Auditing Privacy Mechanisms via Label Inference Attacks

Authors: róbert  istván busa-fekete
travis  dick
claudio  gentile
andrés muñoz medina
adam  smith
marika  swanberg
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 22:16:07

Abstract: We propose reconstruction advantage measures to audit label privatization mechanisms. A reconstruction advantage measure quantifies the increase in an attacker's ability to infer the true label of an unlabeled example when provided with a private version of the labels in a dataset (e.g., aggregate of labels from different users or noisy labels output by randomized response), compared to an attacker that only observes the feature vectors, but may have prior knowledge of the correlation between features and labels. We consider two such auditing measures: one additive, and on multiplicative. These cover previous approaches taken in the literature on empirical auditing and differential privacy. These measures allow us to place a variety of proposed privatization schemes---some differentially private, some not---on the same footing. We analyze these measures theoretically under a distributional model which, we claim, encapsulates reasonable adversarial settings. We also quantify their behavior empirically on real and simulated  prediction tasks. Across a range of experimental settings, we find that differentially private schemes dominate or match the privacy-utility tradeoff of more heuristic approaches.

Main paper: 
 introduction: 
 Data sharing and processing provides an undeniable utility to individuals and the society at large. The modern data ecosystem has played a significant role in scientific advancement, economic growth, and technologies that benefit individuals in their daily lives. Yet, the collection, processing, and sharing of sensitive information can lead to real privacy harms. Understanding the theoretical and empirical risks associated with different types of data disclosure is a rich area of research and topic of discussion. The privacy community has studied a number of empirical measures of disclosure risks against several types of attacks, including: inference attacks [29] , reconstruction attacks [5] , re-identification attacks [14] and label inference attacks [13, 31] .While differential privacy gives rigorous guarantees against these attacks by worst-case adversaries, it is natural to wonder whether non-DP (deterministic) mechanisms give any empirical privacy protections. This line of inquiry necessitates empirical measures of privacy that take adversarial uncertainty into account. At a fundamental level, whether one is studying reconstruction or membership inference, nearly every privacy disclosure metric comes down to quantifying the relationship between an adversary's prior and posterior knowledge before and after a data release.In this work, we consider a number of ways to summarize this prior-posterior relationship, and we apply our proposed metrics to a few Privacy Enhancing Technologies (PETs). We consider two families of metrics: As a coarse measure, we consider the expected additive difference between the prior and posterior success rates; while this does not capture unlikely high-disclosure events, it gives a sense for the risk posed to an average individual. To better capture such events, we then consider the finer-grained multiplicative difference between the adversary's prior and posterior. These empirical measures of privacy risk allow us to relate and compare the risks of both aggregation-based and differential privacy-based PETs, and hence compare their privacy-accuracy trade-off curves: What accuracy does one get for a given level of privacy protection? Together, these metrics paint a more balanced and nuanced picture of the disclosure risks and accuracy trade-offs posed by aggregation and differentially private mechanisms.We will focus on the simplest possible scenario for information sharing: the case where a user wishes to disclose a single bit of information (for instance a binary label in a classification problem). While this problem is simple to present, it already highlights a lot of the difficulties in providing measures of privacy that are meaningful for both noise-based and aggregate-based tools. Most notably, it already requires handling the correlations-known to an attacker, but unknown to the mechanism-between the visible features and the hidden, sensitive labels. Similar to [31] , we incorporate adversarial knowledge by a prior, and measure the difference between the prior and the adversary's posterior distribution conditional on observed aggregates.Understanding binary labels already has practical implications. For instance, with Chrome's proposed conversion reporting API [1] , the event of a user converting after clicking on an online ad -buying a product, signing up for a newsletter, installing an app, etc. -or not is considered sensitive, and therefore is reported only with some noise. However, once reported, ad tech providers can use features associated with an ad click (impression information, publisher information, etc.) to train models that can predict future conversions. Our contribution. We make several contributions: (i) We introduce additive reconstruction advantage measures as ways to quantify the amount of leakage associated with a generic (not necessarily differentially private) privacy mechanism. These are variants and extensions of the Expected Attack Utility (EAU) and advantage contained in [31] . (ii) We quantify such measures for randomized response and random label aggregation under different correlation assumptions with public data. (iii) We consider a more demanding multiplicative reconstruction advantage measure (in the spirit of predecessors of differential privacy [17, 10] ), and again quantify this advantage for randomized response and random label aggregation. (iv) We conduct a series of experiments on benchmark and synthetic datasets measuring the privacy-utility tradeoff of a number of basic mechanisms, including randomized response, label aggregation and label aggregation plus Laplace or geometric noise. Remarkably, these experiments show that learning with aggregate labels tends to be strictly harder in practice than learning with randomly perturbed labels. That is, differentially private schemes dominate or match the privacy-utility tradeoff of label aggregate schemes for all or most privacy levels and for both types of advantage measure, a conclusion we have not seen clearly spelled out in prior work on privacy. Even measured "on their own turf", deterministic aggregation-which lack provable guarantees like differential privacy-do not provide a significant advantage. Prior Work. Our measures of privacy risk relate closely to previous work on membership inference and auditing, as well as to variants of differential privacy that assume adversarial uncertainty. We discuss these in Section 3.3, after defining our measures. Reproducibility. For the sake of full reproducibility of our experimental setting and results, our code is available at the link https://github.com/google-research/google-research/ tree/master/auditing_privacy_via_lia.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: The main contribution of the paper is clearly stated in the abstract and introduction, and reflected in the technical and experimental sections (Section 3, 4, and associated appendices). 
preliminaries: 
 For a natural number n, let [n] = {i ∈ N : i ≤ n}. Let X denote a feature (or instance) space and Y be a binary (Y = {0, 1}) or multiclass (Y = [c]) label space. We assume the existence of a joint distribution D on X × Y, encoding the correlation between the input features and the labels. The marginal distribution over X will be denoted by D X , and the conditional distribution of y given x will be denoted by D Y|x . For distributions over binary labels, we denote by η(x) = P y∼D Y|x (y = 1|x) the probability of drawing label 1 conditioned on feature vector x. In the multiclass case, we will use η(x, a) = P y∼D Y|x (y = a|x), for a ∈ Y = [c]. We define a dataset S = ⟨(x 1 , y 1 ), . . . , (x m , y m )⟩ as a sequence of pairs (x i , y i ), each one drawn i.i.d. from D. We use x = (x 1 , . . . , x m ) to denote the features in S and y = (y 1 , . . . , y m ) to denote the corresponding labels. We will sometimes abuse the notation in the math, and write for brevity (x, y) instead of (x 1 , y 1 ), . . . , (x m , y m ). Consistent with this notation abuse, we will often use X m × Y m and (X × Y) m interchangeably-no ambiguity will arise.In order to unify the study of privacy enhancing technologies, for the rest of the paper we model PETs as (possibly randomized) functions M : (X × Y) m → Z. These functions map a collection of m labeled examples to a privacy protected representation in the domain Z. The domain Z depends on the PET but, throughout the paper, it will be clear from the context.In our setting, we consider PETs that protect labels and release features in the clear (so, strictly speaking, the output of a PET should also include x itself; we leave implicit in our notation). Despite this, our reconstruction advantage measures can be applied to any PET, we recall below two standard (and very basic) PETs that have been proposed for various Ads metrics APIs: one which satisfies label differential privacy, and one which relies on random aggregation. These are the two PETs on which we will also be able to give theoretical guarantees. We formulate them for binary classification. Definition 2.1. Given ϵ ≥ 0, we say that a (randomized) algorithm A that takes as input S is ϵ-Label Differentially Private (ϵ-Label DP) if for any two datasets S and S ′ that differ in the label of a single sample we have P(A(S) ∈ B) ≤ e ϵ P(A(S ′ ) ∈ B), where B is any subset of the output space of A.Randomized Response (RR) is a classical [30] way of achieving ϵ-Label DP. In the binary classification case, RR with privacy parameter π = 1/(1 + e ϵ ) simply works by randomly flipping each label y j in the dataset with independent probability π before revealing it to the learning algorithm. For a fixed ϵ > 0, RR corresponds to the function M RR (x, y) = ỹ = (ỹ 1 , . . . , ỹm ) ∈ {0, 1} m , where ỹi = 1 -y i with probability π = 1/(1 + e ϵ ) and equal to y i with probability 1 -π, independently for each i ∈ [m].A completely different label privacy mechanism, still very utilized in practice (see, e.g., the Privacy Sandbox API in Google [1] and the Apple SKAN initiative [2] ) is one based on (random) label aggregation, whereby the dataset S gets partitioned uniformly at random into bags of a given size k, S = ⟨(x 11 , y 11 ), . . . , (x 1k , y 1k ), . . . , (x n1 , y n1 ), . . . , (x nk , y nk )⟩, and only feature vectors and the fraction of positive labels in each bag are revealed to the attacker/learning algorithm. In other words, the attacker has access to S via a collection {(B i , α i ), i ∈ [n]} of n labeled bags of size k, with m = nk, whereB i = {x ij : j ∈ [k]}, α i = 1 k k j=1 y ijis the label proportion (fraction of labels "1") in the i-th bag, and all the involved samples (x ij , y ij ) are drawn i.i.d. from D. Thus, the attacker receives information about the m labels y ij of the m instances x ij from dataset S only in the aggregate form determined by the n label proportions α i . Note, however, that the feature vectors x ij are individually observed. From an attacker viewpoint, this setting is sometimes called Learning from Label Proportions (LLP). Hence, LLP corresponds to the functionM LLP (x, y) = (α 1 , . . . , α n ) ∈ [0, 1] n .Learning from privatized labels. Randomized response is especially appealing from a practical point of view, since privatized data with label flipping can be handled by many prominent learning algorithms as is, with some tuning of their hyper-parameters. Often the theoretical guarantees of these learners in terms of sample complexity are only deteriorated by some constant that depends on the label noise level, see, for example [27] . To improve accuracy, we debias the gradients in a post-processing step similar to Equation ( 7) in [19] ; we discuss the debiasing details in Appendix C.On the other hand, a simple and very well-known method for learning from aggregate labels is the one that of [33] call Empirical Proportion Risk Minimization. In fact, different versions of this algorithm are discussed in the literature without a clear reference to its origin. In [12] , the authors simply call this algorithm the Proportion Matching algorithm (PROPMATCH), and we shall adopt their terminology here. Given a loss function ℓ : R × R → R + , a hypothesis set of functions H ⊂ R X , mapping X to a (convex) prediction space Y ⊆ R, and a collection {(B i , α i ), i ∈ [n]} of n labeled bags of size k, PROPMATCH minimizes the empirical proportion matching loss, i.e., it solves the following optimization problem:min h∈H n i=1 ℓ 1 k k j=1 h(x ij ), α i .It is known that the above method is consistent (see for instance [12] ). That is, training a model this way -under some mild conditions on ℓ and H and for a large enough sample -results in learning a model h that minimizes the expected event level loss E (x,y)∼D [ℓ(h(x), y)]. Finally, for our baselines, it will be helpful to consider a PET that reveals no label information at all: M ⊥ (x, y) = ⊥.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: Limitations are discussed in Section 5.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. 
auditing large-scale label inference: 
 In this section, we propose a number of auditing metrics to measure the risk of large-scale label reconstruction. Unlike DP auditing techniques which focus on worst-case guarantees, we will focus here on distributional guarantees. In doing so, we extend the (additive) reconstruction advantage definition introduced in [31] to the LLP setting and propose meaningful variants of it.Our reconstruction advantage measures can be applied to virtually any PET for auditing purposes. For concreteness, we will also provide analytical bounds on such measures when applied to PETs like RR and label aggregation. On one hand, these bounds help illustrate the precise dependence on the data distribution D. On the other, they will pave the way for our experimental findings in Section 4.A reconstruction advantage metric is grounded in the following natural privacy question: How much does releasing the output of a PET increase the risk of label inference compared to not releasing anything? This corresponds naturally to measuring an attacker's prior over a target person's label compared to the attacker's posterior after viewing the mechanism output.We model the attacker (often called 'adversary' later on) as a function A : X m × Z → Y m that maps the features x and the output of a PET M(x, y) to a vector of predicted labels, one for each example. We compare this attacker's success to the the uninformed or prior attacker that gets x and the PET that reveals no label information M ⊥ (x, y) = ⊥. To measure the efficacy of an attacker compared to its prior, we define a number of attack utility variants.The Expected Attack Utility of adversary A using information from PET M on a collection of m examples drawn i.i.d. from a distribution D is defined as:EAU(A, M, D) = P (x,y)∼D m , i∼Uniform([m]), coins of M A(x, M(x, y)) i = y i , where A(•, •) i is the i-th component of vector A(•, •).In words, the expected attack utility of adversary A is the probability that A correctly guesses the label of a randomly chosen example when provided the features and the output of M. Equivalently, this is the expected fraction of the m examples that the adversary predicts the correct label for. The adversary's success rate may depend on the distribution over features and labels. For example, if labels are entirely determined by features, then our metric should reflect that privatized labels (for any mechanism M) reveal no additional information about the true labels. To control for the information that features inherently reveal about labels, we assume that the adversary has knowledge of the data distribution D over X × Y, either completely (e.g., Definition 3.1) or approximately through learning on disjoint data (as in our experiments in Section 4).Further, we define the Individual Expected Attack Utility on input data x i asIEAU i (A, M, D, x i ) = P yi∼D Y|x i (x (-i) ,y (-i) )∼D m-1 , coins of M A(x, M(x, y)) i = y i x i ,where x (-i) is x with the i-th item x i dropped, and likewise for y (-i) . The quantity IEAU i (A, M, D, x i ) emphasizes the (expected) attack utility on a specific piece of data x i when the associated label y i is drawn from the conditional distribution D Y|xi . For instance, IEAU i (A, M LLP , D, x i ) measures, for a given x i , the chance that an attacker is able to reconstruct the associated label y i , if y i generated from D Y|xi and, by virtue of mechanism M LLP , y i becomes part of a bag (of some size k), only the bag label proportion α being revealed to the attacker.In order to measure the increase in risk incurred by releasing the output of a PET, we consider the attack utility of an optimal adversary in two scenarios: one in which the adversary gets the features, x, together with the output of the PET, M(x, y), and an alternate setting where the adversary gets only the features (which is equivalent to using M ⊥ ). We call the difference in attack utility between the informed and uninformed adversary the attack advantage. Intuitively, the attack advantage measures the label reidentification risk that can be attributed to the PET rather than to correlations between the features x and labels y which are inherent in the distribution D. Since we have given two notions of attack utility above, we have two corresponding notions of attack advantage. Below is an additive version, based on utility differences, later on (Section 3.2) we will also consider a multiplicative version, based on utility ratios. Similarly, when the i-th item x i is kept frozen, the Individual Expected attack Advantage is defined asIEAdv(M, D, x i ) = sup Ainformed IEAU(A informed , M, D, x i ) -sup Auninformed IEAU(A uninformed , M ⊥ , D, x i ).Note that RR is generally described in terms of its behavior on a single example (x, y), rather than a collection of m i.i.d. samples. Likewise, label aggregation operates on each bag of size k << m independently. However, since the data is i.i.d., it is easy to see that the attack advantage for RR is independent of the number of examples m, while the attack advantage for (random) label aggregation will only depend on k, rather than m.Appendix A.5 contains further variants of attack advantage, like one that accounts for the tail of the attack advantage distribution, as well as associated bounds. These can be stretched to the point where the feature vector x = (x 1 , . . . , x m ) is arbitrary, and only the properties of the condition distribution D Y|x is factored in.As a warm up, we begin by studying the additive expected attack advantage for LLP when the labels are independent of the features. Note that, since the features x do not play any role in this simplified setting, the notions of advantage coincide. We recall again that all results involving M LLP (Theorems 3.2, 3.3 and 3.4 below) deal with a single bag when computing advantage measures (hence we will write α instead of α i ). This is because distribution D is known to the adversary, and the bags the adversary observes are independent of one another, hence there is no extra advantage from operating on an entire dataset of bags at once. All proofs are given in the appendix. Theorem 3.2. Fix a data distribution D, let p = P (x,y)∼D (y = 1), and fix an arbitrary threshold β ∈ [0, 1/2]. If labels are independent of features (i.e., D is a product of distributions over X and Y), then for all bag sizes k ≥ 1 we have:EAdv(M LLP , D) = min{p, 1 -p} -E α [min{α, 1 -α}] ≤ p(1-p) k if p ∈ [0, 1] e -Ω(β 2 k) if |p -1/2| ≥ β,where Ω(•) hides constants independent of β and k.A couple of remarks are in order. First, observe that, as expected, the advantage EAdv(M LLP , D) is always non-negative. This can be easily derived by noting that E[α] = p and then applying Jensen's inequality to the concave function x → min{x, 1 -x}, for x ∈ [0, 1]. Second, despite being non-negative, Theorem 3.2 also proves the desirable property that EAdv(M LLP , D) goes to zero as the bag size k increases. The convergence rate is in general of the form 1/k, but it becomes negative exponential in k when p is bounded away from 1/2. We now investigate general distributions over features and labels and consider in turn EAdv and IEAdv. Theorem 3.3. Let D be an arbitrary distribution on X ×Y, p = E[η(x)], andµ = E[η(x)(1-η(x))].Then, for all bag sizes k ≥ 2 we have:EAdv(M LLP , D) = O µ 1/4 (p(1 -p)) 1/4 √ k + µ 1/4 k ,where O hides logarithmic factors in k.Hence, also in this more general case of LLP, the advantage converges to zero as the bag size k grows large. Compared to the rate in Theorem 3.2, we are only losing the log k factors implicit in the O notation. This is because, when applied to the scenario where labels and features are independent, η(x) = p is constant with x, so that µ = p(1 -p), and the first term becomes p(1-p) k , while the second one reads (p(1-p)) 1/4 k , which is lower order when k is large. We strongly believe that the tighter gap-dependent analysis we carried out for Theorem 3.2 extends to the more general scenario of Theorem 3.3, but we leave this as an open question.The corresponding bound for the individual expected attack advantage is given next. Theorem 3.4. Under the same assumptions and notation as in Theorem 3.3, we have, for µ > 0, and k ≥ 2 µ log(1/µ),IEAdv(M LLP , D, x i ) = µ i O (p(1 -p)) 1/4 µ k + 1 µ 3/2 k + µ k 2 + 1 µ 3/2 k 2 ,where µ i = η(x i )(1 -η(x i )), and O hides logarithmic factors in k.In the bounds of Theorems 3.2, 3.3 and 3.4 the dependence on the data distribution D is encoded in p(1 -p) and µ; as p(1 -p) and/or µ gets smaller we should naturally expect a smaller advantage, as the adversary is facing an easier label prediction problem. Appendix A.5 contains further distributiondependent results, like a bound on the advantage that, conditioned on x = (x 1 , . . . , x m ), is of the form1 k k i=1 µi √ j:j̸ =i µj , where µ i = η(x i )(1 -η(x i )). We now provide the corresponding expression for the additive attack advantage for RR (at a given level ϵ ≥ 0). We recall that the results of [31] imply that every ϵ-label-DP PET M has advantage bounded as EAdv(M, D) ≤ 1 -2 1+e ϵ . However, one drawback of this bound is that it is distribution independent (or, rather, worst case over distribution D). Yet, the attack advantage depends heavily on D. For an extreme example, if we have P (x,y)∼D (y = 1) = 1, the attack advantage is zero for every PET, which is not directly captured by only relying on the properties of ϵ-DP.Recall that, since RR operates on each example independently, the advantage is independent of the number of examples m (a formal proof is given in Appendix A.6). Hence we derive a bound for IEAdv(M RR , D, x 1 ), and an expression for the optimal adversary under RR (see Appendix A.7). Theorem 3.5. For any data distribution D, the individual expected attack advantage IEAdv(M RR , D, x 1 ) for randomized response with privacy parameter π = 1 1+e ϵ is equal tomin{η(x 1 ), 1 -η(x 1 )} -π •I{η(x 1 ) ∈ [π, 1 -π]} .We use the above expression in our experiments (Section 4) to estimate the attack advantage of RR for various values of ϵ. Being distribution dependent, this expression leads to much tighter bounds on the attack advantage. For example at ϵ = 1, the bound from [31] is 1 -2 1+ϵ ≈ 0.46, while for one dataset used in our experiments we see that the attack advantage for RR at ϵ = 1 is only 0.00095.The definitions of attack advantage given so far measure the absolute change in successful reconstruction, not the relative change. Moreover, they do not fully capture the different levels of confidence in the reconstruction, since they involve an expectation over either (x, y) or y given x. Next, we give an additional set of definitions that capture these important nuances. For the sake of brevity, in this section we directly instantiate the adversaries A informed and A uninformed to Bayes optimal predictors. Definition 3.6. Given data distribution D, mechanism M, index i, and pair of labels a, b ∈ Y, the multiplicative advantage is the difference of log odds ratios:I a,b (M, D, x, z, i) = log π i (x, z, a) π i (x, z, b) -log η(x i , a) η(x i , b)where η(x i , a) = P(y i = a | x i ) and π i (x, z, a) = P y∼D m Y|x (y i = a | x, M(x, y) = z) is the probability of y i = a after observing the output z of M. We denote the binary case by I 1,0 (•) = I(•) and shorten η(x i ) := η(x i , 1) and π i (x, z) := π i (x, z, 1).This particular formulation has the advantage that I a,b (M, D, x, z, i) is large in absolute value whenever there is a large relative change in either label probability. This also entails that a multiplicativestyle advantage metric is generally stronger than an additive one.When η(x i ) and π i (x, z) are both less than 1/2 (in the binary setting), we have I(M, D, x, z, i) = Θ(log πi(x,z) η(xi) ), with a symmetric expression for the case that both are close to 1. Because we assume i.i.d. sampling, one can easily see thatI a,b (M, D, x, z, i) = log P(M(x, y) = z | y i = a, x) P(M(x, y) = z | y i = b, x), which makes it clear that for ϵ-label DP mechanisms like M RR (at level ϵ) the above log ratio is at most ϵ in absolute value for all x and y. On the other hand, for M LLP , there is always a small chance that all the examples in the bag will have the same label making the above log ratio infinite. However, for large k and distributions D in which the η(x, a) values are not too close to 0 or 1, the leakage I a,b (M, D, x, z, i) might be small in most cases.What values of multiplicative advantage should be considered acceptable? We argue that this probability should be viewed as a probability of system failure and set appropriately small. (For example, when running with M LLP on modern-scale data sets, we might create billions of bags; even a tiny probability of failure can lead to many bags whose individuals have their labels revealed exactly). The next theorem gives high probability bounds for M LLP in the simple case when the distribution D is a product distribution. Theorem 3.7. Fix a data distribution D over features and binary labels, let p = P (x,y)∼D (y = 1), and assume the labels are independent of features (i.e., D is a product of distributions over X and Y).Then there are universal constants c 1 , c 2 > 0 such that for p ∈ (0, 1), all bag sizes k ≥ c1 ln(1/δ) p(1-p) , and all i ∈ [k] we haveP (x,y)∼D m Y|x i I 1,0 (M LLP , D, x, M LLP (x, y), i) > c 2 ln(1/δ) p(1 -p)k ≤ δ.Then, when k is sufficiently large and the conditional label probabilities η(x i ) are all equal to some constant p ∈ (0, 1), the probability of failure drops off exponentially as k increases. When working with a data set of size m, we can substitute δ = δ ′ • k m to get a bound on the probability that any bag in the dataset exhibits extreme values of leakage I a,b . For the setting covered by Theorem 3.7, one gets that the bags must be of size k ≈ log(n)/(p(1 -p)) for the probability of extreme bags to converge towards 0. In the experiments in Section 4, we report results on both the additive and the multiplicative reconstruction advantage criteria.Our approach on quantifying attack advantage can be viewed as fitting into a recent line of work on auditing learning algorithms via membership inference attacks. This class of attacks was introduced by [21] and subsequently studied in theory [11, 18] and practice [29] . Whether used as attacks or empirical lower bounds on differential privacy parameters [22] , these approaches set up a hypothesis test for the presence or absence of a particular target record in the training data [15] . They diverge in whether they consider an adversary with access to all the rest of the training data ("fully informed"), or with only distributional knowledge of the rest of the training data (usually made available to the attacker as an independent sample drawn from the same distribution).In our setting, testing for the presence of an individual makes no sense, since individual records' features are known. In the binary case, our measures are success metrics for testing the hypothesis that a particular individual's label is 1 as opposed to 0: Multiplicative advantage bounds the ratio of trueto false-positive rates of the Bayes' optimal test, while additive advantage measures the difference of true-and false-positive rates. This perspective was also taken in previous work: for example, [26] use such a test to lower-bound the differential privacy parameters of label-DP algorithms. However, their modeling assumes a fully informed adversary (that knows all labels). In contrast, we posit a reasonable model of adversarial uncertainty in order to to look at the risks of mechanisms such as label aggregation. In this respect, our approach follows more in the line of membership inference attacks that use only distributional knowledge, or to variants of differential privacy that assume adversarial uncertainty such as [9, 7, 16] .Further relevant references include [25, 32] , which have been developed more in the Cryptography literature. Like here, those papers propose to measure prediction advantage by comparing the information status of an agent before and after seen the obfuscated data. Yet, they do not consider the public features/private label mixed setting we study here, which is clearly necessary when investigating, for instance, the LLP mechanism. As a result, the measures proposed in [25, 32] generally fail to distinguish between the information gained as a result of: (1) correlation to the public features, and (2) the mechanism output. Hence, they overestimate the revelation of the mechanism, at times to the extreme. Consider, for clarity, the case where η(x) is either 0 or 1 for all x. In this case the adversary's inference advantage is zero for any reasonable measure of advantage, since the adversary has perfect knowledge of the labels without the mechanism output.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?• The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper. • 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 reviewed the three main categories of concerns in the Code of Ethics (harms of the process, societal impact, and impact mitigation). We believe we have complied with the spirit and letter of the requirements in each category. In particular, no human subjects research was involved in this project, and the data sets we use are both open-source, standard ML data sets which do not come with privacy or intellectual property concerns. Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [No] . Justification: This work aims to advance the science of privacy-aware machine learning. As such, its goal is to get the benefits of training models on sensitive data sets while measuring and mitigating harmful effects on individuals. Putting protections such as the ones we develop into real-world use is delicate since privacy protections can, in addition to helping data contributors, have side effects such as limiting data access and model quality for sparse sub-populations. These issues, while important, are not specific to our work and lie outside the scope of the paper. That is the main reason why we do not explicitly discuss in the paper the societal impact of this work. 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). 
experiments: 
 In this section, we demonstrate how our advantage measures can be used to quantify and compare the potential privacy leakage of RR, LLP, and two additional PETs with a range of privacy parameters. See Appendix C for full details on our PETs, experimental setup, and additional results. Mechanisms. We empirically evaluate a number of proposed mechanisms for label privacy: randomized response (RR), Label Proportions (LLP), and two further PETs: LLP+Lap, where (zero-mean) Laplace noise is added to the label aggregate in each bag, and LLP+Geom, where geometric noise is added and then clipped so the estimated proportion lies in [0, 1]. The noise scale for both LLP+Lap and LLP+Geom is chosen so that the PETs satisfy ϵ label differential privacy. Postprocessed clipped geometric noise is the mechanism for releasing a binary sum that is optimal among all differentially private mechanisms for several loss measures [20] . To minimize squared error, the optimal postprocessing is to correct for the bias introduced by clipping. We empirically study these advantage measures on a variety of synthetic and benchmark datasets, reporting either the AUC vs. advantage trade-off or the prior-vs-posterior scatter plots, that help shed light on the distribution of our advantage measures on different points in the dataset for the various PETs. Estimating class conditionals for advantage and PET Utility. The advantage measures for all PETs we consider can all be computed as a function of the class conditional probabilities η(x) = P(y = 1|x). In our synthetic datasets we compute the advantage measures for different distributions on η(x); however, for non-synthetic data sets, the value of η(x) is not explicitly known. Instead, we estimate η(x) for each x by training a classifier h (without any PETs) and use the prediction probability h(x) as a proxy for η(x). We compute the figure of merit (additive or multiplicative advantage) for the optimal informed and uninformed attackers using these estimates.Figure 1 : Prior-posterior scatter plots for LLP, RR, and LLP+Geom from two synthetic datasets (where the prior η(x) is drawn) and the two real-world datasets (where η(x) is approximated). The colors of the dots correspond to different parameter values for the PETs. For each bag size k and distribution, we did 1000 independent runs. The further a point is from the y = x dotted line, the more is revealed about its label as a result of the PET.We also measure the utility of each PET for trained models. For each dataset, PET, and privacy parameters, we apply the PET to the training labels to produce a privatized version. We then train a model on the privatized data using minibatch gradient descent with the Adam optimizer [24] and a loss function designed for the PET. For RR, the loss debiases the binary crossentropy loss when evaluated on the RR labels, and for LLP, LLP+Laplace, and LLP+Geom, we minimize the Empirical Proportion Risk defined in Section 2. For each dataset, PET, and privacy parameters, we perform a grid search over the learning rate parameter and report the test AUC of the best performing learning rate. All utility results are averaged over multiple runs. The maximum standard error in the mean for the reported AUCs is 0.0076 and the vast majority are less than 0.002.Computing posterior distributions for each PET. Given a list of priors (or estimated priors) for each data point {η(x i )} n i=1 , we analytically compute the posterior probabilities for each of the PETs we consider.First, we apply Bayes' theorem and use the fact that the x i 's are independent:EQUATIONFor RR with flipping probability p, Equation 1 evaluates topη(xi) pη(xi)-(1-p)(1-η(xi) if the outcome of M RR (y i ) = 0, and to (1-p)η(xi) (1-p)η(xi)-p(1-η(xi) if the outcome is 1.For label aggregation (LLP), the bags are independent, so we consider only a single bag at a time. M LLP is deterministic, so in the numerator P(M LLP (x, y) = z | x, y i = 1) is the probability density at z of a Poisson binomial distribution with flipping probabilities (η(x 1 ), . . . , η(x i-1 ), 1, η(x i+1 ), . . . , η(x k )). Similarly, the denominator is the probability density at z of a Poisson binomial distribution with flipping probabilities (η(x 1 ), . . . , η(x k )).For M Lap-LLP and M Geom-LLP , the terms in Equation 1 are convolutions of the posterior for LLP and the Laplace or Geometric distributions, respectively. Results. Our results are reported in Figures 1, 2 , and 3, as well as in Appendix C. We used two synthentic datasets and two real-world datasets (Higgs and KDD12). Notice that Higgs is a relatively balanced dataset, while KDD12 is a quite imbalanced one.Figure 1 give scatter plots of prior P(y = 1 | x) vs. posterior P(y = 1 | x, M(x, y) = z) distribution on the four datasets for LLP, RR and LLP+Geom with different colors for each value of parameters k (for LLP and LLP+Geom) and ϵ (for RR and LLP+Geom). The behavior of LLP+Lap is reported in Appendix C for completeness, but it turned out to be similar to that of LLP+Geom.It is instructive to observe how the points spread w.r.t. the main diagonal y = x. Points that are on the diagonal have no label privacy loss as a result of the PET, because the posterior is identical to the prior. On the other hand, points with a posterior of 0 or 1 have complete privacy loss, since the posterior on the private label is deterministic. In general, a wider spread away from y = x indicates more privacy loss. The scatter plots tend to form spindle shapes whose width is determined by the privacy parameters of the PETs. Yet, there is a substantial difference between LLP and RR. While RR generates points on the boundaries of the spindles (middle row), LLP tends to spread such points more uniformly. Moreover, some of the points for LLP lie on the edges of the square [0, 1] 2 , which correspond to an infinite multiplicative advantage. LLP+Lap (bottom row) is somewhere in between, in that the points are also located inside the spindles, but never on the edges of the square. Note that a spindle boundary is the set of points having the same multiplicative advantage measure. Moreover, these differences become more pronounced on skewed datasets like KDD12 as the probability of homogenous bags (bags of all 0 labels or 1 labels) become much more likely. These scatter plots are already suggestive of the expected behavior of the utility-privacy tradeoff curves that will come next in Figure 3 . Given an allowed level of privacy (as measured by either additive or multiplicative advantage) each point in the interior of the spindle associated with that privacy level will also lie in the boundary of a smaller (thus higher privacy) spindle. This will make inference harder, thereby reducing (average) utility at that level of privacy. Following this intuition, we expect RR to achieve a higher utility-privacy curve than LLP, with LLP+Geom somewhere in between. A more detailed comparison between RR and LLP+Geom (or LLP+Lap) is in Appendix D.Figure 2 helps further illustrate the different behavior of the considered PETs vis-à-vis the advantage measure. On the four datasets, we pick here a set of parameters that make the PETs AUC-comparable on Higgs. We then plot the empirical Cumulative Distribution Functions (CDFs) on the four datasets for both multiplicative (middle row) and additive (last row) measures. While the CDFs of additive measures are roughly similar across the three PETs, this is not the case for the highly skewed Beta(2,30) and KDD12 datasets, where one can easily spot for LLP (blue line) the presence of a significant mass of points with large multiplicative advantage. Note that in the middle row, the CDF of RR (grey line) is just a vertical line (x = 1) while the CDF of LLP+Geom (orange line) first follows the blue line and then the grey one. These high multiplicative advantage points cannot be detected when only relying on the CDFs of the additive measure. Utility vs. advantage tradeoff on benchmark datasets. The experiments so far have compared the distribution of individual additive and multiplicative advantage induced by RR, LLP, and LLP+Geom for various parameter settings and datasets. However, the use of PETs is always a tradeoff between utility and privacy, since if we cared about privacy alone, the best strategy would be to release no information at all. Figure 3 plots the AUC vs advantage for each PET as we vary the PET's privacy parameters. Note that, since the x-axis is advantage, we are able to put the normally incomparable privacy parameters of the PETs on equal footing. For RR and LLP, the single privacy parameter (ϵ and k, respectively) traces out an AUC vs Advantage curve. Since LLP+Lap and LLP+Geom have two parameters, there is an area of achievable AUC vs Advantage pairs. For these mechanisms, we plot a separate curve for each bag size k showing the tradeoff when varying ϵ for that k. Utility is measured by AUC on test set, while privacy is either the additive measure (bottom row) or the 98th-percentile of the multiplicative measure (so as to rule out the infinite multiplicative advantage cases that can occur for LLP). Each point corresponds to a setting of the privacy parameter for the PET (ϵ for RR, k for LLP, and both for LLP+Geom). The x-coordinate is the advantage (either additive or multiplicative) value for that PET, while the y-coordinate is the test AUC of a model trained from the output of that PET. The AUC of the model trained without a PET roughly corresponds to the top value achieved by these curves.When measuring privacy loss via multiplicative advantage, RR has the best privacy vs. accuracy tradeoff compared to all other PETs. This is consistent with the observations in Fig. 2 . In particular, the three mechanisms (with the given parameters) have almost the same AUC on Higgs. Yet when looking at the CDFs of the multiplicative measure one sees that the two DP mechanisms have bounded MAs whereas LLP has a significant number of extreme values. This trend is more pronounced for KDD12 than Higgs. Thus, for the same AUC, LLP has a higher multiplicative advantage, with a more pronounced gap for KDD12 than for Higgs. On the other hand, Fig. 2 also shows that the CDFs of the additive advantage are roughly the same for all three mechanisms (again, at the same AUC). Thus, in Fig. 3 the mechanisms have similar AUC vs advantage tradeoffs. 
discussion and conclusions: 
 We study ways to audit label privatization mechanisms for commonly used PETs: randomized response, random label aggregation, and combinations thereof. Together, the additive and multiplicative advantage measures we introduced paint a richer picture of the reconstruction risks posed by different parameter settings, and for the first time allow us to compare their privacy-accuracy trade-off curves.The measures we propose are tailored to settings where the data are sampled i.i.d. from a distribution, and each record consists of public features together with one sensitive binary feature. Computing these measures empirically requires estimates of the adversary's label uncertainty, which won't be correct if the adversary has significant side information or, crucially, if the same data are reused in multiple mechanisms. Handling such complex settings requires more general concepts like differential privacy. Another complexity emerges when we consider settings like click prediction, where a minority label (a click) is viewed as qualitatively more revelatory than the majority one (no click). While the measures we consider are agnostic to the label semantics, principled ways to incorporate complex semantics might be valuable and shed light on heuristics used in practice. 
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., pretrained language models, image generators, or scraped datasets)? Answer: [NA] . Justification: We don't see any such risks. 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: [NA] . 
 
