Bounding Training Data Reconstruction in DP-SGD

Differentially private training offers a protection which is usually interpreted as a guarantee against membership inference attacks. By proxy, this guarantee extends to other threats like reconstruction attacks attempting to extract complete training examples. Recent works provide evidence that if one does not need to protect against membership attacks but instead only wants to protect against training data reconstruction, then utility of private models can be improved because less noise is required to protect against these more ambitious attacks. We investigate this further in the context of DP-SGD, a standard algorithm for private deep learning, and provide an upper bound on the success of any reconstruction attack against DP-SGD together with an attack that empirically matches the predictions of our bound. Together, these two results open the door to fine-grained investigations on how to set the privacy parameters of DP-SGD in practice to protect against reconstruction attacks. Finally, we use our methods to demonstrate that different settings of the DP-SGD parameters leading to the same DP guarantees can result in significantly different success rates for reconstruction, indicating that the DP guarantee alone might not be a good proxy for controlling the protection against reconstruction attacks.

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Peer review

Reviewer rZtg6/10 · confidence 2/52023-07-05

Summary

The paper studies the training data reconstruction robustness of DP-SGD. The paper observes that there is a huge gap between the existing lower bound and upper bound (attacks) on reconstruction error. Motivated by this observation, the paper proposes a tighter lower bound and a stronger attack, which empirically has a smaller gap. The paper further demonstrates how the hyper-parameters in DP-SGD affect the training data reconstruction error, and shows that the error can vary a lot even with the same DP guarantee.

Strengths

* Originality: The paper proposes new theoretical analyses and attacks for training data reconstruction in the context of DP-SGD. The results are new to the best of my knowledge. * Clarity: The paper is well-written. * Significance: Given that training data reconstruction is a practical concern, understanding ML models' robustness to it is an important topic. The paper is a solid contribution to this topic. * Quality: Overall, the paper is of good quality.

Weaknesses

* I do not see any major weakness of the paper.

Questions

* Eq 1: The first term is an inner product whereas the second term is l1 error. Intuitively, we want the first term to be large, and the second term to be small. It is unclear why it makes sense to "sum up" these two terms as the loss. Do we want to maximize it or minimize it? Other minor questions: * Line 97: should "<=" be "<" or "<<" instead? Randomly releasing one sample gives (0, 1/|D|)-DP. So delta=1/|D| is too weak. * Line 311. It says Balle et al. give a trivial bound for epsilon=100. But the bound is indeed tight given that the attack achieves a success probability that is very close to the bound.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

The authors did not discuss the limitation or the potential negative societal impact.

Reviewer AUNf5/10 · confidence 3/52023-07-05

Summary

This paper explores reconstruction attacks within the threat model associated with the usage of DP-SGD. The study provides a tighter upper bound on the success rate of the reconstruction attacks than previous works. Additionally, the paper presents an attack that exploits a strong discrete prior and aligns closely with the provided bound.

Strengths

- The investigation into the bounds against reconstruction attacks (under the usage of DP-SGD) is both valid and of practical relevance. - The proposal of a tighter bound and a strong attack is intriguing.

Weaknesses

- The quality of the presentation could be largely improved. Currently, the main assumptions of the proposed attack, the primary algorithm, the evaluation setup, and the key novelty are vague. - From the current submission, the bound appears to be a straightforward adaptation of the general DP bound to the reconstruction attack. - The proposed attack seems to operate only under a strong, possibly unrealistic assumption. For instance, knowing the "discrete prior" (specific points from the dataset already) might be quite a strong assumption, potentially transforming the problem from "reconstruction" to a simpler "matching" task. And the assumption of batch gradient descent (instead of stochastic gradient descent) further introduces additional complications. While this could be useful in certain scenarios (e.g., auditing the privacy cost), explicit argumentation is needed (e.g., why the proposed method is superior to existing techniques for such tasks) to justify its practicality. - The scope of this submission seems rather narrow, particularly given the attack assumption which restricts the application scenario to be focused only on DP-SGD (may not even be applicable to SGD due to the "batch" GD assumption). This makes the attack itself not stand-alone useful.

