Nearly Tight Black-Box Auditing of Differentially Private Machine Learning

Authors: meenatchi sundaram muthu
selva  annamalai
emiliano  de cristofaro
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 05:03:58

Abstract: This paper presents an auditing procedure for the Differentially Private Stochastic Gradient Descent (DP-SGD) algorithm in the black-box threat model that is substantially tighter than prior work.
The main intuition is to craft worst-case initial model parameters, as DP-SGD's privacy analysis is agnostic to the choice of the initial model parameters.
For models trained on MNIST and CIFAR-10 at theoretical $\varepsilon=10.0$, our auditing procedure yields empirical estimates of $\varepsilon_{emp} = 7.21$ and $6.95$, respectively, on a 1,000-record sample and $\varepsilon_{emp} = 6.48$ and $4.96$ on the full datasets.
By contrast, previous audits were only (relatively) tight in stronger white-box models, where the adversary can access the model's inner parameters and insert arbitrary gradients.
Overall, our auditing procedure can offer valuable insight into how the privacy analysis of DP-SGD could be improved and detect bugs and DP violations in real-world implementations.
The source code needed to reproduce our experiments is available from https://github.com/spalabucr/bb-audit-dpsgd.

Main paper: 
 introduction: 
 Differentially Private Stochastic Gradient Descent (DP-SGD) [1] allows training machine learning models while providing formal Differential Privacy (DP) guarantees [15] . DP-SGD is a popular algorithm supported in many open-source libraries like Opacus [40] , TensorFlow [19] , and JAX [4] . Progress in privacy accounting and careful hyper-parameter tuning has helped reduce the gap in model performance compared to non-private training [11] . Furthermore, pre-training models on large public datasets and then fine-tuning them on private datasets using DP-SGD can also be used to improve utility [11] .Implementing DP-SGD correctly can be challenging -as with most DP algorithms [5, 26] -and bugs that lead to DP violations have been found in several DP-SGD implementations [7, 22] . These violations do not only break the formal DP guarantees but can also result in realistic privacy threats [17, 33] . This prompts the need to audit DP-SGD implementations to verify whether the theoretical DP guarantees hold in practice. This usually involves running membership inference attacks [32] , whereby an adversary infers whether or not a sample was used to train the model. The adversary's success estimates the empirical privacy leakage from DP-SGD, and that is compared to the theoretical DP bound. If the former is "far" from the latter, it means that the auditing procedure is loose and may not be exploiting the maximum possible privacy leakage. Whereas, when empirical and theoretical estimates are "close," auditing is tight and can be used to identify bugs and violations [28] .Although techniques to tightly audit DP-SGD exist in literature [28, 29] , they do so only in active white-box threat models, where the adversary can observe and insert arbitrary gradients into the intermediate DP-SGD steps. By contrast, in the more restrictive black-box threat model, where an adversary can only insert an input canary and observe the final trained model, prior work has only achieved loose audits [20, 28, 29] . In fact, prior work has suggested that the privacy analysis of DP-SGD in the black-box threat model can be tightened but only in limited settings (i.e., strongly convex smooth loss functions) [3, 9, 39] . Therefore, it has so far remained an open research question [28] whether or not it is possible to tightly audit DP-SGD in the black-box model.In this paper, we introduce a novel auditing procedure that achieves substantially tighter black-box audits of DP-SGD than prior work, even for natural (i.e., not adversarially crafted) datasets. The main intuition behind our auditing procedure is to craft worst-case initial model parameters, as DP-SGD's privacy analysis is agnostic to the choice of initial model parameters. We show that the gap between worst-case empirical and theoretical privacy leakage in the black-box threat model is much smaller than previously thought [20, 28, 29] . When auditing a convolutional neural network trained on MNIST and CIFAR-10 at theoretical ε = 10, our audits achieve an empirical privacy leakage estimate of ε emp = 6.48 and 4.96, respectively, compared to ε emp = 3.41 and 0.69 when the models were initialized to average-case initial model parameters, as done in prior work [11, 20, 28] .In the process, we identify and analyze the key factors affecting the tightness of black-box auditing: 1) dataset size and 2) gradient clipping norm. Specifically, we find that the audits are looser for larger datasets and larger gradient clipping norms. We believe this is due to more noise, which reduces the empirical privacy leakage. When considering smaller datasets with 1,000 samples, our audits are tighter -for ε = 10.0, we obtain ε emp = 7.21 and 6.95, respectively, for MNIST and CIFAR-10.Finally, we present tight audits for models with only the last layer fine-tuned using DP-SGD. Specifically, we audit a 28-layer Wide-ResNet model pre-trained on ImageNet-32 [11] with the last layer privately fine-tuned on CIFAR-10. With ε = 10.0, we achieve an empirical privacy leakage estimate of ε emp = 8.30 compared to ε emp = 7.69 when the models are initialized to average-case initial parameters (as in previous work [11, 20, 28] ).Overall, the ability to perform rigorous audits of differentially private machine learning techniques allows shedding light on the tightness of theoretical guarantees in different settings. At the same time, it provides critical diagnostic tools to verify the correctness of implementations and libraries.2 Related Work DP-SGD. Abadi et al. [1] introduce DP-SGD and use it to train a feed-forward neural network model classifying digits on the MNIST dataset. On the more complex CIFAR-10 dataset, they instead fine-tune a pre-trained convolutional neural network, incurring a significant performance loss. Tramèr and Boneh [35] improve on [1] using hand-crafted feature extraction techniques but also suggest that end-to-end differentially private training may be inherently difficult without additional data. Papernot et al. [30] and Dörmann et al. [14] also present improvements considering different activation functions and larger batch sizes. Finally, De et al. [11] achieve state-of-the-art model performance on CIFAR-10 both while training from scratch and in fine-tuning regimes; they do so by using large batch sizes, augmenting training data, and parameter averaging. Our work focuses on auditing realistic models with high utility that might be used in practice; thus, we audit models with utility similar to [11] and the same models pre-trained in [11] in the fine-tuning regime.Membership Inference Attacks (MIAs). Shokri et al. [32] propose the first black-box MIAs against machine learning models using shadow models. Carlini et al. [6] improve by fitting a Gaussian distribution on the loss of the target sample and using fewer models to achieve equivalent attack accuracies. Sablayrolles et al. [31] consider per-example thresholds to improve the performance of the attacks in a low false-positive rate setting [6] . By factoring in various uncertainties in the MIA "game," Ye et al. [38] construct an optimal generic attack. Concurrent work [16, 25, 37] use poisoned pre-trained models, which is somewhat similar to our approach of using worst-case initial model parameters, but with a different focus. Specifically, Liu et al. [25] and Wen et al. [37] show that privacy leakage from MIAs is amplified when pre-trained models are poisoned but, crucially, do not consider DP auditing. Additionally, while Feng and Tramèr [16] achieve tight audits with poisoned pre-trained models, their strategy is geared towards the last-layer only fine-tuning regime. By contrast, we focus on auditing DP-SGD with full model training to estimate the empirical worst-case privacy leakage in the black-box setting.Auditing DP-SGD. Jayaraman and Evans [21] derive empirical privacy guarantees from DP-SGD, observing a large gap from the theoretical upper bounds. Jagielski et al. [20] use data poisoning in the black-box model to audit Logistic Regression and Fully Connected Neural Network models and produce tighter audits than using MIAs. They initialize models to fixed (average-case) initial parameters, but the resulting empirical privacy guarantees are still far from the theoretical bounds. They also adapt their procedure to the fine-tuning regime, again producing loose empirical estimates.Nasr et al. [29] are the first to audit DP-SGD tightly; they do so by using adversarially crafted datasets and active white-box adversaries that insert canary gradients into the intermediate steps of DP-SGD. In follow-up work, [28] presents a tight auditing procedure also for natural datasets, again in the active white-box model. Zanella-Béguelin et al. [42] present tighter auditing procedures using Bayesian techniques, needing fewer models to audit larger privacy levels. Finally, De et al. [11] perform black-box auditing of their implementation of DP-SGD in the JAX framework [4] ; they also use fixed (average-case) initialization of models as in [20] , similarly achieving loose audits.Overall, while prior work [11, 20, 29] showed that reducing the randomness in the initial model parameters results in tighter empirical privacy leakage estimates, the audits remained loose as the fixed model parameters were still initialized randomly. Specifically, designing an effective adversarial strategy that provides tight audits in the black-box setting without destroying utility has remained, to the best of our knowledge, an open problem. 
limitations: 
 Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We cover the main limitation of our work in the Conclusion section. Specifically, we discuss that tight black-box audits need hundreds of models to be trained, which forms a computational limitation for our approach. We also highlight some advancements in auditing that require much fewer models, but unfortunately they are not applicable to our work as those techniques do not achieve tight audits. Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.• The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions). • The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.) • The assumptions made should be given (e.g., Normally distributed errors). • It should be clear whether the error bar is the standard deviation or the standard error of the mean. Justification: We discuss how as DP tools and libraries are increasingly used in production settings, performing such audits becomes increasingly necessary, to precisely prevent societal harm, in the Introduction.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
differential privacy (dp): 
 Definition 1 (Differential Privacy (DP) [15] ). A randomized mechanism M : D → R is (ε, δ)differentially private if, for any two neighboring datasets D, D ′ ∈ D and S ⊆ R, it holds:Pr[M(D) ∈ S] ≤ e ε Pr[M(D ′ ) ∈ S] + δPut simply, DP guarantees a formal upper bound (constrained by the privacy parameter ε) on the probability that any adversary observing the output of the mechanism M can distinguish between two neighboring inputs to M -i.e., two datasets differing in only one record.DP-SGD. Differentially Private Stochastic Gradient Descent (DP-SGD) [1] is a popular algorithm for training machine learning models while satisfying DP. In this paper, we consider classification tasks on samples from the domain X × Y, where X is the features domain and Y is the labels domain. DP-SGD, reviewed in Algorithm 1, introduces two hyper-parameters: gradient clipping norm C and noise multiplier σ. Typically, C is set to 1, while σ is calibrated to the required privacy level (ε, δ) based on the number of iterations, T , and batch size, B. Early DP-SGD work [1] set batch sizes to be small and similar to those of non-private SGD; however, recent work has highlighted the advantage of setting large batch sizes [11, 14] by training state-of-the-art models with them. Following this trend, and to ease auditing, we set B = N , as also done in prior work [28] . Finally, while the initial set of model parameters θ 0 is typically sampled randomly (e.g., using Xavier initialization [18] ), it can also be set arbitrarily (and even adversarially), without affecting the privacy guarantees provided by DP-SGD.Fine-tuning regime. In this setting, one first pre-trains a model on a (large) public dataset and then fine-tunes it on the private data. When a suitable public dataset is available, prior work has shown that fine-tuning can achieve much higher utility than training from scratch using DP-SGD [11] .There are two methods for fine-tuning: doing it for all layers or only the last one. In both cases, the initial model parameters, θ 0 , are set to the pre-trained model parameters (rather than randomly) before running DP-SGD. However, while the former updates all model parameters during DP-SGD, with the latter, the model parameters until the last (fully connected) layer are "frozen" and not updated during DP-SGD. This setting treats the pre-trained model as a feature extractor and trains a Logistic Algorithm 1 Differentially Private Stochastic Gradient Descent (DP-SGD) [ θt+1 ← θt -ηgt 9: end for 10: return θT Regression model using DP-SGD based on the extracted features. As prior work shows [11] that this setting produces models with accuracy closer to the non-private state of the art (while also incurring smaller computational costs), we also do fine-tuning of just the last layer.As mentioned above, DP provides a theoretical limit on the adversary's ability -bounded by the privacy parameter ε -to distinguish between M(D) and M(D ′ ). With auditing, one assesses this limit empirically to verify whether the theoretical guarantees provided by M are also met in practice. There may be several reasons this does not happen, e.g., the privacy analysis of M is not tight [29] or because of bugs in the implementation of M [28, 36] . Therefore, with DP mechanisms increasingly deployed in real-world settings, auditing them becomes increasingly important.Deriving ε emp . When auditing a DP mechanism M, one runs M repeatedly on two neighboring inputs, D and D ′ , R times each and the outputs(O = {o 1 , ..., o R } and O ′ = {o ′ 1 , ..., o ′ R }) are given to an adversary A. Next, A tries to distinguish between O and O ′ , which determines a false positive rate (FPR) and false negative rate (FNR). For a given number of outputs 2R and confidence level α, empirical upper bounds FPR and FNR can be computed using Clopper-Pearson