 Appendix: We start off by providing a preliminary result that will be useful throughout all proofs.Consider, for a specific i ∈ [m], the random variablesf 1i (x) = sup Ainformed P y∼D k Y|x (A informed (x, M(x, y)) i = y i | x) (2) f 2i (x) = sup Aunininformed P y∼D k Y|x (A uninformed (x, M ⊥ (x, y)) i = y i | x) .The relevance of f 1i and f 2i stems from the fact that for the mechanisms M we consider in this paper, the advantages EAdv, IEAdv, and HPAdv will be defined in terms of f 1i and f 2i . Lemma A.1. For any PET M, any x ∈ X , any conditional data distribution D Y|x , number of examples k, and i ∈ [m], we havef 1i (x) -f 2i (x) = min{η(x i ), 1 -η(x i )} -E y∼D Y|x min b∈{0,1} P(y i = b | x, M(x, y)) | x ,where η : x → P(y = 1 | x). Moreover, the optimal (informed) adversary A * is:A * (x, M(x, y)) i := 1 if P(y i = 1 | x, M(x, y)) ≥ 1/2 0 otherwise .Proof. Set for brevity z = M ⊥ (x, y). Since M ⊥ outputs null, and the (x j , y j ) pairs for j ∈ [m] are independent, we haveP(y i = 1 | x, z) = P(y i = 1 | x i ) = η(x i ),It immediately follows that the best attacker A unininformed involved in the computation of f 2i (x) is the Bayes optimal predictor ŷi = arg max b∈{0,1}P(y i = b | x i ) so that f 2i (x) = 1 -min{η(x i ), 1 -η(x i )} .Let us now turn to f 1i and fix any adversary A informed . We lower bound the probability that the adversary makes a mistake on y i . Let z = M(x, y) be the output of the PET, and (ŷ 1 , . . . , ŷm ) = A informed (x, z) be the output of the adversary. Since ŷi is (x, z)-measurable, we haveP(ŷ i = y i | x, z) = P(ŷ i = 1 | y i = 1, x, z) P(y i = 1 | x, z) + P(ŷ i = 0 | y i = 0, x, z) P(y i = 0 | x, z) = 1{ŷ i = 1} P(y i = 1 | x, z) + 1{ŷ i = 0} P(y i = 0 | x, z) . Hence P(ŷ i = y i | x) = E z|x 1{ŷ i = 1} P(y i = 1 | x, z) + 1{ŷ i = 0} P(y i = 0 | x, z) | x ≤ E z|x max{P(y i = 1 | x, z), P(y i = 0 | x, z)} | x = E y∼D Y|x max{P(y i = 1 | x, M(x, y)), P(y i = 0 | x, M(x, y))} | x = 1 -E y∼D Y|x min{P(y i = 1 | x, M(x, y)), P(y i = 0 | x, M(x, y))} | x = f 1i (x) ,where the inequality holds because the maximum probability term is never smaller than the probability term selected by the indicator variables. Incidentally, the above also shows that the optimal informed adversary is the one that selects the larger of the two probability terms with probability one, which makes the inequality above hold with equality.At this point, we use the fact thatEQUATIONholds for all i ∈ [m] since the data sequence (x 1 , y 1 ), . . . , (x k , y k ) is i.i.d.A.2 Proof of Theorem 3.2 Theorem 3.2. Fix a data distribution D, let p = P (x,y)∼D (y = 1), and fix an arbitrary threshold β ∈ [0, 1/2]. If labels are independent of features (i.e., D is a product of distributions over X and Y), then for all bag sizes k ≥ 1 we have:EAdv(M LLP , D) = min{p, 1 -p} -E α [min{α, 1 -α}] ≤ p(1-p) k if p ∈ [0, 1] e -Ω(β 2 k) if |p -1/2| ≥ β,where Ω(•) hides constants independent of β and k.Proof. For a given (x j , y j ) in the dataset, we can consider without loss of generality only the size-k bag that x j falls into, and then refer the indexing to this bag only.Then set for brevity Σ = kα = k i=1 y i . From (3), we can writeEAdv(M LLP , D) = E x [min{η(x i ), 1 -η(x i )}] -E Σ min b∈{0,1} P(y i = b | Σ) = min{p, 1 -p} -E Σ min b∈{0,1} P(y i = b | Σ) .Since P(y i = 1 | Σ) = α independent of i and p, the minimum value in the second expectation isE α [min{α, 1 -α}] ,and the claimed equality for EAdv(M LLP , D) follows.EQUATIONIf applied to the expressionmin{p, 1 -p} -E α [min{α, 1 -α}]this givesEAdv(M LLP , D) ≤ E α [|α -p|] ≤ E α [(α -p) 2 ] = p(1 -p) k ,where the second inequality is Jensen's.Finally, in the case where |p -1/2| ≥ β, for some gap β > 0, we can proceed through a more direct analysis. Assume p ≤ 1/2 -β. We can writemin{α, 1 -α} = min{α, 1 -α}I{α ≤ 1/2} + min{α, 1 -α}I{α > 1/2} + αI{α > 1/2} -αI{α > 1/2} = αI{α ≤ 1/2} + (1 -α)I{α > 1/2} + αI{α > 1/2} -αI{α > 1/2} = α -(2α -1)I{α > 1/2} ≥ α -I{α > 1/2} . Hence E α [min{α, 1 -α}] ≥ p -P(α > 1/2) . Now, p < 1/2 implies min{p, 1 -p} = p, which leads us to min{p, 1 -p} -E α [min{α, 1 -α}] ≤ P(α > 1/2) .Finally, by the standard Bernstein inequality we haveP(α > 1/2) ≤ exp - k(1/2 -p) 2 2p(1 -p) + (1 -2p)/3 = e -Ω(kβ 2 ) ,which gives the second inequality.A similar argument holds if we reverse the assumption on p to p ≥ 1/2 + β.Again, as in the proof of Theorem 3.2, for a given (x j , y j ) in the dataset, we can consider with no loss of generality only the size-k bag that x j falls into, and then refer the indexing to this bag only.Recall that, for a general distribution D over X ×Y the distribution of random variable kα = k j=1 y j conditioned on x = (x 1 , . . . , x k ) is Poisson Binomial (PBin) with parameters {η(x j )} k j=1 , that is, the distribution of the sum of k independent Bernoulli random variables y j , each with its own bias η(x j ), where η(x) = P(y = 1 | x).Notation. For simplicity of notation we let η i = η(x i ) be the conditional positive probability of a label given feature vector x and let p = E[η i ]. For fixed feature vectors x = (x 1 , . . . , x k , x k+1 ), let A i denote a Bernoulli random variable with mean η i , and letZ k = k j=1 A jWe recall the statement of the theorem we want to prove. Theorem 3.3. Let D be an arbitrary distribution on X ×Y, p = E[η(x)], andµ = E[η(x)(1-η(x))].Then, for all bag sizes k ≥ 2 we have:EAdv(M LLP , D) = O µ 1/4 (p(1 -p)) 1/4 √ k + µ 1/4 k ,where O hides logarithmic factors in k.The above theorem is a direct consequence of the following Lemma. Lemma A.2. Let B denote a bag with k + 1 elements. Letc k = 1 √ k 1 3 log 8k + 1 6 2 log 8k + 12kp(1 -p) log 8k = O p(1 -p) + 1 √ k .ThenEAdv(M LLP , D) ≤ k 1/4 √ 2c k 1 e 3/2 + π 4 + π e E[η 1 (1 -η 1 )] 1/2 k 3/2 + E[η 1 (1 -η 1 )] k .Proof. Set for brevity M = M LLP (x, y). From (3) we haveEAdv(M LLP , D) = E x k+1 [min{η(x k+1 ), 1 -η(x k+1 )}] -E (x,y)[min {P(y k+1 = 1|x, M), P(y k+1 = 0|x, M)}] ≤ E (x,y) [|η(x k+1 ) -P(y k+1 = 1|x, M)|] ,where we have again used (4) . We now focus on calculating P(y k+1 = 1|x, M). Let Σ = k+1 j=1 y j . Note that for a given realization of feature vector x, y k+1 is distributed like A k+1 and the output M is distributed like Z k+1 . Therefore:P(y k+1 = 1 | x, M) = P(A k+1 = 1 | x, Z k+1 = Σ) = P(A k+1 = 1, Z k+1 = Σ | x) P(Z k+1 = Σ | x) = η k+1 P(Z k = Σ -1 | x) P(Z k+1 = Σ | x).Using this expression in the original expectation we see that we can bound the advantage asE (x,y) η k+1 1 - P(Z k = Σ -1 | x) P(Z k+1 = Σ | x)Again, note that for a fixed x, the variable Σ is distributed like Z k+1 . Therefore. taking expectation over y the above expression can be rewritten asE x η k+1 k+1 s=0 P(Z k+1 = s | x)× 1 - P(Z k = s -1 | x) P(Z k+1 = s|x) = E x η k+1 k+1 s=0 P(Z k+1 = s | x) -P(Z k = s -1 | x)Finally, note that since Z k+1 = Z k + A k+1 we also haveP(Z k+1 = s | x) = P(A k+1 = 1 | x) P(Z k = s -1 | x) + P(A k+1 = 0 | x) P(Z k = s | x) = η k+1 P(Z k = s -1 | x) + (1 -η k+1 ) P(Z k = s | x). Therefore, we conclude that the advantage can be bounded byEQUATIONwhere we have used the fact that the random variables η i are independent from each other. Using also the fact that η k+1 has the same distribution as η 1 combined with Lemma A.3 below, we have that the above quantity is bounded by:EQUATIONMoreover, notice thatE[η 2 1 + (1 -η 1 ) 2 ] + 2 E[η 1 (1 -η 1 )] = 1. Therefore E[η 2 1 + (1 -η 1 ) 2 ] = 1 -2 E[η 1 (1 -η 1 )],and using the fact that η 1 (1 -η 1 ) ≤ 1 4 we haveE[η 1 (1 -η 1 )] 2 E[η 2 1 + (1 -η 1 ) 2 ] k = E[η 1 (1 -η 1 )] 2 (1 -2 E[η 1 (1 -η 1 )]) k ≤ max 1 4 ≥x≥0 x 2 (1 -2x) k .But a simple calculation shows that the above function is maximized atx ⋆ k = min{ 1 k+2 , 1/4}, thus we must have E[η 1 (1 -η 1 )] 2 E[η 2 1 + (1 -η 1 ) 2 ] k ≤ (x ⋆ k ) 2 (1 -2x ⋆ k ) k ≤ 1 (ek) 2the last inequality holding for all k ≥ 1. In addition, we have the trivial boundE[η 1 (1 -η 1 )] 2 E[η 2 1 + (1 -η 1 ) 2 ] ≤ E[η 1 (1 -η 1 )] 2 , so that E[η 1 (1 -η 1 )] 2 E[η 2 1 + (1 -η 1 ) 2 ] k ≤ min E[η 1 (1 -η 1 )] 2 , 1 e 2 k 2 ≤ E[η 1 (1 -η 1 )] 2 E[η 1 (1 -η 1 )] 2 e 2 k 2 + 1 (using min{a, b} ≤ ab a+b , with a = E[η 1 (1 -η 1 )] 2 and b = 1 e 2 k 2 ) ≤ E[η 1 (1 -η 1 )] e 3/2 k 3/2 (using x 2 -x 3/2 + 1 ≥ 0, with x = ek E[η 1 (1 -η 1 )]) .Replacing this bound in (6) we obtain the following upper bound on the advantage:k 1/4 √ 2c k 1 e 3/2 + π 4 E[η 1 (1 -η 1 )] 1/2 k 3/2 + π E[η 1 (1 -η 1 )] ek 2 + E[η 1 (1 -η 1 )] k ≤ k 1/4 √ 2c k 1 e 3/2 + π 4 + π e E[η 1 (1 -η 1 )] 1/2 k 3/2 + E[η 1 (1 -η 1 )] k ,as claimed.Lemma A.3. Let c k be as in Lemma A.2. Then the following bound holds:E x k+1 s=0 | P(Z k = s | x) -P(Z k = s -1 | x)| ≤ 1 k + k 1/4 √ 2c k E[η 2 1 + (1 -η 1 ) 2 ] k + π (4 E[η 1 (1 -η) 1 )k) 3/2 + π e E[η 1 (1 -η 1 )]k 2Proof. Let a > 0 and b < k, and let [a, b] = {j ∈ N|a ≤ j ≤ b}. For any x we then havek+1 s=0 | P(Z k = s | x) -P(Z k = s -1 | x)| = s∈[a,b] | P(Z k = s | x) -P(Z k = s -1 | x)| + s / ∈[a,b] | P(Z k = s | x) -P(Z k = s -1)| ≤ s∈[a,b] | P(Z k = s | x) -P(Z k = s -1 | x)| + s / ∈[a,b] P(Z k = s | x) + P(Z k = s -1 | x) ≤ s∈[a,b] | P(Z k = s | x) -P(Z k = s -1 | x)| + 2 P(Z k / ∈ [a, b] | x)Taking expectation over both sides with respect to x we haveEQUATIONLet now Q k denote the probability measure associated with a binomial random variable with parameters (k, p). Since Z k is a Poisson-Binomial random variable with parameters η 1 , . . . , η k , the probabilityP(Z k / ∈ [a, b] | x) is a linear function in each individual η i , so that E x [P(Z k / ∈ [a, b] | x)] = Q k ([a, b] c). We now proceed to bound the first expectation in (7) . By Cauchy-Schwartz inequality we haveE x   s∈[a,b] | P(Z k = s | x) -P(Z k = s -1 | x)|   ≤ E x   (b -a) s∈[a,b] (P(Z k = s | x) -P(Z k = s -1 | x)) 2   ≤ E x   (b -a) k+1 s=0 (P(Z k = s | x) -P(Z k = s -1 | x)) 2   ≤ E x (b -a) k+1 s=0 (P(Z k = s | x) -P(Z k = s -1 | x)) 2 ,where the last inequality holds by Jensen's inequality. LetA = E[η 2 1 + (1 -η 1 ) 2 ] and B = 2 E[η 1 (1 -η 1 )] .By Lemma A.4 below we have that the above term is bounded by(b -a) A k + π (4Bk) 3/2 + π eBk 2 ,so that (7) givesE x k+1 s=0 | P(Z k = s | x) -P(Z k = s -1 | x)| ≤ 2Q k ([a, b] c )+ (b -a) A k + π (4Bk) 3/2 + π eBk 2 . Let a = max{kp - √ kc k , 0} and b = min{kp + √ kc k , 1}. By Bernstein's inequality applied to binomial random variables we have thatQ k ([a, b] c ) = 1 2k. Hence, with this choice of a and b we obtainE x k+1 s=0 | P(Z k = s | x) -P(Z k = s -1 | x)| ≤ 1 k + 2 √ kc k A k + π (4Bk) 3/2 + π eBk 2 .Lemma A.4. The following inequality holdsE x k+1 s=0 (P(Z k = s | x) -P(Z k = s -1 | x)) 2 ≤ E[η 2 1 + (1 -η 1 ) 2 ] k + π (8 E[η 1 (1 -η 1 )]k) 3/2 + π 2e E[η 1 (1 -η 1 )]k 2 . Proof. Let p s (x) = P(Z k = s | x) -P(Z k = s -1 | x), for s = 0, . . . , k + 1. Further, for k + 1 ≥ u ≥ 0 let g u (x) = 1 √ k+2 k+1 s=0 p s (x)e 2πi us k+2denote the discrete Fourier transform. Since, for any x, the mapping p(x) := (p 0 (x), . . . , p k+1 (x)) → (g 1 (x), . . . , g k+1 (x) := g(x) is a unitary linear transformation [8] we can write:k+1 s=0 (P(Z k = s | x) -P(Z k = s -1 | x)) 2 = ∥p(x)∥ 2 = ∥g(x)∥ 2 = k+1 u=0 |g u (x)| 2 .Moreover, by Lemma A.5 below we have thatg u (x) = (1 -e 2πiu k+2 ) 1 √ k + 2 k j=1 (1 -η j + η j e 2πiu k+2 )and therefore|g u (x)| 2 = g u (x)g u (x) = 1 k + 2 1 -cos 2πu k + 2 k j=1 (1 -η j ) 2 + η 2 j + 2η j (1 -η j ) cos 2πu k + 2 .Therefore we can writeEQUATIONWhere we have used the fact that the random variables η j are independent. Finally by linearity of expectation and the fact that η j is distributed as η 1 for all j, we haveEQUATIONApplying Proposition A.6 below with a = E[η 2 1 + (1 -η 1 ) 2 ] and b = 2 E[η 1 (1 -η 1 )] we have that the above expression is bounded byE[η 2 1 + (1 -η 1 ) 2 ] k + π (8 E[η 1 (1 -η 1 )]k) 3/2 + π 2e E[η 1 (1 -η 1 )]k 2 which gives the claimed result. Lemma A.5. Let p s (x) = P(Z k = s | x) -P(Z k = s -1 | x) and let g u (x) = 1 √ k+2 k+1 s=0 p s (x)e 2πiusk+2 denote the discrete Fourier transform of p(x) = (p 1 (x), . . . , p s (x)). Theng u (x) = 1 √ k + 2 (1 -e 2πiu k+2 ) k j=1 (1 -η j + η j e 2πiu k+2 )Proof. By definition of g u (x) we have:1 √ k + 2 k+1 s=0 e 2πius k+2 P(Z k = s | x) - k+1 s=0 e 2πius k+2 P(Z k = s -1 | x) = 1 √ k + 2 k+1 s=0 e 2πius k+2 P(Z k = s | x) -e 2πiu k+2k+1 s=0 e 2πiu(s-1) k+2P(Z k = s -1 | x) = 1 √ k + 2 (1 -e 2πiu k+2 )E Z k [e 2πiu k+2 Z k | x] = 1 √ k + 2 (1 -e 2πiu k+2 )ϕ Z k | x 2πu k + 2where ϕ Z k | x denotes the characteristic function of Z k conditioned on x. Using the fact that Z k = k j=1 A j and that A 1 , . . . , A k are independent given x, we have ϕZ k | x = k j=1 ϕ Aj | xj .The result follows from the fact that A j is a Bernoulli random variable and therefore ϕ Aj | xj (z) = (1 -η j + η j e iz ).Proposition A.6. For any a, b, k > 0 such that a + b = 1 we have1 k + 2 k+1 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k ≤ a k + π (4kb) 3/2 + π ebk 2Proof. Using the fact that cos 2πu k+2 ≤ 0 for u ∈ [(k + 2)/4, 3(k + 2)/4] we have thatk+1 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k = k+2/4 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k + 3(k+2)/4 u=(k+2)/4+1 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k + k+1 u=3(k+2)/4+1 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k ≤ (k + 2)a k + (k+2)/4 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k + k+1 u=3(k+2)/4+1 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k = (k + 2)a k + 2 (k+2)/4 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k ,where we used the fact that (1 -cos t) ≤ 2 for the first inequality and the symmetry of the cosine function for the last equality. We now apply the result of Proposition A.8 to the above expression to see that(k+2)/4 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k ≤ 2π 2 (k+2)/4 u=0 u 2 (k + 2) 2 1 -4πb u 2 (k + 2) 2 ) k ≤ 2π 2 (k+2)/4 u=0 u 2 (k + 2) 2 e -4πkb u 2 (k+2) 2Therefore we conclude that1 k + 2 k+1 u=0 1 -cos 2πu k + 2 a + b cos 2πu k + 2 k ≤ a k +4π 2 (k+2)/4 u=0 u 2 (k + 2) 2 e -4πkb u 2 (k+2) 2 1 k + 2 .Finally, applying Proposition A.7 with m = k + 2 and α = 4πkb we can upper bound the above quantity by:a k + 4π 2 √ π 4(4πkb) 3/2 + 1 4eπbk(k + 2) ≤ a k + π (4kb) 3/2 + π ebk 2 ,which concludes the proof.Proposition A.7. Let m > 0 and α > 0. Then m/4 u=0 u 2 m 2 e -α u 2 m 2 1 m ≤ √ π 4α 3/2 + 1 eαm Proof. Let f : R → R be given by x → x 2 e -αx 2. Note that the sum we are attempting to bound is then given by: m ⌈x 0 ⌉-1 m f (x)dx. Therefore we have m/4 u=0 f u m 1 m ≤ 1/4-1/m 0 x 2 e -αx 2 dx + ⌊x 0 ⌋+1 m ⌈x 0 ⌉-1 m f (x)dx ≤ ∞ 0 x 2 e -αx 2 + 1 m max x f (x) = √ π 4α 3/2 + 1 eαm ,as claimed.Proposition A.8. The following inequality holds for any t ∈ [0, 1/4]:1 -2(πt) 2 ≤ cos 2πt ≤ 1 -4πt 2Proof. For the lower bound we start from the fact that for any x ≥ 0 it is well known that sin 2πx ≤ 2πx.Integrating this inequality from [0, t] we have that t 0 sin 2πx ≤ πt 2 . Since t 0 sin 2πx = 1 2π (1cos 2πt) the lower bound follows.For the upper bound we proceed in a similar fashion. By the fact that sin 2πx is concave for x ∈ [0, 1/4] we have thatsin 2πx = sin 2π((1 -4x) • 0 + 4x • 1 4 ) ≥ (1 -4x) sin 0 + 4x sin π 2 = 4x.Again integrating the above inequality from 0 to t we have 1-cos 2πt 2π ≥ 2t 2 .A.4 Proof of Theorem 3.4Proof. The proof follows the same argument of the proof