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Uncertainties associated with GANgenerated datasets in high energy physics
by Konstantin T. Matchev, Alexander Roman, Prasanth Shyamsundar
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Submission summary
Authors (as registered SciPost users):  Konstantin Matchev · Prasanth Shyamsundar 
Submission information  

Preprint Link:  https://arxiv.org/abs/2002.06307v3 (pdf) 
Date submitted:  20210625 07:36 
Submitted by:  Shyamsundar, Prasanth 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Phenomenological 
Abstract
Recently, Generative Adversarial Networks (GANs) trained on samples of traditionally simulated collider events have been proposed as a way of generating larger simulated datasets at a reduced computational cost. In this paper we point out that data generated by a GAN cannot statistically be better than the data it was trained on, and critically examine the applicability of GANs in various situations, including a) for replacing the entire Monte Carlo pipeline or parts of it, and b) to produce datasets for usage in highly sensitive analyses or suboptimal ones. We present our arguments using information theoretic demonstrations, a toy example, as well as in the form of a theorem, and identify some potential valid uses of GANs in collider simulations.
List of changes
The manuscript has undergone several major changes. We have made our arguments more quantitative by
* Formulating the main argument as a theorem in Section 2 and proving it in Section 4.
* Providing three different information theoretic demonstrations (using mutual information, Fisher information, and KL divergence) in Section 3, which show that the GAN generated dataset cannot contain any more information than the training dataset it is based on.
* Providing a toy example in an appendix which demonstrates our claims.
In addition, we also
* Address the argument in favor of GANs based on their ability to be good function approximators (Section 3.4)
* Discuss a recent work in the literature which suggested the possibility of amplifying datasets using GANs (Section 5)
* Reconcile our results with those of earlier studies, providing an explanation for the seeming incompatibility of our claims with the evidence in the literature (Section 6)
* Identify some applications of GANs that are not subverted by the arguments presented in the paper (Section 7)
While we have improved the presentation of our arguments in this version and added new material, our claims (along the caveats) presented in the previous version remain unchanged and have not been weakened.
Current status:
Reports on this Submission
Report #3 by Anonymous (Referee 5) on 202198 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2002.06307v3, delivered 20210907, doi: 10.21468/SciPost.Report.3504
Strengths
1  The manuscript points out a logical flaw in some highenergy physics applications of GANs. The problem is very basic in nature, but its consequences are often underappreciated.
Weaknesses
1  The paper is, at times, too provocative. While the statements are factually correct, they should be presented in a more productive way.
Report
The manuscript deals with the question whether machine learning can be used to increase the statistical significance of simulated data sets based on a known underlying physics model. Using basic information theoretical arguments, the authors find this not to be the case, and conclude that various existing works on this topic may draw the wrong conclusions.
While the analysis of the problem is correct, I find the presentation rather problematic. The authors should consider rephrasing some of their statements, in particular
 The last paragraph of Sec.3: This comment is polemic, but it can in fact be used to bolster the case for the manuscript if it is taken as an explicit example of how not to use a GAN. It should be rephrased and moved from Sec.3 to the introduction.
 I would caution against calling "Theorem 1" a theorem. The paper in its present form has drawn attention for the wrong reason, with several theorists criticizing its lack of content. This could be avoided by not pretending there to be significant math behind the theorem, but instead giving very explicit examples of unintended consequences when GANs are used to enhance statistical significance.
 I would therefore also like to see App.A moved into the main body of the text. This explicit example can be utilized to make the argument in a clear and convincing manner. Figure 6 in particular will be helpful for this. It should be plotted on a loglog scale, the same as Fig.2. Figure 2 itself could be removed.
 The authors should make it clear that the main problem is not the usage of GANs, but the lack of error propagation. If the statistical uncertainty of the training data was quoted as a systematic uncertainty on the final prediction, there would not be a problem in the first place.
Requested changes
See report above
Report #2 by Andy Buckley (Referee 4) on 2021819 (Invited Report)
 Cite as: Andy Buckley, Report on arXiv:2002.06307v3, delivered 20210819, doi: 10.21468/SciPost.Report.3417
Strengths
1. Highlighting a wellknown but not always appreciated fundamental limitation in use of generative models to address statistical limitations in full MC eventgeneration chains.