Questions

- A more structured presentation of the threat model should be provided for each method, including the baselines and the proposed one. It would be better to place such information in separate section(s), rather than interspersed among small, unstructured paragraphs in Section 2. - What defines a successful attack in the context of previous works aiming for approximate reconstruction instead of exact retrieval? - Providing more details in Section 3.2 would be beneficial, such as explicitly presenting the definitions of $\mu$ and $\nu$ in Algorithm 1. I would also expect that a relatively large number of samples (e.g., for $N$ and $n$) would be required to make the estimation precise, but such information seems to be missing in the submission.

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

2 fair

Contribution

2 fair

Limitations

- Possibly due to the presentation, I may have underestimated the technical and practical value of this submission. A clear claim about its key advantages over existing work should be provided in the revision. - I quickly scanned through the proofs and did not spot any obvious errors. - I didn't notice any discussion regarding the computational cost of the proposed method. I would expect such information to be included in the revision.

Reviewer j4Wo4/10 · confidence 3/52023-07-06

Summary

This paper pertains to a line of work exploring reconstruction attacks (RA) as a complement to the standard membership inference attacks (MIAs) for evaluating the privacy leakage of ML models. The idea is that a model can be made resistant to RAs for large values of the privacy budget epsilon for which it would not be resistant to MIAs. In cases where membership leakage is not considered a privacy issue, the model might be both “safe” as per low RA risk and useful, since less noise would have to be added. Resistance to RAs can be studied theoretically (by upper bounding the RA success in the worst case) and empirically by developing ever stronger attacks. This work proposes contributions on both fronts for models trained using DP-SGD. On the attack front, two new reconstruction attacks are proposed. The first one is based on the gradients of the model after every iteration and the second one tries to infer the correct record among n candidates. On the theoretical front, a new upper bound for RA success is proposed, which is much better than the previous bound. Finally, the attack initially developed for models trained with gradient descent (no mini-batches) is extended to the mini-batch case. The attack success is shown to be sensitive to the mini-batch size.

Strengths

The gradient-based and prior-aware attack methodologies both seem sound, are well explained and motivated, and the authors explored different alternatives to the proposed formulations. The attacks present sufficient novelty. The empirical evaluation is extensive which is highly appreciated. Empirically, the new theoretical upper bound on the success of reconstruction significantly improves upon the previous bound of Balle et al. (2022) as shown on Figure 2. Theoretically, the contribution seems significant too, although I did not verify the proof and I found the hypotheses somewhat opaque.

Weaknesses

1. Prior-aware attack: Conceptually, I am not convinced the 1 out of n classification is the right approach to frame the reconstruction problem. The goal of reconstruction is to extract a target record with limited prior knowledge about the record, i.e., reconstruct the record from scratch. The prior-aware attack changes the task to something quite different, by turning it into a distinguishing game, i.e., finding the right record among 2, 5, 10 etc. possible candidates (specifically, for 2 candidates, the task becomes equivalent to MIA under bounded DP). This means that the adversary already knows the record and the only uncertainty left is which one it is among a limited set of options. 2. One problem with the prior-aware attack is that the prior can be made arbitrarily easy or hard. For instance, if the other records in the prior have a different label than the target record, which the adversary is assumed to already know (Appendix A), it is very easy to guess the right image. In the current experiments, do all of the records in the prior belong to the same class as the target record? Similarly, the authors mention that the prior could be motivated by the adversary narrowing down the list of candidates before running the attack. Have the authors explored a list of candidates that are the most similar to the target itself? 3. The research gap addressed by the paper and the significance of the contributions aren’t clear. The empirical results seem solid, but how the findings translate to the real world is unclear. Perhaps the introduction could explain more the real-world implications of the findings. Eg what does it mean that when training a model with epsilon=10, an extremely strong attack (the prior-aware attack) exists that succeeds 60% of the time, 6x better than the random guess (Figure 2)? In light of the results of this paper, does there exist an epsilon such that the newly proposed, state-of-the-art RAs fail but utility remains very good? 4. Clarity: I found Sec. 3.1 and 3.2 hard to follow. Theorem 2 makes some hypotheses but it’s very difficult for me to understand what these mean and whether they’re validated under real-world scenarios. Hopefully, a reviewer with expertise in DP can check these assumptions as well as the proofs, but regardless of this I think the authors should make these sections clearer and more accessible. 5. Tightness of the upper bound: the authors claim that the upper bound is tight, and indeed Fig.2 shows that for epsilon=1, the upper bound and prior-aware attacks perform similarly. However, the upper bound is also very close to the random guess baseline, such that any better than random attack would appear to be tight. I am not convinced that we can conclude that the upper bound is tight, considering the large gaps shown on Figure 3. At least, some analysis or explanations are needed to better understand when the upper bound is tight and when it isn’t. 6. Minor: Can the authors equalize the color maps on Fig. 5 and 15 so that e.g., yellow on the left becomes comparable with yellow on the right?