used in 3.3, where we see that, in view of the factorization in Equation 5we may bound the individual expected advantage as:IEAdv(M, D, x k+1 ) ≤ η(x k+1 )(1 -η(x k+1 ) E x1,...,x k k s=0 |P (Z k = s|x) -P (Z k = s -1|x)| .But from Lemma A.3 it follows that this can be bounded byµ i O (p(1 -p)) 1/4 (1 -2µ) k k 1/2 + 1 µ 3/2 k + (1 -2µ) k k 1/2 + 1 µ 3/2 k 2 . Now, since (1 -2µ) k k 1/2 < µ k for k ≥ 2 µ log 1 µ, we upper bound accordingly the two terms involving (1 -2µ) k , and the result follows.We are also interested in the tail of the distribution of the attack advantage, that is, the tail of the difference betweenf 1 (x) = sup Ainformed P y∼D m Y|x i∼Uniform([m]) coins of M (A informed (x, M(x, y)) i = y i | x) and f 2 (x) = sup Auninformed P y∼D m Y|x i∼Uniform([m]) coins of M (A uninformed (x, M ⊥ (x, y)) i = y i | x)Specifically, we defineHPAdv(M, D, θ) = P x∼D m X (f 1 (x) -f 2 (x)) > θ) ,viewed as a function of θ ∈ [0, 1], which we call the High Probability attack Advantage (at level θ). This is simply a high-probability version of EAdv(A, M, D) over the generation of x. A similar high-probability version can be defined for the individual version of the attack advantage.We begin by observing that in the special case of M RR , we have IEAdv 1 (M RR , D, x 1 ) = f 1 (x)f 2 (x), so that the high probability bound involved in the computation of HPAdv(M RR , D, θ) is a direct consequence of the tail properties of the function η(x 1 ) = P(y 1 = 1 | x 1 ) as applied to the expression for IEAdv 1,1 (M RR , D, x 1 ) in Theorem A.17. Thus, for M RR , the connection from IEAdv to HPAdv is immediate.We continue with studying HPAdv for M LLP . We have the following bound.Theorem A.9. Under the same assumptions and notation as in Theorem 3.3 (main body) there are universal constants c 1 , c 2 > 0 such that, for k = Ω(1/µ),HPAdv M LLP , D, c 1 µ k ≤ (k + 1) e -c2 kµ .In order to prove this theorem, we first need ancillary results related to Poisson Binomial distributions.We denote a Poisson Binomial distribution with parameters {η 1 , . . . , η k } by PBin(η 1 , . . . , η k ). Lemma A.10. Let s, Σ ∼ PBin(η 1 , . . . , η k ), Σ (1) ∼ PBin(η 2 , . . . , η k ), where η 1 , . . . , η k ∈ [0, 1] are such thatk i=1 η i (1 -η i ) ≥ 1 .ThenE s P(Σ (1) = s) -P(Σ (1) = s -1) P(Σ = s) ≤ 9 k i=2 η i (1 -η i ).Proof. We will use the following distribution to approximate Poisson Binomial distributions.Definition A.11. [28] We say that an integer random variable Y is distributed according to the translated Poisson distribution with parameters µ and σ 2 , denoted TP(µ, σ 2 ), if and only if Y can be written as Y = ⌊µ + σ 2 ⌋ + Z, where Z is a random variable distributed according to Poisson(σ 2 + {µ -σ 2 }), being {µ -σ 2 } the fractional part of µ -σ 2 , and ⌊µ + σ 2 ⌋ the integer part of µ + σ 2 .Let us then define two random variables Γ and Γ (1) . Variable Γ is TP(µ, σ 2 ) with µ = k i=1 η i and σ 2 = k i=1 η i (1 -η i ), while Γ (1) is TP(µ 1 , σ 2 1 ), where µ 1 = k i=2 η i and σ 2 1 = k i=2 η i (1 -η i ).Let d TV denote the total variation distance. We can write E s P(Σ (1) = s) -P(Σ (1) = s -1) 1) , Γ (1) )EQUATIONEQUATIONEQUATIONEQUATIONwhere (10) follows from the triangle inequality, (11) follows from Lemma A.12 below, and ( 12) follows from the overapproximationk i=1 η 3 i (1 -η i ) + 2 k i=1 η i (1 -η i ) ≤ k i=1 η i (1 -η i ) + 2 k i=1 η i (1 -η i ) ≤ 3 k i=1 η i (1 -η i ), the latter inequality exploiting the conditionk i=1 η i (1 -η i ) ≥ 1. Lemma A.12. [ [28] , see (3.4 ) therein] Let J 1 , . . . , J k be independent random Bernoulli with parameters p 1 , . . . , p k . Thend TV k i=1 J i , TP(µ, σ 2 ) ≤ k i=1 p 3 i (1 -p i ) + 2 k i=1 p i (1 -p i ) where µ = k i=1 p i and σ 2 = k i=1 p i (1 -p i ).We recall a lemma which will be useful to upper bound the last term in (12) .Lemma A.13. [6] For µ 1 , µ 2 ∈ R and σ 2 1 , σ 2 2 ∈ R + such that ⌊µ 1 -σ 2 1 ⌋ ≤ ⌊µ 2 -σ 2 2 ⌋, it holds that d TV TP(µ 1 , σ 2 1 ), TP(µ 2 , σ 2 2 ) ≤ |µ 1 -µ 2 | σ 1 + |σ 2 1 -σ 2 2 | + 1 σ 2 1 .We will need the following technical lemma as well.Lemma A.14. For µ ∈ R and σ 2 1 , σ 2 2 ∈ R + , let us define X 1 ∼ TP(µ, σ 2 1 ) and X 2 ∼ TP(µ + 1, σ 2 2 ). Then P(X 1 = ℓ -1) = P(X 2 = ℓ) for all ℓ ∈ Z.Proof of Lemma A.14. One has to observe that Translated Poisson distribution is equivalent to a Poisson distribution with parameter δ where δ = µ -γ and γ = ⌊µ -σ 2 ⌋, but it is shifted by γ. Moreover, note that γ + 1 = ⌊µ + 1 -σ 2 ⌋, thus δ + 1 = µ -γ + 1. This observation implies the claim with µ + 1.We continue by using both Lemma A.13 and Lemma A.14. Define Γ (2) ∼ TP(µ 1 + 1, σ 2 1 ). We can writek s=0 P(Γ = s) -P(Γ (1) = s -1) = k s=0 P(Γ = s) -P(Γ (2) = s) = d TV TP(µ, σ 2 ), TP(µ 1 + 1, σ 2 1 ) ≤ |µ -µ 1 -1| σ + |σ 2 -σ 2 1 | + 1 σ 2 (from Lemma A.13) = 1 -η 1 σ + η 1 (1 -η 1 ) + 1 σ 2 ≤ 3 √ σ ,the latter inequality using again the conditionk i=1 η i (1 -η i ) ≥ 1.E s 1 - P(Σ (1) = s -1) P(Σ = s) ≤ 6 √ σ + 3 √ σ 1 ≤ 9 √ σ 1 .This concludes the proof.With these results handy, we are ready to prove Theorem A.9. Recall the functions f 1 (x) and f 2 (x) defined in the main body.Proof of Theorem A.9. Let Σ = k j=1 y j . From Lemma A.1, we see thatf 1 (x) = 1 - 1 k k i=1 E y∼D Y|x min b∈{0,1} P(y i = b | x, Σ | x and f 2 (x) = 1 - 1 k k i=1 min{η(x i ), 1 -η(x i )} so that f 1 (x) -f 2 (x) = 1 k k i=1 min{η(x i ), 1 -η(x i )} -E y∼D Y|x min b∈{0,1} P(y i = b | x, Σ | x ≤ 1 k k i=1 η(x i ) -P Σ|x (y i = 1 | Σ, x) ,where in the inequality we have again used (4).But, for any given s ∈ {0, 1, . . . , k},P(y i = 1 | Σ = s, x) = P(y i = 1 | x) P(Σ = s | y i = 1, x) P(Σ = s | x) = η(x i ) P(Σ (-i) = s -1 | x) P(Σ = s | x) = η(x i ) P(Σ (-i) = s -1 | x) η(x i ) P(Σ (-i) = s -1 | x) + (1 -η(x i )) P(Σ (-i) = s | x)where Σ (-i) = k j:j=1,j̸ =i y j . HenceEQUATIONwhere s (conditioned on x) is distributed as PBin(η(x 1 ), . . . , η(x k )). Set for brevity µ = E[η(x)(1η(x)], and introduce the short-hand notationC β,i = C β (x (-i) ) = I k j:j=1,j̸ =i η(x j )(1 -η(x j )) ≤ (k -1)µ -β ,for some β ∈ [0, (k -1)µ) to be specified. In the above x (-i) denotes x with its i-th item x i removed.We haveη(x i )(1 -η(x i ))E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | x = C β,i η(x i )(1 -η(x i ))E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | x + (1 -C β,i )η(x i )(1 -η(x i ))E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | x ≤ C β,i + (1 -C β,i )η(x i )(1 -η(x i ))E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | x , since η(x i )(1 -η(x i ))E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | x ≤ 1 .for every i and every realization x.On the other hand, a closer look at term(1 -C β,i )E s P(Σ (-i) = s | x) -P(Σ (-i) = s -1 | x) P(Σ = s | x) | xreveals that we are in a position to apply Lemma A.10. This allows us to conclude that this expectation is upper bounded by 19(1 -C β,i ) k j:j̸ =i,j=1 η(x j )(1 -η(x j )) for every realization of x. Using the definition of C β,i the above can be further upper bounded by 9(k -1)µ -β .Putting together, we have obtainedf 1 (x) -f 2 (x) ≤ 1 k k i=1 C β,i + 9η(x i )(1 -η(x i )) (k -1)µ -β = 1 k k i=1 C β,i + 9 k i=1 η(x i )(1 -η(x i )) k (k -1)µ -βfor every x. Now, from the standard Bernstein inequality,2 P x∼D X (C β,i = 0 ∀ i) ≥ 1 -k e -β 2 2kµ+β/6andP x∼D X k i=1 η(x i )(1 -η(x i )) ≤ kµ + β ≥ 1 -e -β 2 2kµ+β/6for every β > 0. This yieldsP x∼D X f 1 (x) -f 2 (x) > 9(kµ + β) k (k -1)µ -β ≤ (k + 1) e -β 22kµ+β/6 (14) holding for 0 ≤ β < (k -1)µ.One can easily see that choosing β = (k -1)µ/2 yields, when k = Ω(1/µ),P x∼D X f 1 (x) -f 2 (x) = Ω µ k ≤ k e -Θ(kµ) ,which concludes the proof.Remark A.15. The analysis contained above also allows us to provide bounds on the additive advantage that are fully conditional on x = (x 1 , . . . , x m ). For instance, combining Lemma A.10 with Eq. ( 13), one can easily derive a bound of the formf 1 (x) -f 2 (x) = O   1 k k i=1 η(x i )(1 -η(x i )) j:j̸ =i η(x j )(1 -η(x j ))   .Then, in order for this bound to be of the form 1/ √ k, we need to make further assumptions on the function x → P(y = 1|x), the most obvious one being P(y = 1|x) ∈ (0, 1) (bounded away from 0 and 1) for all x.Lemma A. 16 . For all data distributions D, all x i , and all datasets S of size m, IEAdv(M RR , D, x i ) is independent of m, and so is EAdv(M RR , D).Proof. Fix any m, any x i , and any data distribution D. Then, by Lemma A.1,EQUATION-Eyi∼D Y|x i min P (x (-i) ,y (-i) ) (y i = 1 | x, M RR (x, y)), P (x (-i) ,y (-i) ) (y i = 0 | x, M RR (x, y)) x i .The first term already has no dependence on m. We focus on the second term.The key to this proof is that a noisy label RR(y i ) = ỹi is independent of the other true labels y j , j ̸ = i. Thus, we haveP(y i = 1 | x, M RR (x, y)) = P(y i = 1 | x, M RR (x, y) i ) = P(y i = 1 | x i , M RR (x i , y i )) .Applying this to (15), we haveE yi∼D Y|x i min P (x (-i) ,y (-i) ) (y i = 1 | x, M RR (x, y)), P (x (-i) ,y (-i) ) (y i = 0 | x, M RR (x, y)) | x i = E yi∼D Y|x i min P(y i = 1 | x i , M RR (x i , y i )), P(y i = 0 | x i , M RR (x i , y i )) | x i .Plugging this back into (15) gives us EAdv 1 (M RR , D) as desired. Taking the expectation over x i ∼ D X completes the proof of the second claim.A.7 Proof of Theorem 3.5We prove a more verbose version of Theorem 3.5 which includes the optimal adversary: Theorem A.17. For any data distribution D, the individual expected attack advantage for randomized response with privacy parameter π = 1 1+e ϵ isIEAdv(M RR , D, x 1 ) = min{η(x 1 ), 1 -η(x 1 )} -π •I{η(x 1 ) ∈ [π, 1 -π]}.The optimal adversary maximizing IEAdv 1 (M RR , D, x 1 ) isA * (x 1 , ỹ1 ) =    1, if η(x 1 ) > 1 -π 0, if η(x 1 ) < π ỹ1 , otherwise .In words, upon receiving x 1 , the optimal attacker A * predicts the associated label y 1 as the noisy label ỹ1 if the distribution D contains enough uncertainty about y 1 (the condition η(x 1 ) ∈ [π, 1 -π] holds) given the extra noise level π injected. Otherwise, it predicts deterministically (as y 1 is indeed close to being deterministic when η(x 1 ) / ∈ [π, 1 -π]).Proof. First we characterize the optimal attacker for which the individual expected advantage is maximal. The Bayes optimal decision which minimizes the loss I{A(x 1 , ỹ1 ) ̸ = y} conditioned on x 1 and ỹ1 isA ′ (x 1 , ỹ1 ) = I{P(y 1 = 1|x 1 , ỹ1 ) > P(y 1 = 0|x 1 , ỹ1)} which can be written asEQUATIONAssume that ỹ1 = 1, in which case ( 16) becomesEQUATIONwhich is true whenever η(x 1 ) > π, and on the other hand, if η(x 1 ) < π then (17) does not hold anymore, thus A ′ (x 1 , ỹ1 ) = 0. A similar argument holds for ỹ1 = 0: The Bayes optimal decision A ′ (x 1 , ỹ1 ) = 1 if and only if η(x 1 ) > 1 -π. Putting together, and noting that π ≤ 1/2, this implies that the optimal attack isA * (x 1 , ỹ1 ) =    1, if η(x 1 ) > 1 -π 0, if η(x 1 ) < π ỹ1 , otherwise .To compute the reconstruction advantage of optimal attacker A * , we may decompose the feature set into parts as G(π) = {x 1 : η(x 1 ) ∈ [π, 1 -π]} and its complement set G C (π). It is clear the advantage of the optimal attacker A * restricted to G C (π) is 0 since ỹ1 gives no extra information on y 1 when x 1 ∈ G C (π), that is, the value of A * (x 1 , ỹ1 ) is independent of ỹ1 when x 1 ∈ G C (π). On the other hand, the advantage of A * for anyx 1 ∈ G(π) is (1 -π) -(1 -min{η(x 1 ), 1 -η(x 1 )}) = min{η(x 1), 1 -η(x 1 )} -π , since M RR reveals the true label with probability 1 -π, while the attack utility (conditioned on x 1 ) of the uninformed attacker 1 -min{η(x 1 ), 1 -η(x 1 )}. This concludes the proof.B Proofs for Section 3.2 B.1 Proof of Theorem 3.7Proof. Consider the difference of log-odds ratio I k,i (M, D, x, z, i), where z = M LLP (x, y) = 1 k k i=1 y i is the output of the aggregation PET. Since y is independent of x, we can writeI 1,0 (M, D, x, z, i) = I(z) = ln P(y 1 = 1 | z) P(y 1 = 0 | z) -ln P(y 1 = 1) P(y 1 = 0) = ln z 1 -z -ln p 1 -p = ln z(1 -p) (1 -z)p ,where we used the fact that, since kz is a binomial random variable, P(y 1 = 1 | z) = z, independent of p. From the standard Bernstein inequality we have, with probability ≥ 1 -δ, p -B ≤ z ≤ p + B , whereB = B(k, p, δ) = 2p(1 -p) ln(1/δ) k + 2 3k ln(1/δ) . Since z → ln z 1-z is monotonically increasing in z ∈ [0, 1], this yields ln 1 - B (1 -p + B)p = ln (p -B)(1 -p) (1 -p + B)p ≤ I(z) ≤ ln (p + B)(1 -p) (1 -p -B)p = ln 1 + B (1 -p -B)p with the same probability. The condition k ≥ 32 ln(1/δ) p(1-p) implies 2B ≤ p -p 2 , which in turn implies B (1-p+B)p ≤ 12 , as well asB (1-p-B)p > 0.We further bound the right-most side through ln(1 + x) ≤ x for all x ≥ 0, and the left-most side via ln(1-x) ≥ -x(ln 2) for x ∈ [0, 1/2]. This gives - B ln 2 (1 -p + B)p ≤ I(z) ≤ B (1 -p -B)p ,which, after further overapproximations, allows us to write|I(z)| ≤ B ln 2 (1 -p -B)p .Next, the condition 2B ≤ p -p 2 implies (1 -p -B) ≥ (1 -p)/2 so that, overall,|I(z)| ≤ 2B(k, p, δ)(ln 2) p(1 -p) = O ln(1/δ) p(1 -p)kwith probability at least 1 -δ. This concludes the proof.Datasets. We conduct our experiments on the following datasets: KDD2012: We use the click prediction data from the KDD Cup 2012, Track 2 [3] with the feature processing performed in [23] . The learning task for this data is to predict the click through rate for an advertisement based on a number of features related to the advertisement, the page that it appears on, and the user viewing it. There are 11 categorical features that are each one-hot encoded, resulting in a sparse feature vector with 11 non-zero entries in 54,686,452 dimensions. The label for the example is 1 if the ad was clicked, and 0 otherwise.We also use the Higgs dataset [4] , which is a binary classification dataset where the goal is to distinguish between a signal process that produces Higgs bosons, and a background process which does not. The dataset has 28 real-valued features, which are a mix of kinematic properties measured by particle detectors and hand-crafted high level features designed by physicists. The dataset consists of 11,000,000 examples produced via Monte Carlo simulation. We use the first 10,000 examples as our testing data, and the remaining examples for training.Model Architectures. Next, we describe the models we use for each dataset. We use the same model architecture regardless of which PET has been used to privatize the labels.KDD2012 Model: The model we use for KDD2012 is a deep embedding network. We reduce the dimension of each example from 54,686,452 dimensions to 500,000 dimensions by hashing feature indices. Each hashed feature index is associated with a learned embedding vector in R 50 , and the representation vector for an example is the sum of the learned embeddings for each of its non-zero hashed feature indices. This representation vector is passed through two dense layers with 100 and 50 units, respectively, and ReLU activation functions. The final output is a single unit with sigmoid activation that is interpreted as the click probability for each example.Higgs Model: For the Higgs data, we do no feature pre-processing. We use a fully connected feed forward neural network with 4 hidden layers all with 300 units and ReLU activation. The final output is a single unit with sigmoid activation that is interpreted as the probability that the example has the positive class.Training Setup. When training a model on the output of any PET, we always use minibatch gradient descent together with the Adam optimizer [24] . PET-specific training details are presented in the following paragraphs.Training with Randomized Response. In our experiments we found that the bias in randomized response can affect prediction accuracy for the model trained on these labels, so we post-process the RR output to remove the bias introduced (see for example [19] , Equation (7) therein).Let f : Y → R be any function (for example the gradient) of a label, and let y ∈ {0, 1}. For plain RR,E[f (RR(y))] = e ϵ -1 e ϵ + 1 • f (y) + 1 e ϵ + 1 • (f (0) + f (1)).Thus,f (y) = E e ϵ + 1 e ϵ -1 • f (RR(y)) - 1 e ϵ -1 • (f (0) + f (1)) .Therefore, we can post-process the output of RR to obtain an unbiased estimate of f (y) for any function f . This has no effect on the advantage, however it does help the accuracy substantially. We use this debiasing procedure to obtain an unbiased estimate of the binary cross-entropy loss of the model, and perform minibatch gradient descent on the estimated loss. In all of our figures, we use "RR" to denote this debiased version of the mechanism.Training with LLP. For LLP we use minibatch gradient descent to optimize the Empirical Proportion Risk defined in Section 2. We use the binary cross-entropy loss to compare the predicted label proportion against the observed label proportion. As discussed in Appendix D, since the binary cross-entropy loss is a proper loss, the minimizer of the population-level proportion risk for cross-entropy loss is also a minimizer of the population-level cross-entropy loss, as long as the Bayes optimal classifier can be represented by the model architecture.Training with LLP+Lap. LLP+Lap is parametrized by two parameters, k ∈ {1, 2, . . .