2. Pedagogically useful and maybe novel applications of the dataprocessing inequality to show the intuitive result that GANs trained on a modelderived dataset cannot increase information about the model via a GANgenerated dataset.
3. Useful contextual discussion of claims for GAN interpolation as inference, and proposed procedural improvements for reports on GANbased HEP generator performance.
Weaknesses
1. The central theorem is shown in its "proof" to be phrased tautologically and hence add nothing to the discussion (which is qualitative, but valid and useful)
2. The main thrust of argument  that GANs do not improve statistical convergence to the true model  is already wellknown and appreciated among the MC generator community (and I'm sure more widely)
3. Much of the discussion is nicely phrased, but seems more verbose than necessary: some simple things are explained to the point of confusion, and the conclusions/recommendations from each section's argument are not always clear.
Report
A wellpresented paper, but the main message is already wellknown based on knowledge or intuition of the dataprocessing inequality, as the authors admit in Section 6. The discussions are interesting and detailed, if rather verbose, and as such I feel it contributes well to the discourse around where generative models can add value, and how to quantify it. However, I do not see the point in spending pages on "proving" a tautological theorem: "allowing GANs in an analysis chain cannot improve on the best performance if GANs were not previously forbidden" is not an important result and hence distracts from the useful comments elsewhere.
What is left is pedagogically useful, particularly the explicit demonstrations of the dataprocessing inequality as applied to Markovian generative models, and I am glad it is in the public literature, but I am not sure it is suitable for publication as a novel article as opposed to a commentary or review. I do not think it meets the stringent acceptance conditions of SciPost Physics, but could be suitable for SciPost Physics Core.
Detailed comments:
• p3: the statement of the theorem is vague as to the meaning of "discriminating power". While an upfront statement of the result is welcome, postponing the detail of what it actually means until the proof is unhelpful. I would add that this theorem is unsurprising to all I am aware of working in this area (to the point of having been assumed true), although I'm not aware of a previous explicit proof. Indeed, the assumption made in this proof that the GAN will replace the entire chain from the fundamental process onward is the very reason that MCgenerator authors have been critical of generation GANs, and encouraged focusing such methods on less critical downstream simulation, cf. Section 7.
• p5: this seems to be a wordy way of saying that model inferences require a good estimation of the mean rates of bin/signalregion population by the model, and the estimates of these mean rates have an undesirable uncertainty due to finite MC statistics. It is then clear that the accuracy of such estimates are set by the training sample, as any generative model based on it has no information with which to systematically improve on the estimate. As later commented in Section 6, this fact is well appreciated in the MC community.
• eq 2: this analysis is sound, but it gives the impression that the fully specified likelihood ratio is the real quantity of interest, while in practice the intermediate states are not of interest. If the likelihood ratio were considered on partially specified states, e.g. the ratio of P(D_GAN  theta_i) between i=1 and i=0  the probability of the same GANgenerated dataset having resulted from two different input models  then the summation over intermediate "paths" makes the cancellation less obvious and hence the information theoretic proofs more interesting. The first proof (on the mutual information) I think does not rely on the likelihood ratio or score, and so the structure would maybe be better with these results postponed to where they are used in the KL and Fisher proofs.
• 3.3: while nicely explained, I think most readers probably approach this backward, being familiar with the fact that generation by sampling from training data will not improve convergence to the true model, but not having seen the preceding proofs.
• 3.4: eq. 24 and the equivalent points above is an assumption, not always true. The point that interpolations or smoothings (including GANs) are neither unqiue nor a priori closer to the truth than the unsmoothed dataset is well made, however. An additional issue, mentioned in the context of Section 7 but not here, is with emergent features due to lowrate physics effects that are likely to be completely unrepresented in the training sample: there is no reason at all to expect that such features can be "interpolated" into existence by the latent forms in the GAN machinery.