confidence intervals [29, 28] . Finally, the empirical upper bounds are converted to an empirical lower bound ε emp using the privacy region of M (see below) and ε emp can be compared to the theoretical ε. We consider an audit tight if ε emp ≈ ε. To ease presentation, we abstract away the details and use ε emp ← EstimateEps(FPR, FNR, 2R, α, δ) to denote the process of estimating an ε emp from a given FPR and FNR. 1Privacy region of DP-SGD. Given a mechanism M, the privacy region of the mechanism, R M (ε, δ) = {(FPR, FNR)|•} defines the FPR and FNR values attainable for any adversary aiming to distinguish between M(D) and M(D ′ ). Nasr et al. [28] note that the privacy region of DP-SGD corresponds (tightly) to that of µ-Gaussian Differential Privacy [13] , which can be used to first compute a lower bound on µ:EQUATIONThen, a lower bound on µ corresponds to a lower bound on ε by means of the following theorem:Theorem 1 (µ-GDP to (ε, δ)-DP conversion [13]). A mechanism is µ-GDP iff it is (ε, δ(ε))-DP for all ε ≥ 0, where:EQUATION4 Auditing ProcedureIn this section, we present our auditing procedure. We discuss the threat model in which we operate, the key intuition for using worst-case initial parameters, and how we craft them to maximize the empirical privacy leakage. : Sample D ∼ D n-1 2: Craft target sample (xT , yT ) 3: D ′ = D ∪ {(xT , yT )} 4: Observations O ← {}, O ′ ← {} 5: for r ∈ [R] do 6: θr ← DP-SGD(D; θ0, ℓ) 7: θ ′ r ← DP-SGD(D ′ ; θ0, ℓ) 8: O[t] ← ℓ(θr; (xT , yT )) 9: O ′ [t] ← ℓ(θ ′ r ; (xT , yT )) 10: end for 11: FPR ← 1 R |{o|o ∈ O, o ≥ τ }| 12: FNR ← 1 R |{o|o ∈ O ′ , o < τ }| 13: εemp ← EstimateEps(FPR, FNR, 2R, α, δ) 14: return εempThreat model. We consider a simple black-box model where the adversary only sees the final model parameters, θ T , from DP-SGD. Specifically, unlike prior work on tight audits [28, 29] , the adversary cannot observe intermediate model parameters or insert canary gradients. We do so to consider more realistic adversaries that may exist in practice. Additionally, we assume that the adversary can choose a worst-case target sample as is standard for auditing DP mechanisms [20] . Finally, we assume that the adversary can choose (possibly adversarial) initial model parameters. Note that this is not only an assumption allowed by DP-SGD's privacy analysis, but, in the context of the popular public pre-training, private fine-tuning regime [11] , this is also a reasonable assumption.Auditing. Our DP-SGD auditing procedure is outlined in Algorithm 2. As we operate in the black-box threat model, the procedure only makes a call to the DP-SGD algorithm-once again, they cannot insert canary gradients, etc. As mentioned, the adversary can set the initial model parameters (θ 0 ), which is fixed for all models. Finally, we assume that the final model parameters are given to the adversary, who calculates the loss of the target sample as "observations." These are then thresholded to compute FPRs and FNRs and estimate the ε emp using the EstimateEps process (see Section 3.3). This threshold, which we denote as τ , must be computed on a separate set of observations (e.g., a validation set) for the ε emp to constitute a technically valid lower bound. However, as any decision threshold for GDP is equally likely to maximize ε emp [28] , we follow common practice [27, 29, 28] to find the threshold that maximizes the value ofε emp from O ∪ O ′ .Crafting worst-case initial parameters. Recall that DP-SGD's privacy analysis holds not only for randomly initialized models but also for arbitrary fixed parameters. Indeed, this was noted in prior work [11, 20, 28] and led to using initial model parameters that are fixed for all models. In our work, we consider worst-case, rather than average-case, initial parameters that minimize the gradients of normal samples in the dataset. This makes the target sample's gradient much more distinguishable. To do so, we pre-train the model using non-private SGD on an auxiliary dataset that follows the same distribution as the target dataset. Specifically, for MNIST, we pre-train the model on half of the full dataset (reserving the other half to be used for DP-SGD) for 5 epochs with a batch size of 32 and a learning rate of 0.01. For CIFAR-10, we first pre-train the model on the CIFAR-100 dataset for 300 epochs with batch size 128 and a learning rate of 0.1. Then, we (non-privately) fine-tune the model on half of the full dataset for 100 epochs, with a batch size of 256 and a learning rate of 0.1. Ultimately, we show that the gradients of normal samples are indeed minimized, achieving significantly tighter audits of DP-SGD in the black-box model. 
experiments: 
 In this section, we evaluate our auditing procedure using image datasets commonly used to benchmark differentially private classifiers and CNNs that achieve reasonable utility on these datasets. Table 1 : Accuracy of models for average-case initial model parameters (θ 0 ).Due to computational constraints, we only train 2R = 200 models (100 with the target sample and 100 without) to determine the empirical privacy estimates ε emp in the full model training setting and train 2R = 1000 models for the last-layer only fine-tuning setting. In theory, we could achieve tighter audits by increasing the number of models, however, this would require significantly more computational resources and time (see below). Since our work focuses on the (broader) research questions of whether tight audits are possible in the black-box setting and how various factors affect this tightness, we believe 2R = 200 provides a reasonable tradeoff between tightness and computational overhead.In our evaluation, we report lower bounds with 95% confidence (Clopper-Pearson [10] ) and report the mean and standard deviation values of ε emp over five independent runs. 2Datasets. We experiment with the MNIST [24] and CIFAR-10 [23] datasets. The former includes 60,000 training and 10,000 testing 28x28 grayscale images in one of ten classes (hand-written digits), Often referred to as a "toy dataset," is commonly used to benchmark DP machine learning models. CIFAR-10 is a more complex dataset containing 50,000 training and 10,000 testing 32x32 RGB images, also consisting of ten classes. Note that the accuracy of DP models trained on CIFAR-10 has only recently started to approach non-private model performance [11, 35] . Whether on MNIST or CIFAR-10, prior black-box audits (using average-case initial parameters) have not been tight. More precisely, for theoretical ε = 8.0, De et al. [11] and Nasr et al. [28] only achieve empirical estimates of ε emp ≈ 2 and ε emp ≈ 1.6, respectively, on MNIST and CIFAR-10.To ensure a fair comparison between the average-case and worst-case initial parameter settings, we split the training data in two and privately train on only half of each dataset (30,000 images for MNIST and 25,000 for CIFAR-10). The other half of the training dataset is used as the auxiliary dataset to non-privately pre-train the models and craft the worst-case initial parameters (see Section 4). We note that for CIFAR-10, we additionally augment the pre-training dataset with the CIFAR-100 dataset to boost the model accuracy further (see Appendix A for further details).When training the model fully, we use a (shallow) Convolutional Neural Network (CNN) for both datasets. We do so as we find accuracy to be poor with models like Logistic