} and ϵ > 0.For each bag B i the mechanism discloses α i + Z i , where α i is the label proportion in that bag, and Z i ∼ Lap( 1 kϵ ), is an independent zero-mean Laplace random variable with parameter 1 kϵ , that is, with variance 2 k 2 ϵ 2 . Theorem D.2 shows that for any minibatch of data, the expected gradient of the proportion matching risk is unaffected by adding mean-zero noise to the label proportion. Therefore, using Proportion Matching to train with LLP+Lap is similar to training with LLP, except the variance of each gradient step is increased.Training with LLP+Geom. LLP+Geom also has parameters, k ∈ {1, 2, . . .} and ϵ > 0, but Z i is replaced by a two-sided Geometric random variable, whereZ i = Z + i -Z - i k, and Z + i and Z - i are geometric distribution with probability of success 1 -e -ϵ . The value α i + Z i is then clipped to [0, 1]. It is not hard to see that, for every given ϵ, LLP+Geom with parameters k = 1 and ϵ coincides with RR with parameter ϵ.Unlike LLP+Laplace, the clipped noisy label proportion is not an unbiased estimate of the bag's true label proportion. We post-process the output of LLP+Geom to obtain unbiased estimates of the bag's label proportion. In particular, let α i be the true label proportion for a bag, α (noise) i = α i + Z i be the unclipped noisy proportion, and α (clip) i = clip(α (noise) i , [0, 1]) be the clipped noisy proportion. Then, using the memoryless property of the Geometric distribution, we have thatE[α (noise) i | α (clip) i ] =        α (clip) i if α (clip) i ∈ (0, 1) -1 k 1 1-e -ϵ -1 if α (clip) i = 0 1 + 1 k 1 1-e -ϵ -1 if α (clip) i = 1.When training with LLP+Geom, we replace the clipped label proportion byE[α (noise) i | α (clip) i] before computing the proportion matching loss. Since α (noise) i is an unbiased estimate of α i , so is the conditional expectation. After this debiasing, Theorem D.2 guarantees that the expected gradient of the proportion matching risk is the same as if the true label proportion had been used. It follows that using Proportion Matching to train with LLP+Geom with proportion debiasing is similar to training with LLP except with higher variance.AUC vs Advantage Experimental Setup. For all PETs, we train models to minimize the binary cross-entropy loss. For RR, we use privacy parameters ϵ in {2 -4 , 2 -3 , . . . , 2 5 }, for LLP we use bag sizes k in {2 0 , 2 1 , . . . , 2 9 }, and for LLP+Laplace and LLP+Geom we use all combinations of ϵ and k from the same sets. For every PET and every value of their privacy parameters, we train the model with each learning rate in {10 -6 , 5 • 10 -6 , 10 -5 , 10 -4 , 5 • 10 -4 , 10 -3 , 5 • 10 -3 , 10 -2 }. Finally, for each combination of privacy parameter and learning rate, we train the model 10 times (each trial corresponds to different model initialization and data shuffling, and RR noise). For each privacy parameter, we report the mean AUC of the learning rate with the highest mean AUC.Estimating Attack Advantage. The attack advantage of the optimal attacker is estimated and reported for every PET with various parameters. The optimal attacker and its advantage have been characterized for RR as well as for LLP, in Lemma A.1 and Theorem A.17, respectively. Crucially, notice that the advantage can be easily calculated with knowledge of the parameters of each PET (label flipping probability or the bag size), and the class conditional distribution η(x) -or at least an accurate estimate of it. We considered two estimates of the class conditionals: one based on Deep Neural Networks (DNN) as described above, and one based of k-Nearest Neighbors (kNN). The kNN estimator can produce accurate estimates for class conditional distributions in the large-scale data regimes, due to its strong consistency properties that has been studied in the 70s and 80s, and that became the part of Machine Learning folklore. DNNs can also produce accurate estimates if the training process and architecture is tuned carefully enough. We found that these two approaches result in very similar results on our large scale benchmark datasets, therefore we relied on the estimate provided by the DNN, and we estimated the attack advantage of the optimal attacker based on the output score of the DNN.Remark. It is known that when trained with event-level (i.e., non-private) labels, DNNs are not guaranteed to give rise to calibrated classifiers, which is important for estimating the probabilities involved in the definition of attack advantage, while a kNN is typically calibrated. Yet, since we Utility is measured by AUC on test set, while privacy is either the additive measure (bottom row) or the 98th-percentile of the multiplicative measure (so as to rule out the infinite multiplicative advantage cases that can occur for LLP). Each point corresponds to a setting of the privacy parameter for the PET (ϵ for RR, k for LLP, and both for LLP+Geom). The x-coordinate is the advantage (either additive or multiplicative) value for that PET, while the y-coordinate is the test AUC of a model trained from the output of that PET. The AUC of the model trained without a PET roughly corresponds to the top value achieved by these curves. This plot is similar to Figure 3 except that the curves for the LLP+Geom PET correspond to a fixed value of ϵ, rather than a fixed value of k. observed in our initial experiments that the two learning methods resulted in similar overall outcomes, we decided go with DNNs as the underlying classifier for estimating the behavior of the optimal attacker.Further experimental results omitted from the main text. Figure 4 contains scatter plots for LLP+Lap in addition to LLP+Geom for comparison. The two groups of scatter plots look very similar on each dataset. Figure 5 shows the privacy-vs-utility plots (the data is identical to Figure 3 ) except in this case, the curves for the LLP+Geom PET correspond to a fixed value of ϵ with the bag size k varying.Computational Resources. We conduct our experiments on a cluster of virtual machines each equipped with a p100 GPU, 16 core CPU, and 16GB of memory. Each training run on Higgs takes approximately 10 minutes, and each training run on KDD12 takes approximately 20 hours.In this appendix we argue that the standard PROPMATCH algorithm is still an effective learning algorithm for the LLP+Lap PET (henceforth denoted M Lap-LLP ). Recall that PROPMATCH generally learns by using stochastic gradient descent (SGD) to minimize the empirical proportion matching loss. Our result below shows that the expected value of the gradient of proportion matching loss is the same whether Laplace noise is added to the label proportion or not. In other words, the expected trajectory of SGD for PROPMATCH when using M Lap-LLP is the same for all values of ϵ, including ϵ = ∞, which corresponds to the special case of M LLP . In fact, the main way that decreasing the value of ϵ (resulting in stronger differential privacy guarantees) affects utility is the increased variance of the gradients. The following result characterizes how much the variance increases compared to the case where ϵ = ∞ (or equivalently, compared to PROPMATCH run with M LLP ).To analyze PROPMATCH, we require a few properties of the loss ℓ : R × R → R and the noise distribution. First, the loss is proper, which means that for every real-valued random variable A, the function f (p) = E(ℓ(p, A)) has a unique minimizer at µ = E(A). We also assume that ℓ(p, y) is differentiable in p, and that its derivative is affine in y, that is This differs from the loss studied in [13] only in that we allow the label proportion to be perturbed. The PROPMATCH algorithm essentially minimizes this loss. Theorem D.1. Let B be a random bag of size k, ℓ : R × R → R + ∪ {∞} be a loss function, and M be a mechanism for releasing the label proportions with M(α) = α = α + Z α , where E[Z α |α] = 0 for all α. Suppose H contains a predictor h * that matches the true label probabilities in D (i.e. h * (x) = P D (y = 1|x). If ℓ is proper (defined above), then h * is a minimizer of the population-level noisy proportion-matching loss on D. It is essentially unique, in that any other minimizer h ′ must satisfy P (x,y)∼D (h ′ (x) = h * (x)) = 1.Theorem D.2. Let B be a random bag of size k, ℓ : R × R → R + ∪ {∞} be a loss function, and M be a mechanism for releasing the label proportions withEQUATIONand furthermoreEQUATIONThat is, for nice loss functions like the logistic and square loss, we can write the variance of the noisy LLP gradient as the sum of two terms: the variance of the LLP-based estimate of the proportionmatching loss, and a noise term that scales roughly as the variance of Z. For Laplace noise, this variance is 2 k 2 ϵ 2 . For Geometric noise, the variance is upper bounded above by 2e -ϵ k 2 (1-e -ϵ ) 2 (which is always smaller than the Laplace variance, most noticeably for large ϵ). For small values of k, this upper bound is loose (because clipping and debiasing reduces the overall variance). For example, with randomized response (which is the same as LLP+Geom with k = 1), the variance is just e ϵ e ϵ +1 . Analyzing this formula helps us understand the performance of the noisy variants of LLP. Let us compare LLP+Geom with bag size k to averaging randomized response estimates over k separate examples. Suppose we fix ϵ and let k increase. There are three effects in play: (1) the noise added for privacy is lower for LLP+Geom (variance 1/k 2 versus 1/k); (2) the bias of the gradient estimator is higher for LLP+Geom, since we are estimating the gradient of the proportion-matching loss and not necessarily the true gradient; (3) the variance due to sampling is about the same in both cases, about 1/k.We thus get a classic bias-variance tradeoff. As a thought experiment, consider what happens when ϵ is very small, and almost all the error comes from the added noise. In that case, setting k a bit larger than 1 will help, since the bias introduced will be smaller than the reduction in noise. However, there is a point of diminishing returns, at least when the proportion-matching optimum is different from the true optimum (that is, when the model cannot express the Bayes-optimal predictor). Looking at the experiments with the Higgs data set, we see a muted version of this effect: for small values of ϵ, there is no real difference in the performance of LLP+Geom for k ∈ {1, 2, 4}, but larger k introduce a real penalty. For larger values of ϵ (over 1) on Higgs, and all values of ϵ on KDD12, the dropoff occurs already for k = 2. This analysis suggests that the advantages of LLP+Geom over RR will be dataset-dependent, and noticeable only for small values of ϵ.Proof of Theorem D.1. This proof is the same as the argument in [12, Thm 3.2] . Fix some bag B of examples, and take expectation over the sampling of their labels to get a label proportion α and its noisy variant α. Because the loss is proper, the unique minimizer of the proportion matching loss is given by the predictions of h * .arg min p E α, α ℓ(p, α) = E(α) = E(α) = 1 k h * (x) .This is true for every bag B, and thus in expectation over the choice of B. Furthermore, this loss is unique since the indicator functions of bags form a spanning set for the indicators for the support of D. If h ̸ = h * with nonzero probability, then the bag-level predictions of h will differ from those of h * with nonzero probability.Proof of Theorem D.2. Let ℓ ′ (p, y) = ∂ ∂p ℓ(p, y) = a p y + b p , where a p = ∂ ∂y ℓ ′ (p, y). Using the chain rule and the fact that ℓ ′ is affine, we get:∇ θ ℓ (h θ (B), α) = ℓ ′ (h θ (B), α)∇ θ h θ (B) = ℓ ′ (h θ (B), α) + Z α a h θ (B) ∇ θ h θ (B) = ∇ θ ℓ(h θ (B), α) + Z α a h θ (B) ∇ θ h θ (B) = ∇ θ ℓ(h θ (B), α) + Z α • ∂ ∂α ∇ θ ℓ(h θ (B), α) .Since the noise is unbiased for every α, the second term above is 0 in expectation for every pair (B, α). This proves that the estimator is unbiased (Equation ( 18)). To analyze its variance, note that the unbiasedness of α means that the two terms are decorrelated, and we can sum their convariances:Covar ∇ θ ℓ(h θ (B), α) + Z α • ∂ ∂α ∇ θ ℓ(h θ (B), α) = Covar (∇ θ ℓ(h θ (B), α)) + Covar Z α • ∂ ∂α ∇ θ ℓ(h θ (B), α) + E ∇ θ ℓ(h θ (B), α) , ∂ ∂α ∇ θ ℓ(h θ (B), α)) = Covar (∇ θ ℓ(h θ (B), α)) + Covar Z α • ∂ ∂α ∇ θ ℓ(h θ (B), α) .The expected squared norm of a random vector is the trace of its covariance. Thus, the expression above gives us the equality in Equation (18) . The final inequality follows again from decorrelation: by conditioning on the pair (B, α), we can pull out E[Z 2 α ] from the outer product ofZ α • ∂ ∂α ∇ θ ℓ(h θ (B), α) with itself.Taking the trace (to get the expected square norm), we can use the fact that ∥∇ θ ℓ(h θ (B), α)∥ 2 is nonegative, to replace E[Z 2 α ] with an upper bound that holds for all α. This yields the final inequality in Equation ( 19), as desired.We have analyzed PROPMATCH in this section since it is the algorithm used in our experiments. Similar results hold for other LLP algorithms, like EASYLLP [12] , though we do not work those out here.• 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.38th Conference on Neural Information Processing Systems (NeurIPS 2024).The condition k i=1 ηi(1 -ηi) ≥ 1 therein is implied by C β,i (x (-i) ) = 0, which is equivalent to (k -1)µ ≥ 1 + β. This, in turn, given our choice of β, will read as k = Ω(1/µ).Note that we are upper bounding the second moments of the random variables η(xi)(1 -η(xi)) with their first moment µ, since the variables η(xi)(1 -η(xi)) are in [0, 1/4]. 