• Section 4: the "proof" is rather a neat thought experiment, but the meaning of the X and X' in Fig 3 are not clear and do not seem to correspond to anything in the text (the proof is not symbolic). However, my fundamental concern is that the proof is really a statement that the theorem's conditions are tautological. This is not a good thing: it effectively removes it from the discussion entirely, leaving the following argument for the limitations of GANs to depend entirely on subjective judgements. As a result I think you have shown Theorem 1 to be a red herring that adds little if anything to the clarity of the paper's core message, and quite possibly detracts by introducing an irrelevant complications. I think the informationtheoretic statements are far stronger, if "obvious" arguments for the fundamental limitations of GANs (or at least those which attempt to cover the fundamental theory model).
• Section 6: "several other studies" absolutely needs a set of corresponding citations. It would also assist greatly, despite seeming perhaps aggressive, if the critiques in the list below also attached citations to clarify which studies are being referred to. To not cite in this criticism section is guilty of the worse crime of passive aggression! I say this as someone sympathetic with your implied criticism that MLapplications papers sometimes do not consider that the paradigm could be imperfect, and excuse suboptimal performance with banalities. The recommendations for publishing GAN performance are good but could be improved by suggesting explicit methods for estimating GAN uncertainties  an obvious one is to train many GANs based on randomly chosen training sets from a larger total set of training events, and calculate the variances in GAN'd results over the set of GANs, but there are maybe smarter ways.
• Section 7: I may be unfamiliar with the ways envisaged for use of simulation+reco GANs, but had not imagined use of a mixed fullsim/reco and GAN'ed event set. Use of GANs to replace finite precalculated shower libraries and similar seems a more obvious approach, as taken by e.g. CaloGAN (https://arxiv.org/abs/1712.10321) . This, however, has the huge caveat that additional methods are needed to adapt the shower generation to the new, continuous parameter space of truthparticle/jet kinematic phasespace. Maybe worth mentioning or focusing on this approach, rather than defeating a strawman  if the "N  M" approach is in active use, a citation would be appropriate. There are many more distinctions between the simulation+reco step as compared to fundamentalmodel sampling, probably too many to mention: the relative regularity and centrallimit nature of detectors, the (un)importance of rare detector phenomena, the roles of cleaning cuts and object calibrations, and the fundamental distinction in importance between accurate inference of detector nuisance parameters vs fundamentalphysics parameters. Whether GANs are an appropriate replacement for elements in the postgeneration chain seems to depend a lot on what they are to be used for, and whether it would depend on the existence of rare configurations in the training sample.
Requested changes
1. Given the tautology, I do not see that Theorem 1 adds any value to the paper. I would suggest removing it except perhaps for including the observation in textual discussion, and focusing more clearly on the key issue as nicely clarified by the informationtheoretic derivations.
2. In the Report section I suggest some possible, optional, improvements to the information theory presentation to make the flow of argument easier to follow.
3. In the Section 6 critiques of earlier studies, it is crucial to add citations to the "other studies" as appropriate, otherwise the critique is flogging a straw man.
4. Optionally expand on the issues for postgeneration uses of GANs to address statistical/CPU bottlenecks. This seems a more realistic use of GANs due to the existing awareness of the issues for physicsmodel sampling, and I feel there is much interesting discussion lying in the distinction between the two modes.
Author: Prasanth Shyamsundar on 20220217 [id 2217]
(in reply to Report 2 by Andy Buckley on 20210819)
 "Given the tautology, I do not see that Theorem 1 adds any value to the paper. I would suggest removing it except perhaps for including the observation in textual discussion, and focusing more clearly on the key issue as nicely clarified by the informationtheoretic derivations."
We thank the referee for the feedback. This is one of several conflicting recommendations in this regard which we have received from referees. The original versions of the paper did not have a formal "theorem", but its content was discussed in the text. However, several referees were skeptical of its validity, which forced us to formalize the statement and prove it. In the new version we have renamed it as simply a "statement".
 "In the Report section I suggest some possible, optional, improvements to the information theory presentation to make the flow of argument easier to follow."