Regression and Fully Connected Neural Networks, and training hundreds of deep neural networks is impractical due to large computational cost. Therefore, we draw on prior work by Dörmann et al. [14] and train shallow CNNs that achieve acceptable model accuracy while being relatively fast to train. (We report exact model architectures in Appendix A). Whereas, for the last layer-only fine-tuning setting, we follow [11] and use the Wide-ResNet (WRN-28-10) [41] model pre-trained on ImageNet-32 [12] .Experimental Testbed. All our experiments are run on a cluster using 4 NVIDIA A100 GPUs, 64 CPU cores, and 100GB of RAM. Auditing just one model took approximately 16 GPU hours on MNIST and 50 GPU hours on CIFAR-10.To train models with an acceptable accuracy on MNIST and CIFAR-10, we first tune the hyperparameters (i.e., learning rate η and number of iterations T ); due to space limitations, we defer discussion to Appendix A.Model Accuracy. In Table 1 , we report the accuracy of the final models for the different values of ε-s we experiment with. While accuracy on MNIST is very good (≥ 95% for all ε), for CIFAR-10, we are relatively far from the state of the art (≈50% at ε = 10.0 vs. ≈80% at ε = 8.0 in [11] ). This is due to two main reasons. First, as we need to train hundreds of models, we are computationally limited to using shallow CNNs rather than deep neural networks. Second, we need to consider a much larger batch size to ease auditing [28] ; thus, the models converge much more slowly. In Appendix B, we show that, given enough iterations (T = 1,000), the model reaches an accuracy of ≈70% at ε = 10.0, almost approaching state-of-the-art. As a result, we audit models striking a reasonable trade-off between utility and computational efficiency, as discussed above.In Figure 1 , we compare the difference in ε emp between worst-case initial parameters and average-case parameters (as done in prior work [11, 20, 28] ). In the latter, all models are initialized to the same, randomly chosen θ 0 (using Xavier initialization [18] ), while, in the former, to a θ 0 from a model pre-trained on an auxiliary dataset. In the rest of this section, the target sample is set as the blank sample, which is known to produce the tightest audits in previous work [11, 28] .With worst-case initial parameters, we achieve significantly tighter audits for both MNIST and CIFAR-10. Specifically, at ε = 1.0, 2.0, 4.0, 10.0, the empirical privacy leakage estimates reach ε emp = 0.08, 0.54, 1.89, 6.48 and ε emp = 0.00, 0.18, 1.43, 4.96 for MNIST and CIFAR-10, respectively. In comparison, with average-case initial parameters as in prior work, the empirical privacy leakage estimates were significantly looser, reaching only ε emp = 0.00, 0.06, 0.61, 3.41 and ε emp = 0.07, 0.02, 0.20, 0.69 for MNIST and CIFAR-10, respectively. Notably, at larger privacy levels ε = 4.0 and ε = 10.0, for both MNIST and CIFAR-10, the empirical privacy leakage estimates when auditing with worst-case initial parameters are not only larger than the average-case, but are outside of the ± s.d. range. This shows that our auditing procedure not only produces significantly tighter audits than prior work, especially at larger εs, but also that, at lower ε-s, our evaluations are currently limited by the number of models used to audit (resulting in large s.d.). To shed light on why worst-case initial parameters improve tightness, we vary the amount of pretraining done to craft θ 0 and, in Figure 2 , plot the corresponding ε emp -s and the average gradient norm of all samples (after clipping) in the first iteration of DP-SGD. For computational efficiency, we focus on MNIST and set the theoretical ε to 10.0. Recall that, regardless of the initial model parameters, all models achieve ≥ 95% accuracy on the test set. As the number of pre-training epochs increases from 0 (no pre-training) to 5, the average gradient norm steadily decreases from 1.00 to 0.51, as expected. On the other hand, the empirical privacy leakage increases from 3.41 to 6.48. In other words, with more pre-training, the impact of the other samples in the dataset reduces, making the target sample much more distinguishable, thus yielding tighter audits. Although there appears to be an anomaly in the ε emp obtained when the model is pre-trained for 3 epochs, this is within the standard deviation.Remarks. With lower ε values, the empirical ε emp estimates are still relatively far away from the theoretical bounds. This is because, in each run, we train a small number of models and report the average ε emp across all five runs. In fact, the maximum ε emp obtained over the five runs amount to ε emp = 0.27, 1.48, 3.23, 8.28 and ε emp = 0.00, 0.54, 2.33, 6.70, respectively, on MNIST and CIFAR-10, for ε = 1.0, 2.0, 4.0, 10.0. Nevertheless, even on the averages, our audits are still appreciably tighter than prior work-approximately by a factor of 3 for MNIST [11] and CIFAR-10 [28] . Overall, this confirms that the tightness of the audits can vary depending on the datasets, possibly due to differences in the difficulty of the classification task on the datasets. For instance, MNIST classification is known to be easy, with DP-SGD already reaching close to non-private model performance, even at low ε-s. By contrast, CIFAR-10 is much more challenging, with a considerable gap still existing between private and non-private model accuracy [11] .Impact of size of dataset. In the auditing literature, empirical privacy leakage is often commonly estimated on (smaller) dataset samples both to achieve tighter audits and for computational efficiency reasons [11] . In Figure 3 , we evaluate the impact of the dataset size (n) on the tightness of auditing. While smaller datasets generally yield tighter audits, the size's impact also depends on the dataset itself. For instance, at ε = 10.0, auditing with n = 100 results in ε emp = 7.41 and 7.01, for MNIST and CIFAR-10, respectively, compared to ε emp = 6.48 and 4.96 on the full dataset. In other words, with MNIST, the dataset size does not significantly affect tightness. However, for CIFAR-10, smaller datasets significantly improve the audit's tightness significantly.Sensitivity to clipping norm. Finally, we evaluate the impact of the hyper-parameters on the tightness of black-box audits. In the black-box setting, unlike in white-box, the adversary cannot arbitrarily insert gradients that are of the same scale as the gradient clipping norm, C [28] . Thus, they are restricted to gradients that naturally arise from samples. In Figure 4 , we report the ε emp values obtained with varying clipping norms, C. For computational efficiency, the results on CIFAR-10 are for models trained on datasets of size n = 1,000.When the gradient clipping norm is small (C = 0.1, 1.0), the audits are tighter. This is because the gradient norms are typically very small in the worst-case initial parameter setting, leading to a much higher "signal-to-noise" ratio. While this might suggest that black-box audits may not be as effective with larger gradient clipping norms, we observe that, with C = 10.0, model accuracy is poor too-more precisely, with ε = 1.0, 2.0, 4.0, 10.0, accuracy reaches 85.6%, 93.1%, 95.0%, 96.1%, respectively, compared to 97.2%, 97.4%, 97.5%, 97.5% with C = 1.0 on MNIST. This indicates that audits are looser with C = 10.0 because the model is not learning effectively, as there is too