__ Summary and Contributions__: The paper discusses fast approximation to cross-validation for Markov
and conditional random field models when marginal maximum likelihood
estimation is used for inference.

__ Strengths__: For the type of models and inference, the proposed approach is
sensible extension of the previous work. The experiments demonstrate
the practically useful gain computation time, which will make it more
feasible to use more elaborate models for bigger data sets.

__ Weaknesses__: The limitations when the approach is applicable are not well stated. Marginal maximum likelihood is mentioned in the end of section 2. At this point I assumed that exact gradients and deterministic optimization would be used. In one of the experiments stochastic gradient methods is used, but it's not clearly stated whether stochastic gradients can be used also in Algorithm 1. This point affects my overall conclusion a lot. I expect that the authors may have good answer clarifying the conditions when the approach is applicable which they could include in the final version.

__ Correctness__: Mostly correct with some misleading wording used as explained below.
"But this existing ACV work is restricted to simpler models by the
assumptions that (i) data are independent and (ii) an exact initial
model fit is available. In structured data analyses, (i) is always
untrue, and (ii) is often untrue."
This is slightly misleading. If we assume complete independence, there is no common model. It would be better to discuss conditional independence. In case of structured models there can also be conditional independence (see, e.g. Figure 1 in https://arxiv.org/abs/1810.10559). Having a structured model, does not automatically invalidate, for example, leave-one-out cross-validation. The paper is now missing the distinction between the model structure and the prediction task, such that given certain prediction task, something like leave-one-out cross-validation may be inappropriate. For example, in case of time series data and structured model for the time dependency, the observations can be conditionally independent given the latent values, and leave-one-out cross-validation is valid, but it is the prediction task reasons why to leave out all observations at the end of a time series (see, e.g. discussion in Bürkner et al, 2020 cited in your paper and https://doi.org/10.1007/s42113-018-0020-6). Thus, it would be more accurate to write "approximate cross-validation for structured prediction in structured models". I don't mean you should change the title, but I suggest to consider some clarifications of this issue especially in the abstract and the introduction and some sentences later. The first paragraph of section 3 in your paper has discussion about extrapolation of time series and pictures and this discussion is towards the correct direction in sense of acknowledging that the structure in the prediction task is the crucial point why leave-one-out is not sufficient. You could use that example already in the introduction to introduce the concept of the prediction ta

__ Clarity__: Paper is mostly well written. The title, the abstract and the introduction all mention that the approach is for "structured models", but how to approximate CV depends also on how the inference is made and this is mentioned first time in the end of section 2.1 at the end of the page 2. As there are other common choices for making inference for type of structured models discussed in the paper, I would suggest to mention already in the abstract and in the introduction that the ACV discussed in the paper is specifically for the marginal maximum likelihood estimation or other well defined optimization--feel free to specify what are the requirements--and not for, e.g., MCMC.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: No

__ Additional Feedback__: ### After rebuttal
I have read the other reviews and the author rebuttal. My main concern was that the assumptions were not clearly stated. The authors have responded well. Both the recap (lines 1-9 in the rebuttal) and lines 31-41 are exactly what I wanted to see and these are presented in such compact form that I'm confident that the authors are able to include them in the revision. Due to the excellent rebuttal I'm increasing my score to 8.

__ Summary and Contributions__: The paper proposes a methodological extension of the infinitesimal jackknife (IJ) approximate cross-validation to data that are modeled via hidden Markov models or conditional random fields. This is done by applying an HMM or CRF likelihood to the general ACV objective function and algorithm. Theory for finite sample error bounds and algorithmic computational performance are shown. Several real-data examples are presented comparing their ACV method to exact CV; it is shown that the two are comparable in terms of loss, while ACV only has a small fraction of the runtime compared to exact CV.

__ Strengths__: The claims in the paper seem to be correct. They appear to be fairly direct extensions of methodology and theory from previous generalized cross validation literature. The empirical evaluations presented appear to be interpreted correctly as well.
The paper is relevant to the NeurIPS community. The method introduced in this paper could be used for any machine learning method involving latent variable estimation and cross-validation for hyperparameter selection.

__ Weaknesses__: The major limitation of this paper is the empirical studies section. A greatly expanded experimental simulation section could be useful for studying the method under controlled or known conditions and for comparing the approach to other methods in the literature. The paper does not compare the performance of their ACV method to cited previous ACV methods, such as the naive Newton step (NS) or IJ methods explicitly mentioned in the paper.
The novelty of the work is relatively minor. Its main contribution is the application of a likelihood from a family of models that is fairly well-known to a general framework for ACV from a previous paper. There are some new theoretical results with respect to the error bounds of IJ ACV specifically with respect to HMMs and CRFs.
Some of the specific contributions of the work presented are somewhat oversold. The authors discuss the societal impact of their method in very broad terms; it may be more pertinent to consider specific applications or fields where ACV may be necessary for data with dependency structures. The authors also introduce their method as widely applicable to data with dependent observations, but their specific studies only apply to methods that model these dependencies via latent processes.

__ Correctness__: Assuming that the cited previous literature on generalized cross-validation and approximate cross-validation are correct, the claims and methods here should be correct as well.

__ Clarity__: The mathematical notation is understandable and fairly standard. The proofs are well-formatted Grammatical errors appear in the paper, but they do not severely detract from the understanding of the main ideas of the paper.

__ Relation to Prior Work__: The authors clearly discuss how their ACV method extends work from previous ACV methods by applying it to models that don't assume i.i.d. data.

__ Reproducibility__: Yes

__ Additional Feedback__: There are a few minor grammatical errors throughout the paper. Figures in the real-data experiment section could be better labeled/presented, e.g. error as a percentage of original model MSE. The ACV vs. exact CV runtime graphs could either be bar graphs or just plotted like a normal line graph. Figure 2 placement needs to be adjusted in section 4, as the first line of a paragraph is split off. The paper refers to two different methods of leaving out data for the HMM model as "case A" and case "B"; these could be better named as currently it can be difficult for the reader to remember which case is which. The terminology in Proposition 1 is a little difficult to understand.

__ Summary and Contributions__: This paper proposes algorithms for approximate cross validation for structured data by developing infinitesimal jackknife approximations to circumvent an exact initial fit which is considered intractable in general for structured data. The paper provides some theoretical analysis on the approximation error. The paper empirically verifies the effectiveness of the proposed methods on multiple settings.

__ Strengths__: 1) The problem of extending approximate cross validation to structured data is very important.
2) The paper provides both theoretical and empirical justifications.

__ Weaknesses__: 1) The presentation of the paper can be substantially improved. For example, frequent switching between LSCV and LWCV can confuse readers.
2) In the empirical evaluation, it lacks quantitative results, which makes it hard to assess the performance of the proposed algorithms.

__ Correctness__: looks correct to me

__ Clarity__: okay but has room to improve.

__ Relation to Prior Work__: yes

__ Reproducibility__: Yes

__ Additional Feedback__: 1) The authors are suggested to put key algorithms (e.g., Alg. 2) in the main text to make it self-contained.
2) Why do the authors choose 500 folds in the experiment? It sounds too large to use in practice.
3) Line 11-12: it reads that the authors have a solution for (ii), but it is not clear to me what the solution is.