We thank the referee for the numerous optional suggestions. We went through the comments and optional suggestions in the report section and in the revised version incorporated some of them as follows:

"p3: the statement of the theorem is vague as to the meaning of "discriminating power". While an upfront statement of the result is welcome, postponing the detail of what it actually means until the proof is unhelpful. I would add that this theorem is unsurprising to all I am aware of working in this area (to the point of having been assumed true), although I'm not aware of a previous explicit proof. Indeed, the assumption made in this proof that the GAN will replace the entire chain from the fundamental process onward is the very reason that MCgenerator authors have been critical of generation GANs, and encouraged focusing such methods on less critical downstream simulation, cf. Section 7."
In the revised version we have clarified that the statement holds for any agreedupon evaluation metric capturing the discriminating power. We also agree with the referee that there is a portion of the community (us included) to whom the theorem is unsurprising, but, as the referee points out, there is also a substantial literature on generation GANs which this paper is addressing. The goal of this paper is to formalize the criticism the referee is alluding to, and place it in the literature.

"Section 4: the "proof" is rather a neat thought experiment, but the meaning of the X and X' in Fig 3 are not clear and do not seem to correspond to anything in the text (the proof is not symbolic). However, my fundamental concern is that the proof is really a statement that the theorem's conditions are tautological. This is not a good thing: it effectively removes it from the discussion entirely, leaving the following argument for the limitations of GANs to depend entirely on subjective judgements. As a result I think you have shown Theorem 1 to be a red herring that adds little if anything to the clarity of the paper's core message, and quite possibly detracts by introducing an irrelevant complications. I think the informationtheoretic statements are far stronger, if "obvious" arguments for the fundamental limitations of GANs (or at least those which attempt to cover the fundamental theory model)."
Such thoughtexperimentbased proofs for arguably obvious results are not uncommon in science, e.g., the "no free lunch" theorem for search and optimization problems. Any application of GANs that falls within the assumptions of our statement, is objectively limited by it.

"Section 6: "several other studies" absolutely needs a set of corresponding citations. It would also assist greatly, despite seeming perhaps aggressive, if the critiques in the list below also attached citations to clarify which studies are being referred to. To not cite in this criticism section is guilty of the worse crime of passive aggression! I say this as someone sympathetic with your implied criticism that MLapplications papers sometimes do not consider that the paradigm could be imperfect, and excuse suboptimal performance with banalities. The recommendations for publishing GAN performance are good but could be improved by suggesting explicit methods for estimating GAN uncertainties  an obvious one is to train many GANs based on randomly chosen training sets from a larger total set of training events, and calculate the variances in GAN'd results over the set of GANs, but there are maybe smarter ways."
We followed the referee's advice and added the corresponding citations to section 6.

"Section 7: I may be unfamiliar with the ways envisaged for use of simulation+reco GANs, but had not imagined use of a mixed fullsim/reco and GAN'ed event set. Use of GANs to replace finite precalculated shower libraries and similar seems a more obvious approach, as taken by e.g. CaloGAN (https://arxiv.org/abs/1712.10321) . This, however, has the huge caveat that additional methods are needed to adapt the shower generation to the new, continuous parameter space of truthparticle/jet kinematic phasespace. Maybe worth mentioning or focusing on this approach, rather than defeating a strawman  if the "N  M" approach is in active use, a citation would be appropriate. There are many more distinctions between the simulation+reco step as compared to fundamentalmodel sampling, probably too many to mention: the relative regularity and centrallimit nature of detectors, the (un)importance of rare detector phenomena, the roles of cleaning cuts and object calibrations, and the fundamental distinction in importance between accurate inference of detector nuisance parameters vs fundamentalphysics parameters. Whether GANs are an appropriate replacement for elements in the postgeneration chain seems to depend a lot on what they are to be used for, and whether it would depend on the existence of rare configurations in the training sample."
We thank the referee for the feedback. The purpose of this section was not to critize any particular usage of GANs, but rather to indicate a potentially valid usage of GANs. We have changed the title of section 7 from "Potential uses of GANs in collider simulations" to "Potential valid uses of GANs in collider simulations". We have also rewritten parts of section 7.1 to be clearer.

