Summary
Influence functions allow one to measure the effect of up-weighting a training sample on the test loss (or functions of the model parameters) of a test example. This paper studies several of the assumptions needed for hessian-based influence functions to produce valid leave-out-one estimates of the effect on a test sample's loss. This assumptions include: convexity, stability of the hessian, and additivity of training trajectory. The paper shows that some of these assumptions are not as problematic, in practice, as previously assumed. They show that several of these assumptions can indeed be satisfied with changes to the formulation; for example, if the hessian becomes degenerate, one can approximate the hessian via arnoldi iteration. It turns out that the most problematic assumption results from a divergence between the initial set of parameters, and another obtained by re-training or even fine-tuning for longer time steps. The paper then bounds this parameter divergence using a discrete version of gronwall's lemma, and gives a set of takeaways for how to evaluate influence functions: the takeaway from this work is that influence scores only predict effect of a training sample over a small number of training steps. In the final portion of the paper, they evaluate how to use influence functions to correct errors.
Strengths
Overall, I really enjoyed reading this paper, and found the breakdown of the assumptions really comprehensive.
**Quality/Clarity**\
I found the paper to be of very high quality. Each assumption is stated clearly and cleanly discussed. For example, I found the resolution of assumption #1 to be interesting. Here the authors show that for hessian-based influence functions one does not need to assume strict convexity but we can relax it to assume that the final gradient steps do not change the hessian by too much. While this might seem like a simple change, it helps provide clarity to this literature. The bigger takeway from this work is that influence scores only predict effect of a training sample over a small number of training steps. This finding is important since it requires rethinking how these approaches are currently used. Overall, this paper sets out key assumptions of influence functions, discusses how to remedy violations to these assumptions, and presents experiments to corroborate the theory presented.
**Significance**\
One important takeaway is that, for deep models, it now makes sense that influence scores obtained via retraining have poor correlation with leave-one-out retraining scores. Secondly, another important takeaway is that one can correct errors simply be reweighting or fine-tuning on opponents or proponents. The findings from the paper should be important for practitioners and others using influence functions as part of their debugging toolbox.
**Originality**\
This paper mostly considers a question that was also studied by Bae et. al. (Neurips 2022), but does so in a different way and using mostly new techniques. I found the use of the discrete version of gronwall's lemma to be quite nice. I am not sure if this is new, but taken together, this work advances the understanding of influence functions quite substantially.
Weaknesses
The main weakness I have with this work are minor and listed below:
**Relation with Bae et. al. from Neurips 2022**: The main related work to this one is this paper. However, this work does not do enough to sufficiently contrast their analyses with theirs. I am mostly familiar with this previous work, and understand that they address exactly the same question as well, but arrive at a different conclusion. Their finding is that hessian-based influence functions approximate what they term the "proximal bregman response function", and that the ability of this quantity to match a sample loo estimate is mostly affected by the linearization error (i.e., due to the taylor expansion), and 2) hessian approximation. This work seems to not agree with theirs, i.e., you suggest that hessian approximation be handled with arnoldi, and that the key discrepancy comes from parameter divergence. It would be great if the authors could spend more time in the draft to contrast this work against that one. I think this is an important issue to address in the paper.
Questions
Some clarification questions:
- For error correction, if I have k examples that I want to correct, then am I correcting each example independently or the taking gradient steps on all k examples at the same time. If it is independent, then isn't this too onerous?
- One use of these influence scores is mislabelled training samples, how does assumption 5 affect that? When in training should the self-influence score be a useful metric for mislabeled label correction?
- This is minor question, going from $\theta_\epsilon \approx \theta_0 + \epsilon^\top\nabla \theta_\epsilon$ to equation 1, where did the second $\nabla_\theta$ come from? Is this the chain rule? I am familiar with other ways to derive influence functions.
Rating
8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
Confidence
4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Limitations
The authors discuss limitations.