much noise. In fact, this could suggest that the privacy analysis for the C = 10.0 setting could be further tightened in the black-box threat model, which we leave to future work.We now experiment with the setting where a model is first non-privately pre-trained on a large public dataset, and only the last layer is fine-tuned on the private dataset. For this setting, we choose the WRN-28-10 model pre-trained on ImageNet-32 and fine-tune the last layer on CIFAR-10 using DP-SGD, as done in [11] . At ε = 1.0, 2.0, 4.0, 10.0, fine-tuning the last layer yields model accuracies of 91.3%, 91.4%, 91.4%, 91.5% and 93.1%, 93.2%, 93.3%, 93.2% for average-case and worst-case initial model parameters, respectively. In this section, we run the ClipBKD procedure from [20] to craft a target sample, as it has been shown to produce tighter empirical privacy guarantees for Logistic Regression models.Once again, we compare the tightness of the audits with respect to the initial model parameters. In Figure 5 , we report the empirical privacy leakage estimates obtained at various levels of ε when the last layer is initialized to average-and worst-case (through pre-training) initial model parameters.Similar to the full model training setting, model initialization can impact audit tightness, albeit not as significantly. Specifically, at ε = 10.0, we obtain ε emp = 9.19 and 9.33, respectively, when the last layers are initialized to average-and worst-case initial model parameters.Overall, the audits in this setting are considerably tight, and increasingly so for larger ε, as we obtain empirical privacy leakage estimates of ε emp = 0.67, 1.62, 3.64, 9.33 for ε = 1.0, 2.0, 4.0, 10.0, respectively. The maximum ε emp across five independent runs is also much tighter, namely, ε emp = 1.37, 2.10, 4.28, 9.94 for ε = 1.0, 2.0, 4.0, 10.0, respectively. Although maximum ε emp exceeds the theoretical ε at 1.0, 2.0, and 4.0 suggests a DP violation, this is, in fact, within the standard deviation of ±0.40, 0.36, and 0.38, respectively.We also evaluated the impact of the dataset size and clipping norm in this setting as well; however, these factors do not significantly impact audit tightness, as this is a much simpler setting compared to training a CNN model fully. For more details, please see Appendix C. 
conclusion: 
 This paper presented a novel auditing procedure for Differentially Private Stochastic Gradient Descent (DP-SGD). By crafting worst-case initial model parameters, we achieved empirical privacy leakage estimates substantially tighter than prior work for DP-SGD in the black-box model and for natural (i.e., not adversarially crafted) datasets. At ε = 10.0, we achieve nearly tight estimates of ε emp = 7.21 and 6.95 for datasets consisting of 1,000 samples from MNIST and CIFAR-10, respectively. While we achieve slightly weaker estimates of ε emp = 6.48 and 4.96 on the full datasets, these still represent a roughly 3x improvement over the estimates achieved in prior work [11, 28] .Naturally, our work is not without limitations -the main one being the computational cost of auditing. Black-box auditing typically requires training hundreds of models to empirically estimate FPRs/FNRs with good accuracy and confidence. This can take hundreds of GPU hours, even for the shallow CNNs we audit in our work. Thus, auditing deep neural networks (e.g., WideResNet) trained on large datasets (e.g., ImageNet) may be computationally challenging for entities that are not huge corporations. However, as our main objective is addressing the open research question of whether tight DP-SGD audits are feasible in the black-box model, we focus on shallow models. Our results are very promising as, thanks to the intuition of using worst-case initial parameters, we do achieve nearly tight audits. Nevertheless, we leave it to future work to reduce the computational cost of the auditing procedure, e.g., by training significantly fewer models.Relatedly, recent work [2, 34] has begun to focus on auditing within a single training run, although it has thus far achieved only loose empirical estimates. One interesting direction would be to combine the insights from this paper (i.e., using worst-case initial model parameters) with one-shot auditing techniques. However, this may not be trivial as these techniques typically employ a large number of canary samples, each with large gradient norms, which can potentially interfere with our goal to reduce the gradient norm of "other" samples.Furthermore, we only consider full batch gradient descent (B = n), i.e., without sub-sampling. Note that this is standard practice and eases auditing by enabling GDP auditing, which requires fewer models and is much less computationally intensive compared to auditing DP-SGD with sub-sampling using Privacy Loss Distributions (PLDs) [28] . Nevertheless, when auditing DP-SGD with subsampling in the "hidden-state" threat model where the adversary can only observe the final trained model but can insert gradient canaries, recent work has shown that there is a significant gap between the theoretical upper bound and the empirical lower bound privacy leakage achieved [8] . Even though this suggests a privacy amplification phenomenon in the hidden-state setting (which would extend to black-box as well), for general non-convex loss functions, prior work has also shown that the privacy analysis of DP-SGD with sub-sampling is tight even in the hidden-state [3] . Therefore, whether audits can be tight in the black-box threat model for DP-SGD with sub-sampling under realistic loss functions and datasets remains an open question for future work.  
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pre-trained language models, image generators, or scraped datasets)?Answer: [NA] Justification: We do not attack any live systems, or propose a new "attack". We use existing attacks to empirically evaluate the privacy of open algorithms using open datasets.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: We cite the original paper for the MNIST and CIFAR-10 datasets used in our work.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 
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 Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?Answer: [NA] Justification: We do not release any new datasets or models with our work.Guidelines:• The answer NA means that the paper does not release new assets.• Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc. • The paper should discuss whether and how consent was obtained from people whose asset is used. • At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file. 
crowdsourcing and research with human subjects: 
 Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?Answer: [NA] Justification: Our work does not involve human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects. • Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper. • According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector. 
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 Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?Answer: [NA] Justification: Our work does not involve human subjects.Guidelines:• The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.• Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper. • We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution. • For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review. 
 