Questions

1. How sensitive is the empirical upper bound to the parameters of Algorithm 1 used to estimate it, e.g, N? 2. Is the gradient-based attack the first one to successfully reconstruct records from models trained using DP-SGD? The authors state that Balle et al (2022)’s attack doesn’t succeed, but it isn’t clear in the text whether other attacks from prior works succeed. Clarifying this aspect would better highlight the significance of the contribution. 3. Fig. 3 suggests that even with a very strong prior (size=2) there is a gap of up to 5% depending on the value of epsilon, and the gap can be much larger for larger prior size. Can the authors give potential reasons why the prior-aware attack is not tight in these scenarios and potential directions for improving the success of attacks? It seems that the threat model cannot be made stronger, what about the method? Or should future work focus on tightening the theoretical upper bound? 4. Prior-aware attack: Do all of the n candidate records belong to the same class as the target record? The attack could be made artificially easier by selecting candidates with a different label. Similarly, the authors mention that the prior could be motivated by the adversary narrowing down the list of candidates before running the attack. Have the authors explored a list of candidates that are the most similar to the target itself?

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

3 good

Contribution

2 fair

Limitations

The paper doesn’t mention limitations or broader societal impacts but studies the privacy of ML models, a topic with positive societal impact.

Reviewer wroj8/10 · confidence 3/52023-07-06

Summary

The paper proposes a novel bound on Reconstruction Robustness from DP training using a notion of a blow-up function. In order to use the bound, one needs to estimate the quantities $\kappa$ (prior probability of reconstruction) and $\gamma$ (blow-up between neighboring noise distributions) using Monte-Carlo methods. Compared to the prior bounds based on RDP, the proposed bound is tighter. The paper also introduces a new gradient-based that is shown to perform close to the bound, and investigates the impact of different DP-SGD hyperparameters on the bound and on the attack success.

Strengths

- The paper proposes a significant improvement in bounding reconstruction attack risk in DP training. This is of practical importance, as **improved bounds on relevant privacy risks enable practitioners to calibrate their DP-SGD parameters to mitigate those risks** as opposed to relying on the $\epsilon$ value which is hard to interpret. - The paper proposes a new reconstruction attack using gradient information.

Weaknesses

*Approximate nature of the bound in practice.* The computation of the bound relies on a Monte Carlo procedure that is only shown to converge to the optimal solution asymptotically. The paper does not provide methods to estimate the accuracy of the resulting numeric value of the bound for a given number of MC algorithm samples. UPDATE (August 10): I update my score to 8 as the weakness above has been clarified in the response.

Questions

- How can one evaluate the accuracy of the upper bound given inherent error in the Monte Carlo estimates? Is there a way to provide an upper confidence bound at a given level of certainty as opposed to an asymptotic guarantee? Is 1M good enough? - What is the computational cost of estimating $\gamma$ with 1M samples? - Is the notion of blow-up function new? Are there citations? Most bounds in DP rely on information-theoretic divergences --- is it related to a known divergence? - How exactly can the bound be used to reduce the required level of noise in DP-SGD if one were only concerned with reconstruction risks? - How can the attack be applied in federated learning scenarios?

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

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

4 excellent

Limitations

Approximate nature of the bound in practice --- see weaknesses.