Decision: The submission introduces two auditing metrics (additive and multiplicative attack advantages) to assess the vulnerability of label privatization mechanisms. The authors effectively demonstrate the application of these metrics to (empirically) auditing both differentially private and non-differentially private privacy mechanisms. Most reviewers praised the work, noting its potential for impact in the growing field of privacy auditing. In particular, during the discussion, Reviewer mygV highlighted the paper’s practical contributions and pragmatic approach, as well as its analysis of privacy-utility trade-offs across various mechanisms. Reviewer kz5F also appreciated the rigorous theoretical analysis and empirical validation. Reviewer enWp also strongly supported acceptance, emphasizing the paper's substantial methodological contributions to auditing privacy mechanisms and noting the novelty of the experimental results.

Given the reviewers' comments and my own read of the paper, this submission is a clear acceptance. While Reviewer Us4Y raised concerns regarding the paper's rigor and its placement relative to the broader privacy literature -- engaging in an extensive back-and-forth with the authors -- these concerns were not shared by other reviewers and the AC. I encourage the authors to incorporate the suggestions provided by the reviewers and promised in their rebuttal to the final paper, including releasing their code, providing more details on how to compute posterior distributions (Reviewer enWp), fixing typos, and clarifying ambiguities in some of the mathematical results (Reviewer Us4Y).