 Appendix: In this work, we use the shallow Convolutional Neural Networks used by Dörmann et al. [14] , who make minor modifications to the CNNs used earlier by Tramèr and Boneh [35] to achieve good model utilities on MNIST and CIFAR-10. Exact model architectures for MNIST and CIFAR-10 are reported in Tables 2 and 3 , respectively. Hyper-parameter tuning. To achieve the best model utilities, we tune the hyper-parameters of DP-SGD (i.e., number of iterations, T and learning rate, η). For MNIST, we train for T = 100 iterations, with a learning rate of η = 4 for both average-case and worst-case initial model parameters. For CIFAR-10, we train for T = 200 iterations, with a learning rate of η = 2 for the average-case initial model parameter setting. For the worst-case initial model parameter setting, we use a learning rate of η = 1 instead, as this resulted in the best model performance for CIFAR-10. When fine-tuning just the last layer, we train for T = 25 iterations, with a learning rate of 10.0. Note that, for the experiments varying the dataset size, the learning rate was scaled accordingly so that the size of the dataset does not affect the "influence" of each sample (i.e., η B ). All clipping norms are set to C = 1.0 as done in prior work [11] , and batch sizes are set to the dataset size B = n to ease auditing [28] . Finally, the noise multiplier σ is calculated from the batch size and number of iterations using the Privacy loss Random Variable accountant provided by Opacus [40] .In Figure 6 , we plot the test accuracies obtained when training the CNN model on the CIFAR-10 dataset for varying number of iterations, T , and learning rates, η, at a fixed theoretical ε = 10.0. Specifically, we train with full batch gradient descent (i.e., B = n) and re-calculate the noise First, we observe that η = 2.0 and η = 4.0 result in comparable test accuracies that increase with the number of iterations, reaching close to 70% accuracy at T = 1,000 (67.3% and 65.4%, respectively). Therefore, even though at T = 200 and ε = 10.0, the models we audit only achieve 53.6% accuracy, this is a computational limitation, and the model can, in fact, achieve good model utilities given enough iterations. However, increasing the learning rate to η = 10.0 results in poor model performance.In their original work, Dörmann et al. achieve a test accuracy of 70.1% at ε = 7.42 by training the model with a batch size of 8,500 (sampling rate, q = 0.17). However, following prior work [28] , we choose not to consider sub-sampling as this makes tight auditing difficult. This is because the theoretical privacy region of the sub-sampled Gaussian mechanism (the underlying mechanism for DP-SGD with sub-sampling) is loose compared to the empirical privacy region observed, thus making audits in this setting inherently weaker.Finally, we evaluate the impact of the clipping norm and dataset size on the tightness of auditing in the last layer only fine-tuning setting. Overall, we find that these factors have little to no impact, as tight audits are achieved in the most difficult settings considered.In Figures 7a and 7b , we report the empirical privacy estimates obtained when using different gradient clipping norms and when training on datasets with different sizes, respectively. While there may be some minor differences between the empirical ε emp obtained in different settings, overall, the gradient clipping norm and dataset size do not significantly affect tightness. We believe this is because it is much simpler to fine-tune the last (Logistic Regression) layer of a CNN than to train the full model. Here, the empirical privacy leakage is already maximized (ε emp ≈ 10.0 for theoretical ε = 10.0) in the most difficult setting (C = 10.0, n = |D|). Therefore, "relaxing the setting" by using smaller clipping norms and dataset sizes does not significantly improve the privacy leakage.1. Claims Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? Answer: [Yes] Justification: The main claim of our paper is that we achieve nearly tight audits of DP-SGD in the black-box setting. Indeed, for a theoretical ε = 10.0, we achieve empirical privacy leakage values that are quite close, especially in settings where the number of samples in the dataset is low. Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.Please refer to https://github.com/spalabucr/bb-audit-dpsgd for details.The source code needed to reproduce our experiments is available from https://github.com/spalabucr /bb-audit-dpsgd. 


