updated in the finetuning tasks. We test our proposed method in various experiments to show its effectiveness and potential applications.

Review:
The authors derive the influence function of models that are first pre-trained and then fine-tuned. This extends influence functions beyond the standard supervised setting that they have been primarily considered in. To do so, the authors make two methodological contributions: 1) working through the calculus for the pre-training setting and deriving a corresponding efficient algorithm, and 2) adding $L_2$ regularization to approximate the effect of fine-tuning for a limited number of gradient steps.
I believe that these are useful technical contributions that will help to broaden the applicability of influence functions beyond the standard supervised setting. For that reason, I recommend a weak accept. I have some questions and reservations about the current paper:
1) Does pretraining actually help in the MNIST/CIFAR settings considered? These seem to be non-standard pretraining settings. More generally, can we relate influence to some objective measure that we care about (say test accuracy), for example by showing that removing the top X% of influential pretraining data hurts test accuracy as much as predicted? Minor: section 4.2 also seems non-standard. Are the exact same bird vs. frog examples being used for both pretraining and finetuning?
2) In what situations might we want to examine the influence of pretraining data, and can we design experiments that show those situations? For example, perhaps we're wondering if different types of sentences in the one-billion-word dataset might be more or less useful. Can we verify those claims using these multi-stage influence functions? It is otherwise difficult to assess the utility of the qualitative results (e.g., Figure 3 and Appendix C).
3) It'd be helpful to get a better understanding of the technical contributions of this paper. Specifically, 
a. What is the impact of $\alpha$ in equation 12 and how does it interact with the number of fine-tuning steps taken?
b. If the Hessian has negative eigenvalues, we can still take $H^{-1}b$ by solving CG with $H^2$, but what does this correspond to? Is the influence equation well defined (or the Taylor approximation justified) if $H$ is not positive definite?