Reviewer paNh6/10 · confidence 2/52023-07-27

Summary

This paper is about the bounds of differentially private training for protection against data reconstruction attacks. This paper provides an upper bound on the success of any reconstruction attack against DP-SGD. Also it includes experiments that match the expected bounds. The experiments also include different settings that lead to various success rates for reconstruction.

Strengths

* This paper is well written. It is easy to follow the paper with the background, main theorems (with proofs in supplementary), and experimental settings. * This paper provides sounded proofs for the theorems. * This paper provides enough context on the problems and how they approach the problems differently. * There are sufficient experiments to support the theorems and bounds.

Weaknesses

The WideResNet in the experiments is pretrained on ImageNet. It would be a stronger paper if the authors also perform experiments on ImageNet. I suppose while the bounds should also apply to ImageNet, the higher resolution images might also affect the reconstruction success rate.

Questions

Unfortunately, I'm not the expert in this field. I'm curious to learn from the authors about how important these results (theorems and experimental results) are, as it is not easy to be interpreted from the written contributions.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

The authors are encouraged to discuss how their findings might be used to help develop more sophisticated reconstruction attacks.

Reviewer wroj2023-08-10

Thanks for the response. I strongly recommend re-parameterizing the results in terms of the trade-off function, which is quite standard now. I also strongly recommend incorporating the error bounds at least in the appendix.

Authorsrebuttal2023-08-10

We will incorporate your suggestions in the next iteration of the paper. Thank you for the careful reading of the paper, your useful comments, and also for increasing your score!

Reviewer j4Wo2023-08-14

Response to authors

Thank you for your response. The response brings much needed clarification on the contribution and I strongly recommend that the authors incorporate the “That we can calibrate our bounds based on the knowledge…” and “To be clear, our main contribution is showing that…” responses to the text. This will facilitate the understanding of the paper. To be honest, I was disappointed to see that the authors did not offer to make these changes, as these discussions (and overall framing) are not currently in the paper. The applicability of the algorithm beyond uniform priors should be discussed, e.g., can I get an upper bound if my prior is that with a large probability the car plate starts with an 1 and with a small probability it starts with a different digit, or for a tabular dataset if I only know one-way marginals and correlations. I agree with modifying the tightness claim with nearly tight, and I strongly recommend revising the paper to explain when and why the gap is larger, as well as adding the statement “We would like to note that it is rare for a privacy attack on DP to achieve such close results…” (or a similar statement) to the paper. Similarly, any insights on why the decrease w.r.t. prior size is convex for small epsilon but concave for large epsilon would be interesting. The remaining issue is the attack definition. The framework presented in this paper studies reconstruction attacks that account for an attacker prior. The key novelty of this paper hinges on using the uniform prior over a discrete set of candidates to analyse the privacy of ML models. However, this framework is very general, and the semantics of the work depend strongly on the prior: for instance, for a binary prior, the “reconstruction” attack is actually a MIA, which conflicts with the usual understanding of a reconstruction attack. In that sense, the results in the main paper should be interpreted as a somewhat generalized MIA (which one of ten records is in the data). Similarly, the example given by the authors (car digits) corresponds to an attribute inference attack. This is not necessarily a problem, but it weakens the novelty of the paper and calling it reconstruction is a bit misleading. The fuzzy boundary between these threat models should at least be discussed in the paper.