Decision: The reviews are generally positive, and the overall sentiment further improved slightly following the author rebuttal and discussion. There was a debate as to whether the limited technical novelty (running existing auditing techniques on pre-trained models) should be a barrier for acceptance. In the end, the consensus was that the work has some value by evaluating a simple idea which leads to significant practical gains for privacy auditing. Therefore, the paper is accepted.

Important note: the authors have promised to tone down their claims of “near tight” audits to specify in which settings the audits are appreciably tighter than prior work, as requested by some reviewers. Additionally, they have been asked to clearly discuss the limitation of auditing only DP-GD and not DP-SGD (i.e., without subsampling). I therefore ask the authors to incorporate these changes in the final version.

Reviews: review: paper_summary: paper_summary: The paper shows that privacy auditing is tight in the threat model where the model initialization is adversarially selected. They find that the decrease in gradient norms over the course of training helps improve the "signal to noise" ratio of the auditing example.
strength_weakness: strength_weakness: The general finding that "when other examples' contributions are smaller, auditing is tighter", has been a feature of other papers on auditing, even in the adversarial dataset experiments from Nasr et al. 2021 or the design of ClipBKD in Jagielski et al 2020.

The paper's claims of "near tightness" seem somewhat exaggerated - there are several parameter settings where the proposed auditing does not even appear to be better than prior work, and all experiments show a rather large gap.

Given that the batch size is an important detail for auditing, the setting of B = n should be mentioned in the main body of the paper.

The experiments are unfortunately on rather small models, making it difficult to know how generalizable the strategy is to more recent advances in DP-ML training.

I would appreciate experiments on other datasets - several conclusions are drawn that seem dataset specific (e.g. dataset size), and it would be useful to know if there are more general trends in these.
questions: questions: What is the gradient norm of the auditing example in your experiments? I wonder if at C=10, the gradient norm is not large enough to "max out" the full clipping norm.

Did you retune hyperparameters when changing the values of clipping norms?

Given the hypothesis that lower gradient norm of other examples leads to tighter audits, have you tried explicitly constructing an initialization where the other gradients are exactly 0? Perhaps even just nonprivate SGD on the private training data will get there.
limitations: limitations: I think the batch size should be better highlighted, but limitations related to model architectures, datasets, auditing trials are all well addressed.
paper_summary: paper_summary: This paper proposes a new method for auditing DP-SGD in a black-box setting, where the auditor can only see the final parameters of the model (rather than intermediate steps). The main idea is to select worst-case initial model parameters; this seems to give a substantial advantage to the black-box analysis.
strength_weakness: strength_weakness: # Motivations
I think the authors should refine their motivations. They use two: 1) their method is good for detecting bugs, and 2) it provides insights into DP-SGD analyses.

1. Bugs.

The ultimate purpose of _black-box_ auditing is unclear. There's two sides to this discussion. On the one hand, if our goal is to audit a DP-SGD implementation, surely we should be making all the worst-case assumptions (i.e., give as much advantage to the auditor as possible) to look for bugs. On the other hand, if our goal is to evaluate real-world attacks against DP-SGD, then it does make sense to consider a black-box assumption. The assumptions this paper makes are a bit of an hybrid: it assumes black-box for auditing, yet the adversary is allowed to specify the initial model parameters. The only semi-practical scenario I can think of is an attacker who creates a model (for subsequent fine-tuning to be carried out by the victim) and wants to augment the leakage in the fine-tuned version of the model. Aside from this scenario, I would recommend leaving out from this paper any motivation arguing that this work is more practical than other auditing methods for finding "bugs"; please refer to "Questions" for a concrete example.

2. New insights.

The authors do provide one very convincing motivation: a big question is whether DP analyses of DP-SGD could be improved if they considered it as a black-box algorithm; currently, they all assume knowledge of intermediate gradients. This work offers some hope: from the analysis, it seems that the difference between black-box and the theoretically estimated "epsilon" may be closer than expected. If I may, I would strongly recommend the authors to center their motivations on this aspect.

# Comparison

I could not find a real comparison to previous auditing methods. Only two numbers appear (within the Datasets paragraph), reporting the performance of previous works; were these computed under the same experimental setting (and code base)? Ideally, they should be replicated under identical conditions to yours.
Secondly, it'd be very interesting to see what similar white-box methods achieve, in comparison. How tight are they, on those datasets/models?
questions: questions: - "Although techniques to tightly audit DP-SGD exist in literature [26, 27], they do so only in active  white-box threat models, where the adversary can observe and insert arbitrary gradients into the  intermediate DP-SGD steps. However, real-world adversaries cannot always access the model’s  inner parameters, let alone insert arbitrary gradients."

As argued above, I don't think this is a valid premise. There's nothing in auditing that requires modelling a real-world adversary. Further, the assumptions that you make (e.g., worst-case initial parameters) do not correspond to a real-world adversary. I think the implications of what you claim here are incorrect, and I don't think it should be part of your motivations.

- Threat model. It does look weird to see a threat model section in an auditing paper, and I would humbly recommend removing it: DP doesn't specify any particular threat model, and auditing is just about ensuring how tightly the definition is being matched empirically. If you do want to keep this section, please note that it's missing the assumption that the auditor (adversary?) can specify the initial model parameters; also, I was unclear as to what the following meant in this context: "assume that the adversary can choose a worst-case target sample as is standard for auditing DP mechanisms".

- "we set B=N".

Can you please clarify whether this is a constraint of your method, and if so what are the potential limitations?

- Sentence "When auditing a DP mechanism [...]". Please define the symbol $R$. (It's clear what it is, but it's undefined.)

- "Crafting worst-case initial parameters". This should be one of the main contributions, but unfortunately the paragraph doesn't provide much information on how exactly these parameters are created. Can you please provide more information in the text (and in rebuttal)?

- Could you please double check: did your cluster only have 1 CPU core?