Reviews: review: paper_summary: paper_summary: This paper proposes two reconstruction advantage metrics to audit label privatization mechanisms. Unlike differentially private (DP) auditing techniques, which focus on worst-case guarantees, the authors of this work focus on distributional guarantees. Concretely, the authors assume the adversary has knowledge of the data distribution $\mathcal{D}$ over $\mathcal{X} \times \mathcal{Y}$ , either completely or approximately through learning on data. This allows the adversary the leverage correlations in the data, which are unknown to the mechanism, to augment their attack. Consequently, the attack advantage metrics proposed in this work measure the label reidentification risk that can be attributed to the mechanism rather than to correlations between the features and labels inherent to the distribution $\mathcal{D}$. Moreover, these metrics apply to both DP mechanisms and deterministic mechanisms commonly used in practice. 

The first metric investigated in the paper is the additive attack advantage, which was introduced in the context of label inference attacks by Wu et al. At a high level, this metric measures the difference between the expected accuracy of two Bayes optimal adversaries: one that can see the output of the mechanism and one that cannot. Notably, the expectation is taken over both the data distribution $\mathcal{D}$ and coin flips in the mechanism. Authors derive various analytical bounds that track how this measure changes under different correlation assumptions and mechanisms (Theorem 3.3, 3.4, 3.5), and demonstrate the privacy-utility trade-off using this metric for various DP mechanisms and a non-DP mechanism (Figure 3).