Authorsrebuttal2023-08-14

Response to Reviewer j4Wo

Thank you for the quick response. We will absolutely incorporate the clarifications into the paper, along with changing the tightness claims. Apologies for not confirming this in our first response. Our framework is flexible enough to handle non-uniform priors. We have a (very brief) comment on this for the attack in Appendix G, but can expand the discussion around non-uniform priors. > However, this framework is very general, and the semantics of the work depend strongly on the prior: for instance, for a binary prior, the “reconstruction” attack is actually a MIA, which conflicts with the usual understanding of a reconstruction attack. In that sense, the results in the main paper should be interpreted as a somewhat generalized MIA (which one of ten records is in the data). Similarly, the example given by the authors (car digits) corresponds to an attribute inference attack. This is not necessarily a problem, but it weakens the novelty of the paper and calling it reconstruction is a bit misleading. The fuzzy boundary between these threat models should at least be discussed in the paper. We completely agree the boundary between membership - attribute - reconstruction is fuzzy and deserves comment, which we will add to the paper. Our framework has two key ingredients, a prior probability of reconstruction, and a function that determines if reconstruction is successful. The framework is flexible enough to incorporate different functions (it could be L2 distance as is used in Haim et al and Balle et al, or it could be choosing a point from a set). As we mentioned, we chose to study "reconstruction" in a setting that gave a lot of power to the adversary, as is standard when modelling attackers in DP, although we do have experiments with L2 in the Appendix. We will add a discussion on the fuzzy boundary between reconstruction and attribute inference, as we agree that these discussions can add a lot of value.

Reviewer rZtg2023-08-16

Thank you for the response!

Reviewer paNh2023-08-20

Thanks to the authors for the response! I don't have further questions.

Area Chair rDfs2023-08-21

Author discussion

Thanks to the authors for their rebuttals and engagement with the author discussion process. In some cases I realise the reviewers have not yet acknowledged these rebuttals - I'd like to assure the authors that the conference has reminded the reviewers of the need to fully engage with author rebuttals, and that there are still opportunities for this to happen during this (almost finished) author discussion period, and then into the reviewer discussion period that comes next. I'd also like to reassure the authors that I've read in full all of the reviews, the rebuttals and all comments to date. I'll focus mostly on concerns/weaknesses (with strengths already noted elsewhere), and of course, with nothing noted being new, there's certainly no expectation of new expectations at this late stage! Some summary highlights include: **paNh** queries evaluation on full ImageNet, avoiding pretraining to which the authors explain real computational challenges, choices that lead to interesting cases (e.g., SOTA accuracy on ImageNet), and there being little advantage in avoiding pretraining. To **wroj**'s query about precise computation of a bound given Monte Carlo estimation, the authors offer the application of two concentration results to provide this w.h.p., and to other questions on computational cost, application to FL, using the bounds to set privacy parameters, the authors address these questions. **j4Wo** probes on the threat model of reconstruction, and a number of important details of the prior-aware attacks, and connections to practice. The authors address many of these queries in a constructive discussion. **rZtg** seeks clarifications and intuition which is addressed by the authors. The discussion has uncovered a number of improvements to the paper's introduction, discussions, etc. that I believe the authors plan to make including those above, trade-off instead of blow-up, etc. I believe these would all improve the paper's presentation and future impact. ### AUNf **W2)** the authors clarify how the new bound was non-trivial to discover with evidence both mathematical (what steps were needed) and historical (improvements over past work in an active area of study) **W3)** the authors motivate the formulation of reconstruction in this discrete/prior setting with a compelling analogy, and pointer to broader experiments; authors clarify that experiments do consider mini-batch and are aligned with existing applications of DP-SGD **W4)** the author's disagree with the reviewer's comments about scope; while I agree with the authors that DP-SGD is of significant interest to the research/practical communities, I'd suggest this is a value judgement **Q1)** I'm enthusiastic about the authors agreeing to write a threat model section. Highlighted by other reviewers, I think this is important. **Q2)** the authors discuss what success has looked like, and use this to motivate the work. I tend to agree with this framing, and see this as another key benefit of clear threat models. **Q3)** thanks to the authors for clarifying details of Sec 3.1, along with clear aims. **L3)** compute cost preview results are much appreciated, promising, and worth adding to the paper in full. Thanks.

Authorsrebuttal2023-08-21

Thank you for your feedback

We'd like to thank you, rDfs, for consolidating the various conversations taking place. We fully agree that an explicit threat model section would help clarify any misunderstandings, and will work to add this to final version along with incorporating all the feedback we've received.

Reviewer AUNf2023-08-21

Thanks for the detailed response. It has effectively addressed the majority of my concerns regarding the novelty and practicality of the attack. I've raised my score accordingly.

Program Chairsdecision2023-09-21

Decision

Accept (poster)

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