- Typo "is is".
limitations: limitations: These were adequately addressed.
paper_summary: paper_summary: The paper considers the problem of auditing DP-SGD, i.e. deriving empirical lower bounds on the DP parameter $\epsilon$. Many previous auditing works operated in the white-box setting, i.e. are allowed to choose arbitrary gradients for the canary (sensitive example) during DP-SGD, or were in the black-box setting (i.e. could determine some aspects of training, but not specific per-round gradients, and also does not get to see intermediate states) but had a sizeable gap between the empirical $\epsilon$ and the theoretical upper bound. The paper gives a black-box auditing procedure that in some cases achieves empirical $\epsilon$ very close to the theoretical upper bound on standard benchmarks of MNIST and CIFAR10. The main algorithmic insight is to use a model pre-trained on in-distribution data, which causes the gradients of in-distribution examples to be quite small even in the first round of private training, hence increasing the impact of the canary on training. In contrast, previous work used 'average-case' initializations without optimizing for their impact on the empirical $\epsilon$. The authors perform a number of experiments using this improvement, on MNIST and CIFAR, varying whether they train a full model vs the last layer, and varying the dataset size and clip norm. The authors' audit generally improves upon the baseline of an average-case initialization, sometimes as much as a 5x increase in the empirical $\epsilon$.
strength_weakness: strength_weakness: * In some cases we may want to do privacy auditing not to e.g. find bugs in an existing DP-SGD pipeline, but to support the claim that a model has much better privacy guarantees than the worst-case theoretical $\epsilon$ we are reporting. In this case, taking half of the dataset and using it as public pre-training data a privacy violation, i.e. it is unclear how to extend the authors' insights to this application of auditing. This might be a limited weakness, as anyway in practice we are probably going to try to pre-train on a distribution as close to the private distribution as possible, which might retrieve the benefits the authors observe.
* The improvements in Figure 5 don't seem to be very significant, and for eps = 1.0 the average empirical epsilon of average-case initialization is in fact double that of worst-case initialization. This did not affect my score as I understand the authors may not have access to enough compute budget to narrow the confidence intervals, and also I don't think they should be punished for reporting insignificant results, but I think the improvement here is somewhat overstated in the paper.
questions: questions: * Do the authors believe the results in Figure 5 would become significant with more trials? i.e. do the authors have some intuition for why an initialization might matter more or less for last-layer-only fine-tuning than for full-model training? If so, it might be nice to include this in the paper.
limitations: limitations: Yes
paper_summary: paper_summary: This paper studies the problem of auditing DP-SGD in the black box threat model, i.e. only black-box access to the last iterate $\theta_T$ of DP-SGD. The paper's main contribution is to show experimentally that pre-training helps to get tighter auditing. The argument behind using this idea is that pre-training makes the average gradient norm of “in-distribution” points small, thus making the target point used for auditing/membership attack more “distinguishable”. The paper explores this worst-case initialisation idea for two datasets, MNIST and CIFAR10, and reports tighter privacy lower bounds.
strength_weakness: strength_weakness: - The main contribution of this work is incremental. In addition, as noted in the paper, the idea that the initialisation of $\theta_0$ affects auditing/membership inference has already been reported in the literature [10,18,26].

- The justification for using pre-training is hand-wavy and not rigorous. See the question below.

- No code is available to reproduce the results. The experimental setting is well-detailed. However, some important factors for reproducing the results have not been reported. See questions below.
questions: questions: - Worst-case vs average-case initialisation: (a) For the worst-case, $\theta_0$ is selected by pretraining on half of the dataset, but what algorithm is used for that? Is it non-private SGD? And what is the random distribution used to generate $\theta_0$ for the average case?

- In Figure 5, the average-case initialisation seems to give tighter privacy and lower bounds, especially for small epsilon. This contradicts this work's main argument, i.e., using worst-case initialization is better. How do you explain this?

- What canary strategy is used for the black-box audit? Do you only use the blank example? Did you test any other black box canaries?

- The justification behind the reported success of pretraining is that the average gradient norm decreases with the number of pre-training epochs, reported in Figure 2. (a) Is the average of gradients computed on all samples (training+ test) or only training samples? (b) Why does this correlate with the success of your membership attack? Specifically, the MIA attack tresholds over the loss of the canary, thus this seems to be the statistic that should be analysed over the pre-training epochs. The gradient average norm is indicative for white-box attacks. (c) Pretraining decreases the gradient average loss of “in distribution” samples, so how would you adapt your auditing procedure for datasets where the empty sample, i.e. your canary, is "in distribution"?

- What are the loss thresholds used for the black-box membership attack? Are they fine-tuned to find the best privacy budget? Also, the function EstimateEps() is never explicitly detailed in the paper to reproduce the results.

- Is it not already possible to use [32]’s framework to run directly your black-box auditing in one run?
limitations: limitations: As noted by the authors in the conclusion, the auditing procedure is computationally heavy and time expensive.

Comments: comment: We once again thank reviewer JVDo for their time and updating their score to reflect their concerns being addressed.
We once again thank reviewer uyKT for their time and constructive feedback. We agree with the reviewer that the first weakness is a limited one and thank the reviewer for acknowledging our rebuttal.
We once again thank reviewer ncD1 for their time and constructive feedback. As the discussion period draws to a close, we hope our rebuttal has sufficiently addressed the reviewer's concerns and would be grateful to hear the reviewer's thoughts on our comments and clarifications.
We once again thank reviewer qSii for their time and constructive feedback. As the discussion period draws to a close, we hope our rebuttal has sufficiently addressed the reviewer's concerns and would be grateful to hear the reviewer's thoughts on our comments and clarifications.
Thanks for the response.

The first weakness is a limited one, so I do not think the authors need to stress about it. Yes, the high-level point is that having in-distribution pretraining data is not feasible in every problem setting, e.g. in the use-case of https://arxiv.org/pdf/2302.03098 who explore auditing the training pipeline in the same run used to generate the final model. I agree with the authors that for the problem setting considered in the paper, having in-distribution public data is a fine assumption.
I would like to thank the authors for addressing most of my concerns. I updated my score accordingly.