Since the first metric measures average-case risk, it does not capture unlikely high-disclosure events. The second metric, the multiplicative attack advantage, is introduced to remedy this shortcoming. At a high level, this metric measures the difference of log odds ratio between the adversary's prior and posterior probability of observing two different labels. This metric can be written as $\\log \\frac{\text{Pr}(\mathcal{M}(\mathbf{x}, \mathbf{y} ) = z \mid y_i = 1, \mathbf{x})}{\text{Pr}(\mathcal{M}(\mathbf{x}, \mathbf{y} ) = z \mid y_i = 0, \mathbf{x})}$, and is therefore analogous to the ''log likelihood ratio'' that is commonly characterized in the DP literature. In contrast to the DP literature, the probability in the likelihood ratio here is taken over both the data distribution $\mathcal{D}$ and coin flips in the mechanism. Authors provide a high probability bound for this second metric for a often utilized deterministic mechanism (Theorem 3.7), and demonstrate the privacy-utility trade-off using this metric for various DP mechanisms and a non-DP mechanism (Figure 3). Further notions of attack advantage are explored in Appendix A.5.  

An interesting result from this work is the empirical observation that learning with deterministic aggregate labels is harder than learning from randomly perturbed labels. In other words, randomized response had the best privacy-utility trade-off out of all the investigated mechanisms.
strength_weakness: strength_weakness: My main critique of this paper is that, given the prominence that the hybrid LLP+Geom mechanism plays in the experimental results, the paper spends comparatively little time explaining how either the proposed metrics or the posterior probability for this mechanism are computed. 

In the Questions section, I go into more detail and ask specific questions regarding this critique, and also give some suggestions to improve organization and readability of the paper.
questions: questions: $\textbf{Questions:}$ 

1. Figure 1 plots the prior and posterior distribution for LLP, RR and LLP+Geom. While I have a guess for how these posterior probabilities were calculated (e.g. using the ''likelihood'' $\text{Pr}(\mathcal{M}(\mathbf{x}, \mathbf{y} ) = z \mid y_i = 1, \mathbf{x})$ and the prior $\text{Pr}(y_i = 1 \mid \mathbf{x})$), the paper is not clear how the posterior probabilities were calculated, particularly for LLP+Geom and LLP+Lap. Could the authors please elaborate on how this was done? 

$\textbf{Suggestions:}$

1. Given how important the spindles are for the discussion in the experimental section, the remark "Note that a spindle boundary is the set of points having the same multiplicative advantage measure" on line 350 should be elaborated upon. I understand what the authors are trying to say, i.e. that the spindles in Figure 1 trace the contours of $\log \frac{y}{1-y} - \log \frac{x}{1-x} = \pm C$, but this was not obvious upon a first reading. 

2. In Theorem 3.2, it was as initially unclear to me why the random variable $\alpha$ appeared without a subscript denoting the ball it belongs to. I think would benefit the reader to remind them that by the iid assumption, the distribution of $\alpha$ over each ball is the same, hence the subscript can be dropped wlog. 

$\textbf{Typos and Related Nitpicks}$

1. Line 224 contains what I believe is a typo. It currently reads: "In the bounds of Theorems 3.2, 3.3 and 3.3”, whereas it should read "In the bounds of Theorems 3.2, 3.3 and 3.4”. 

2. In the following line 225, the statement "as $p$ and/or $\mu$ gets smaller we should naturally expect a smaller advantage" is misleading because of the previous theorems' dependence on $p$ and $1-p$. A more precise statement would be "as $p$ gets further from 1/2". 

3. There is currently an inconsistent use of $\epsilon$ and $\varepsilon$ in the paper. For example, in Definition 2.1 both $\epsilon$ and $\varepsilon$ are used in the same sentence to refer to the privacy parameter. 

4. In Definition 3.6, the arguments of $I\_{a,b}$ are $I_{a,b}(\mathcal{M}, \mathcal{D}, \mathbf{x}, z, i)$, but in the following paragraph the arguments $I_{a,b}(\mathcal{M}, \mathcal{D}, \mathbf{x}, z, x_i)$ are used. 

5. Lines 266 - 267: “What values of MA hould be considered acceptable? We argue that this probability should be viewed as a probability of system failure and set appropriately small” contains a typo with "hould" and an acronym "MA" that is not defined or (as far as I can tell) mentioned again.
limitations: limitations: Authors adequately address the limitations of their work.
paper_summary: paper_summary: The paper introduces new metrics to quantify the amount of label leakage associated with a generic privacy mechanism. It then analyzes two common label privacy mechanisms and quantifies the measures for these two mechanisms. Especially for the mechanism that also has the guarantee of differential privacy, the paper shows that the upper bound of the measures coming from differential privacy can be loose by comparing it to the results analyzed in the paper. In the experiment, the paper empirically evaluates the algorithms with both utility and the proposed privacy measurement.
strength_weakness: strength_weakness: As the measure is defined as the maximum advantage of all label inference attack $\mathcal{A}$ and the paper shows how to calculate the measure exactly for two mechanisms, it would be great to see how the existing label inference attack is bounded. This is because the measure is generally hard to compute, e.g. for the mechanisms that are not analyzed in the paper. If an existing label inference attack is shown to be tight to the exact upper bound, this attack can be used as an empirical indicator of the measure too.
questions: questions: Please see the weaknesses.
limitations: limitations: The paper has well discussed the limitations
paper_summary: paper_summary: The authors present new auditing tools for privacy mechanisms (differentially private or otherwise) with respect to the threat of label inference. They define measures to measure how much an adversary's posterior belief differs from their prior belief after viewing a dataset processed by a PET method, provide bounds on these measures for different methods theoretically, and provide an empirical assessment of privacy risks over multiple datasets. All of this is accompanied by an in-depth discussion about each definition, bound, and result, along with reflections on what this metric means for privacy and related considerations.
strength_weakness: strength_weakness: No clear weaknesses. The theory and experiments are solid, every proof is provided, the motivation is well done, and the discussion has great depth and clarity. This is among those rare papers that I cannot point to something to criticise (at least not right away).

The only weakness, to be pedantic, is the lack of code to reproduce the experiments, which I would appreciate the provision of by the authors.
questions: questions: * Could the authors please provide the code for reproducing the empirical results?
limitations: limitations: Yes, the authors do discuss that and speculate on future directions of work to address them.
paper_summary: paper_summary: This paper proposes a number of measures of information gained by an adversary from a mechanism.  It presents some theorems and performs a number of experiments.
strength_weakness: strength_weakness: The formalization is insufficiently rigorous and often unclear.
The claimed results don't seem to be always sound.
While it is to some extent interesting to define new measures, it is important to understand what we can learn from these measures.  The paper doesn't motivate why the proposed measures are preferable over existing measures.  The experiments are illustrative but don't really answer the question of the value of the measures, and also don't compare to existing alternatives.
Several other measures exist, e.g., there is quite some work using entropy-based measures to quantify how much an adversary can learn, i.e., the information gained by the adversary is the difference between the prior entropy and the posterio entropy.  


Some details:

* Often $\mathcal{M}$ is called a mechanism rather than a PET.  Usually, one uses PET to refer to a strategy to enhance privacy, not to a learning algorithm.
* In line 154, a $(\mathbf{x},\mathbf{y})$ is drawn from $\mathcal{D}^m$.  However, $\mathcal{D}^m$ is a product of $m$ datasets, each of the form $\mathcal{X}\times\mathcal{Y}$.  In contrast, $(\mathcal{x},\mathcal{y})$ is a pair of an element from $\mathcal{X}^m$ and an element from $\mathcal{Y}^m$.  These don't match syntactically.  Similar conversions of object seem to happen at several other places in the paper.
* In line 154, I also assume that $\mathcal{A)(\ldots)_i$ means the $i$-th component of the output of $\mathcal{A}$ (rather than for example $\mathcal{A}$ applied to the $i$-th component of $(\mathbf{x},\mathbf{y})$.
* In line 198, Theorem 3.2:
  - it is unclear what is the distribution of $\alpha$ (In fact, $\alpha$ has been used in various contexts before and it seems hard to unambiguously decide the meaning of $\alpha$ here).
  - it is unclear what is the role of the threshold $\beta$
  - $p$ is a probability, therefore the domain of $p$ is $[0,1]$.  The first alternative in the righthandside of the equation gives $p\in[0,1]$, so this first option will always apply.  It is unclear why then there is a second alternative $|p-1/2|\ge \beta$.  Depending on the constants hidden in $\Omega$ this second alternative may not be relevant, or it may is some cases provide a stronger bound.
  - Earlier text suggests that $\alpha_i=\frac{1}{k}\sum_{j=1}^k y_{i,j}.  In the proof, line 528, it seems now that $\alpha=\alpha_i$, but it is unclear what is the value of $i$ for which this holds.  Maybe you mean $\alpha$ is a random variable taking values $\alpha_i$ where $i$ is drawn uniformly from $[k]$ but even that doesn't solve all questions around here.
  - line 528 sets $\Sigma=k\alpha$ ($\Sigma$ is a real value), but doesn't explain what $\Sigma$ means in the conditional probability $P(\ldots | \Sigma)$ in the next line ($\Sigma$ would be an event or boolean condition).  The proof gets hard to understand at that point.
* in Line 208, when describing the behavior when $k$ increases I assume you are considering a scenario in which $n$ is kept constant.
* Theorem 3.3 suggests that if $\eta(x) \in \{0,1\}$, then $EAdv=0$.  This is unexpected:
  - around line 154, there is the definition of $EAU$.  The prior/uninformed attacker doesn't get any information about the distribution $\mathcal{D}$ as $\mathcal{M}_\bot(x,y)=\bot$, so he can only guess classes with probability proportionally to the frequency of these class labels.  In contrast, the informed attacker gets information from a mechanism $\mathcal{M}$.   One would hope the attacker can learn something from $\mathcal{M}$, especially as learning a deterministic function if easier than learning a probabilistic function.  Still, Theorem 3.3 seems to say that the EAdv will be zero.
questions: questions: --
limitations: limitations: The paper doesn't discuss limitations.
paper_summary: paper_summary: This paper develops measures for auditing privacy-preserving mechanisms that consume datasets with public features and private labels. The auditing measures compare two attackers whose goal is to infer a private label: a weaker attacker who has access to the dataset’s features and prior knowledge of the correlation between features and labels, and a stronger attacker who additionally has access to the privatized labels output by the mechanism. The advantage of the stronger attacker over the weaker attacker can be measured in two ways: additively or multiplicatively. Upper bounds on the additive and multiplicative measures are obtained for two privacy-preserving mechanisms: randomized response (RR) and learning from label proportions (LLP). These mechanisms are otherwise difficult to compare as RR satisfies label differential privacy, but LLP does not. Experiments on synthetic and real datasets, demonstrate that privacy-utility trade-offs can be evaluated using the proposed measures, and that RR dominates LLP.
strength_weakness: strength_weakness: The narrative could be strengthened, perhaps by leaning more on the application to online advertising. At times, I found myself questioning the path taken in the paper. For example, why are the additive and multiplicative measures both needed? It seems the multiplicative measure is superior as it is data-dependent, and a relative comparison is arguably more natural for probabilities. I also wondered why the definitions of expected attack utility and expected attack advantage differ from Wu et al. (2022), who condition on the features $\mathbf{x}$. It would be helpful to explain why conditioning on the features doesn’t make sense here.

Some of the bounds make strong assumptions about the data distribution. For instance, Theorems 3.2 and 3.7 assume the labels and features are uncorrelated, which doesn’t seem to be an interesting case to consider. There are analogues of Theorem 3.2 for general data distributions, but not Theorem 3.7.

The proposed measures are claimed to provide “a more balanced and nuanced picture” (line 40) of privacy/accuracy trade-offs. And the model of the adversary’s uncertainty is claimed to be “reasonable” (line 298). It would be helpful to elaborate on these claims a little more, so that practitioners can better understand any caveats. I understand that the measures are more optimistic because they depend on the adversary’s uncertainty (rather than assuming the worst-case). However the validity of the measures then depends on having a good model of the adversary’s uncertainty. The model adopted in the paper seems to assume the adversary has no side information. It would be great to discuss this assumption in the context of online advertising.


**Minor:**
- Figure 1: It is difficult to distinguish between different coloured points for LLP and LLP+Geom.
- line 224: Incorrect reference to Theorem 3.3.
- line 322: Is the classifier calibrated?
- line 252 and 260: There is an inconsistency in the last argument of the measure: $i$ vs $x_i$.
questions: questions: - How would you envisage these measures being applied? Would they be used to tune privacy parameters? Compare privacy-preserving mechanisms?
- Why is the model of the adversary’s uncertainty reasonable? Could you expand on this?
- Why does the definition of expected attack utility differ from Wu et al. (2022)?
limitations: limitations: Section 5 discusses limitations including the assumption of iid data, the need to estimate the adversary’s label uncertainty and label semantics. Another limitation of the work is that the measures are only instantiated for binary labels.

Comments: title: Response to Reviewer Us4Y's comments (cont'd)
Response to Reviewer Us4Y's comments
Official Comment by Reviewer zE7q
Response to Reviewer enWp's comment
comment: > There are several works that study, say, the mutual information between a mechanism’s input and output as a measure of privacy; however, in our setting these measures generally fail to distinguish between information gained as a result of: (1) correlation to the public features, and (2) the mechanism output.

Such work indeed may exist.  Still, the natural approach is to compare (a) the prior knowledge without the output of the mechanism and (b) the posterior knowledge after seeing the output of the mechanism.  Comparing these two can distinguish between correlation with public features (usually called prior knowledge) and mechanism output (on top of what is expected given the input and public features).

> Privacy vs. Learning

It is true that the authors, me, and in fact also some existing literature sometimes use the word "learning" when they mean "mechanism".  I trust in almost all cases this is for reasons of illustration, i.e., a privacy-preserving machine learning algorithm is an important example of a mechanism, and reasoning about the properties of learning algorithms can help us understand the desirable properties for a mechanism.  For the purpose of the current discussion, please be assured that in all these cases I (and I trust also the authors) mean these statements more generally and also applying to mechanisms in general (minus the learning-specific details).

I insist that none of the following is a PET (at least not the overwhelming majority of papers in the literature):
* a mechanism
* a learning algorithm
* a DP algorithm for probabilistic inference in a Bayesian network
* a DP algorithm for solving a constraint optimization problem

We could use the word PET in the following phrases:
* The PET used in a DP learning algorithm is "differential privacy"
* The PET used in a DP constraint optimization algorithm is "differential privacy"
* The PET used in an algorithm that performs stochastic gradient descent under homomorphic encryption is "homomorphic encryption"
* The PET used in "an algorithm sending a result encrypted using a one-time pad" is "encryption".

Here, encryption, hoimomorphic encryption, differential privacy, ... are (privacy enhancing) technologies (PETs), not mechanisms.
Please also not that
* "an algorithm sending a result encrypted using a one-time pad" is not a mechanism, as it is not randomized (given the pad).

> scope of the paper

The authors are free to choose the scope of the paper, the reviewers are free to have an opinion what is needed to make a work complete.

>. the paper considers an average case over the distribution D, which may have significant implications f

(I omitted here my original "learning theory" as these words made the sentence overly specific)

We should distinguish two cases:

(1) The common case (e.g. $(\epsilon,\delta)$-DP) : Whatever is the distribution and the dataset drawn from it, with high probability (over the randomization of the mechanism) the adversary will not be able to infer significant information from the output (over his prior knowledge).

(2) The case in the current submission:  Whatever is the distribution, with high probability over the dataset drawn from it and the randomization of the mechanism, the adversary will not be able to infer significant information from the output (over his prior knowledge).