"In the Section 6 critiques of earlier studies, it is crucial to add citations to the "other studies" as appropriate, otherwise the critique is flogging a straw man."
We thank the referee for the feedback. We have now included proper references in section 6.
 "Optionally expand on the issues for postgeneration uses of GANs to address statistical/CPU bottlenecks. This seems a more realistic use of GANs due to the existing awareness of the issues for physicsmodel sampling, and I feel there is much interesting discussion lying in the distinction between the two modes."
We thank the referee for highlighting those issues, which are beyond the scope of this paper and we may revisit in a future paper.
Report #1 by Anonymous (Referee 3) on 202177 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2002.06307v3, delivered 20210707, doi: 10.21468/SciPost.Report.3208
Report
First of all, I would like to thank the authors for being much more specific than before. I think their point is quite clear now, even though their separation of generation and analysis appears a little adhoc and I am not sure how it is related to the paper title. Essentially, the authors now say that generative networks are not MC generators, which might not be all that new or insightful, but it is correct. Any kind of improvement beyond standard MC is dubbed analysis, which is weird, given that it also applies to modern methods like event reweighting etc. Only one remaining comment, given that times are moving fast, please comment on the recent papers on networkbased unweighting. Those advertize learning a phase space density on weighted events and sampling unweighted events, maybe with a correction to the true density via postprocessing, so I am curious how they fit into this specific scheme.
Author: Prasanth Shyamsundar on 20220217 [id 2218]
(in reply to Report 1 on 20210707)
 "First of all, I would like to thank the authors for being much more specific than before. I think their point is quite clear now, even though their separation of generation and analysis appears a little adhoc and I am not sure how it is related to the paper title. Essentially, the authors now say that generative networks are not MC generators, which might not be all that new or insightful, but it is correct. Any kind of improvement beyond standard MC is dubbed analysis, which is weird, given that it also applies to modern methods like event reweighting etc."
We thank the referee for the feedback. We disagree that the distinction between generation and analysis is adhoc. A collider analysis can be made more sensitive by 1) increasing the amount of real/simulated data; and/or 2) by improving the techniques used to analyze the data. Although the end result (i.e., improved sensitivity) is the same, neither approach is a substitute for the other. We believe that it is misleading to claim to be solving one problem (insufficient simulation statistics) by addressing the other (improving the analysis). Furthermore, as we showed in the paper, one cannot use GANgenerated data in the same way as one would use the standard MCgenerated data. It is in that sense that the separation between generation and analysis is justified and relevant for the discussion in this paper.
 "Only one remaining comment, given that times are moving fast, please comment on the recent papers on networkbased unweighting. Those advertize learning a phase space density on weighted events and sampling unweighted events, maybe with a correction to the true density via postprocessing, so I am curious how they fit into this specific scheme."
We thank the referee for the suggestion. MLbased event weighting, unweighting, and reweighting strategies is an exciting recent development. In analyses which use MLweighted datasets, it is absolutely critical to account for uncertainties resulting from finiteness of the training data. We are currently working on a separate paper addressing this topic.
Author: Prasanth Shyamsundar on 20220217 [id 2216]
(in reply to Report 3 on 20210908)We thank the referee for the suggestion. As requested, we rephrased and moved that paragraph to the introduction.
This is one of several conflicting recommendations in this regard which we have received from referees in the past. The original versions of the paper did not have a formal "theorem", but its content was discussed in the text. However, several referees were skeptical of its validity, which forced us to formalize the statement and prove it in the revised version. In the new version we have renamed it as simply a "statement".
We thank the referee for the feedback. Figure 2 serves a useful purpose because it is generic and conveys the message that our statement is more universally applicable than a single toy example would suggest. Figure 6 is also useful, as it quantitatively backs up Figure 2 with a concrete example. Following the suggestion of the referee, in the revised version Figure 6 is replotted on a loglog scale.
We thank the referee for the feedback. We agree that propagating all the errors from the usage of machine learning will make the resulting analysis correct. However, when the stated reason for using a GAN is to reduce an uncertainty that will (have to) be propagated under a different name anyway, then there is a more fundamental problem with the approach (and not just an errorpropagation issue).