In the latter case, it is possible that we are unlucky and the specific dataset drawn from the known distribution does not provide any privacy when the mechanism is applied.  (This is possible as the offered guarantee only applied on average over all datasets drawn)

In real-world applications, the current dataset is the only dataset we care about, we don't care about the "average dataset".  
The risk that our current dataset is worse than the average case exists, but it is sometimes hard to evaluate how much worse the situation is for the current dataset than for the average dataset.
**Further responses**

- On the use of quotes.

Apologies. We changed the order of the last two sentences and did some very light paraphrasing when writing "like entropy based" instead of "e.g., there is quite some work using entropy-based measures to ...".  Our paraphrasing came about when editing for length but it was not our intention to change the quotation.

- ``In general, please consider that the list of technical details provided in my review is only a sample. Other sections seem to have a similar level of rigor and require being improved too.”

This comment is rather awkward, since we are not put in a position to respond. If the reviewer has more technical issues to flag, then they are compelled to make these issues explicit.


- `i.i.d. data is required, but I guess that if the data is not i.i.d., in this setting this just means that the test data may be from a different distribution [...] valid worst case bounds? “

As discussed above, the test/train distinction the reviewer is speculating about is irrelevant to our setting, since we are interested in measuring the information revealed about the points in the training dataset not the test dataset. We mainly use the i.i.d. assumption to reason about the attacker’s knowledge, specifically that one person’s label does not reveal any additional information about another person’s label. 

- ``The text says "public features together with one sensitive binary feature", but it is unclear why the proposed measures can't be applied in other cases.”

We don’t fully understand which specific “other cases” the reviewer has in mind. Our measures can clearly be applied, e.g., to multiclass (that is, nonbinary) sensitive features.  Yet, it is not the point of the paper to widen the applicability of the proposed method to the largest possible extent. 

- On the statement of Thm 3.2. 

Thanks, we will consider the reviewer's suggestion here, like splitting the bound into two separate statements and/or giving a specific reference to the appendix. 

- On composability.

LLP does not satisfy composability guarantees in the worst case (in particular if two computations use bags that differ on one person’s data). Thus, any measure that adequately captures LLP also will not satisfy composability guarantees. This is briefly mentioned in Sect. 5.


- ``it seems to be assumed the adversary also has access to the dataset. E.g., Line 164 first applies to the complete training set and then takes the i-th component.” 

No, the adversary only sees the feature parts of the training set. The reason why there is also a probability over the labels $y$ is because the measures also factor in the conditional distribution of $y$ given $x$.


-  ``It doesn't allow to run on some instance not in the training set. This seems to limit the applicability of the proposal.”

As described above, our goal is to measure the risk to a given dataset by publishing the output of the PET run on that data, not to prevent statistical inferences on future data not processed by the PET.
**On related  work, entropy-based measures, etc.**

There are several works that study, say, the mutual information between a mechanism’s input and output as a measure of privacy; however, in our setting these measures generally fail to distinguish between information gained as a result of: (1) correlation to the public features, and (2) the mechanism output. Thus, they overestimate the revelation of the mechanism, at times to the extreme. For clarity, consider the extreme situation where $\eta(x) \in {0,1}$ for all $x$. Then, the mutual information between $M(x,y)$ and $y$ fails to capture the fact that the adversary’s inference advantage is 0 for any reasonable measure of advantage in this setting, since the adversary has perfect knowledge of the labels without the mechanism output. Regarding the specific references the reviewer is pointing out, it is true that they have a similar goal as the one in this paper. Yet, those papers do not consider the public features/private label mixed setting we consider here (which is elucidated by the above simple example, and clearly necessary when investigating, for instance, the LLP mechanism), and so it is unclear how to incorporate the public information. This seems to be the case, e.g., for the two references Makhdoumi et al., and Xiao et al. the reviewer is providing. In any event, we will cite such papers in our related work section, thanks for bringing them up.

**Privacy vs. Learning**

The reviewer makes several comments and raises concerns related to training/testing data and learning formalisms that we believe are orthogonal to the main contributions of our paper, and perhaps indicate a misunderstanding of the goals of our paper. We want to emphasize that our proposed measures try to quantify the risk posed to individuals in a dataset as a result of the output of the PET, whether the released data is used for learning or not. We use machine learning to motivate the experimental section of our work, but our definitions and results make sense in other contexts. For example, if a human analyst uses the PET output to create a report or dashboard, our measures would still be meaningful in that context.

In particular, we believe that several drawbacks from the reviewer’s further thoughts are out of scope for our paper:

- “The evaluation of privacy happens on the same set as the training set.”

Yes, our measures quantify the privacy risk to a dataset due to running a PET on that dataset. Because the data is i.i.d., the only way the PET output can help an adversary infer the labels of data not present in the PET input is through learning statistical correlations between the features and labels. We do not feel that this risk should be attributed to the PET, since those correlations could potentially be learned from other data sources, or the adversary might already know them. Note that Differential Privacy also does not protect against learning correlations in a population (See for example Section 1 of  [Dwork, Roth “The Algorithmic Foundations of Differential Privacy” ‘14] for a full discussion of this concept).

- “The approach assumes adversaries have knowledge to the full distribution [...] This is different from [...] "fixed but unknown distribution" for the data.”

We do not agree that the common assumptions for learning should also be adopted when modeling an adversary. As explained in LL. 158–163, we assume the adversary knows the distribution $D$ to capture the fact that they may be very well informed about statistical patterns in the data. Our measures capture how much risk can be attributed to the PET itself, beyond what could be inferred from the publicly available data by a well-informed adversary.

- “... the paper considers an average case over the distribution D, which may have significant implications from the point of learning theory.”

Even though our risk measures average over the distribution $D$, it is unclear to us why this is listed as a drawback, and what are the “significant implications from the point [of view?] of learning theory” the reviewer is alluding to. As we said in the paper, since we want to put on the same footing deterministic and randomized mechanisms, we have to follow an average case (or high probability) analysis over the data distribution. If the mechanism is randomized then we *do* also consider the average behavior over the randomized mechanism (unlike what the reviewer seems to claim) – please have a closer look, e.g., at the display after L. 154. 

[to continue...]
Thank authors for their responses and they partially addressed my question. I decided to keep my positive score.
#### Technical details

In general, please consider that the list of technical details provided in my review is only a sample.  Other sections seem to have a similar level of rigor and require being improved too.

> There seem to be two general misunderstandings regarding our theoretical setting. First, ...

I agree with what you explain here, but stress that it is important that the reader too understands this.


> >   L. 154, syntactic mismatch.

> Yes, this is a minor abuse of notation (and the specific sampling that we use is described in lines 82-84). In general we identify elements of and if they describe the same sequence of labeled examples. We will fix it throughout so as to avoid this notational inconsistency.

If you abuse notation, it is important to tell this to the reader.  Lines 82-84 do not do that.

> The two bounds we give are generally incomparable, but when k is large and there is a big enough gap $\beta$, the second bound is clearly better.

This may be true, still as my review points out the statement on itself makes things hard to interpret, especially as the big-O notations hiding constant factors make it hard for a user to decide which of both bounds is the best one.   It seems that in this case it would be better to give explicitly the constant factor (or at least a sufficient approximation of it, or a reference to an appendix) so the reader can appreciate the relation.


#### Further thoughts


Thanks for the rebuttal.  Thinking again about the submission, I now see the following drawbacks which maybe I didn't sufficiently realize or not write in sufficient precision in my original review:

* it seems the proposal has no composability, i.e., in contrast to most existing work, given two mechanisms M1 and M2, EAU(A,(M1,M2),D) is hard to upper bound from EAU(A,M1,D) and EAU(A,M2,D).  This is a significant drawback which doesn't seem to be compensated by other advantages.
* The evaluation of privacy happens on the same set as the training set, e.g., in the equation after line 154.  In many use cases, the adversary gets some output (e.g., a learnt model) but not the original training data (feature parts).  These use cases can't be handled by the current proposal, it seems to be assumed the adversary also has access to the dataset.  E.g., Line 164 first applies $\mathcal{A}$ to the complete training set $\mathbf{x}$ and then takes the $i$-th component.  It doesn't allow to run $\mathcal{A}$ on some instance $x_i$ not in the training set.  This seems to limit the applicability of the proposal.
* The approach assumes adversaries have knowledge to the full distribution $\mathcal{D}$ and have the computational power to work with it.  This is different from the common assumption in machine learning where one assumes a "fixed but unknown distribution" for the data.  
* The paper doesn't follow a "worst case approach", which is reasonable.  However, rather than considering average behavior over the randomization of the mechanism, the paper considers an average case over the distribution $\mathcal{D}$, which may have significant implications from the point of learning theory.  

So in conclusion, I still feel the authors propose a new measure without sufficiently demonstrating advantages it has compared to the state of the art techniques.  The paper does compare to related work, but only a narrow segment of most similar work.  The approach clearly has some drawbacks, but the paper doesn't discuss their implications in much depth.
#### Section 5

While one could indeed argue Section 5 points to some limitations, the text in Section 5 does not seem very accurate, nor does it discuss these interesting points in much depth.

* Line 410 makes the overly general claim "for the first time allow us to compare their privacy-accuracy trade-off curves".  It is unclear why this wasn't possible with earlier approaches.
* Line 411-412 state that the approach is tailored towards a specific setting, but it doesn't explain how these limitations work and how it would fail in more general settings:
  - i.i.d. data is required, but I guess that if the data is not i.i.d., in this setting this just means that the test data may be from a different distribution than the training data and hence the adversary can even less reliably infer the label from the output?  So the measures the current submission presents would remain valid worst case bounds?
  - the text says "public features together with one sensitive binary feature", but it is unclear why the proposed measures can't be applied in other cases.
* The latter point may be related to line 415 refers to multiple mechanisms.  Composability of measures has always been an important point in privacy measure work, but it doesn't seem to be discussed here.  In fact, the text seems to suggest that there is no easy way to compose, i.e., if we have audited two mechanisms and now want to audit their composition, we can't easily from the measures on the first two mechanisms compute a worst case bound for the measure applied on the composition.  Not having such property would be a significant drawback which would deserve being discussed (with arguments why the proposed measure is nevertheless still better than alternatives).

#### Terminology

> > Often M is called a mechanism rather than a PET. Usually, one uses PET to refer to a strategy [...] not to a learning algorithm.

> It is true that we use both PET and Mechanism to refer to privacy enhancing technologies. However, we would like to emphasize in all cases, the PETs/Mechanisms we discuss are strategies for enhancing privacy and not learning algorithms. We will make this clearer in the paper.

I disagree.
* Privacy Enhancing Technology (PET) refers to a wide range of strategies to obtain or improve privacy.  Examples include homomorphic encryption, differential privacy, function secret sharing, etc.  One can see from relevant websites such as the PETS symposium and hits of googling "privacy enhancing technology" that a PET is not necessarily a mechanism
* A mechanism is a specific technical term referring to a randomized algorithm with an input and an output.   Literature on differential privacy almost always uses the term "mechanism", literature in cryptology often refers to "randomized algorithm".   It is indeed true that "mechanism" is a more general concept than "randomized learning algorithm", but it still is significantly different than the technologies the word PET refers to.
#### Comparison to related work

> >    “it is important to understand what we can learn from these measures. [...] Other measures exist, like entropy based. The paper doesn't motivate why [...] preferable over existing measures.”

Please, when you quote text of other people, copy it exactly (except for omissions you can indicate with [...]).  Else, please don't use a quotation style and just say "we understand that ... (your own interpretation) ...".

> We do discuss related work extensively in Section 3.3. 

It is important to read the full comment.   This is not only about "comparing to existing work".   The first sentence you quote says "it is important to understand what we can learn from these measures."  Comparison with related work is indeed one way to partially understand the meaning of a new measure, but it is not sufficient on itself.  It is relatively easy to define a new measure which is different from existing measures, but that doesn't necessarily mean that this new measure is interesting.  "What can we learn from the measure?" asks for explanation on the meaning of the output of the measure, and on how this output can be useful to perform tasks readers are familiar with (indeed, ideally in a way which is better than existing measures).

> Indeed, our paper explicitly builds on several other works that aim to measure advantage. We are not aware of entropy-based measures being applied in the specific context of label aggregation, or label privacy more generally. Yet, we would be very interested to add comparisons to such works if the reviewer could suggest some specific starting points.

My original text doesn't claim that there are papers specifically focusing on entropy-based measures for the specific context for label privacy.  I only said that there exist other measures, and that some of these are entropy-based.  I believe that the entropy-based work which exists is sufficiently general to apply (among others) to the specific context of label privacy.
Even if there would not exist a paper whose title is exactly "entropy-based measure in the specific context of label privacy" this shouldn't prevent authors from applying general methods to the specific context of label privacy for the purpose of comparison.

I'm not an expert myself in the specific combination of entropy based measures and label privacy, but it is easy to find papers which discuss measuring privacy using mutual information strategies, e.g.,

* Konstantinos Chatzikokolakis, Tom Chothia, and Apratim Guha. Statistical measurement of information leakage. In Tools and Algorithms for the Construction and Analysis of Systems, 16th International Conference, TACAS 2010, volume 6015 of Lecture Notes in Computer Science, pages 390–404. Springer, 2010.
* Ali Makhdoumi, Salman Salamatian, Nadia Fawaz, and Muriel Médard. From the information bottleneck to the privacy funnel. In 2014 IEEE Information Theory Workshop, ITW 2014, 501–505. IEEE, 2014.
* Lejla Batina, Benedikt Gierlichs, Emmanuel Prouff, Matthieu Rivain, François-Xavier Standaert, and Nicolas Veyrat-Charvillon. Mutual information analysis: a comprehensive study. J. Cryptol., 24(2):269–291, 2011.

As said, some existing measures are not entropy-based, e.g., 

Hanshen Xiao, Srinivas Devadas. PAC privacy: Automatic privacy measurement and
control of data processing. In Advances in Cryptology - CRYPTO 2023 - 43rd Annual Inter-
national Cryptology Conference, CRYPTO 2023, volume 14082 of Lecture Notes in Computer
Science, 611–644. 2023.

Most of this work aims at understanding how much the release of some output increases the risk of inferring sensitive information (similar to the key question in lines 144-146).  Hence, while my task is not to provide an exhaustive list of only relevant references, I conclude that there at least a few lines of thinking with goals similar to the current submission which are not mentioned in the current discussion of related work (nor in the comparative experiments or qualitative argumentation why the current proposal would be better).

> One possibly relevant point is that multiplicative measures such as the one we consider here generally upper bound entropy-based quantities like mutual information (which corresponds to the reviewer’s suggestion of entropy drop from prior to posterior). We can add a general discussion of that point to Section 3.2.

Indeed the specific example of mutual information is an exact, hence hard to compute, hence most authors approximate it.  It may indeed be useful to point out that the proposed new measure is an upper bound of some existing measure (if that would be the case, please provide a proof).

> We do discuss related work extensively in Section 3.3.

Sec 3.3 extensively discusses a narrow segment of related work.  It discusses "a recent line of work on auditing learning algorithms via membership inference attacks", but virtually no other related work.  It doesn't argue whether the new proposal has advantages.
Thanks, the authors' response has helped me better appreciate the context/objectives of the paper. I have decided to increase my score for soundness, contribution and the overall rating.
Yes, we agree with that suggestion. We generally did the convolution numerically, and will explain the calculation. We also plan to post the exact code -- Thanks !
I read through the authors' responses to my own comments and to the other reviewers. In response, I have increased my score to strong accept (8). 

My one comment is that, if computing the posterior probabilities in Figures 1/4 for the LLP-Geom/Laplace mechanism involves convolving (either numerically or analytically) a Geometric/Laplace random variable with a PoissonBinomial to obtain the likelihood $\text{Pr}(\mathcal{M}(\mathbf{x},\mathbf{y}) = z \mid y_i=1, \mathbf{x})$, that this convolution step should be explicitly mentioned somewhere in the Appendix.