Scalable DP-SGD: Shuffling vs. Poisson Subsampling

Authors: lynn  chua
ravi  kumar
pasin  manurangsi
amer  sinha
chiyuan  zhang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 19:17:46

Abstract: We provide new lower bounds on the privacy guarantee of _multi-epoch_ Adaptive Batch Linear Queries (ABLQ) mechanism with _shuffled batch sampling_, demonstrating substantial gaps when compared to _Poisson subsampling_; prior analysis was limited to a single epoch.
Since the privacy analysis of Differentially Private Stochastic Gradient Descent (DP-SGD) is obtained by analyzing the ABLQ mechanism, this brings into serious question the common practice of implementing Shuffling based DP-SGD, but reporting privacy parameters as if Poisson subsampling was used.
To understand the impact of this gap on the utility of trained machine learning models, we introduce a novel practical approach to implement Poisson subsampling _at scale_ using massively parallel computation, and efficiently train models with the same.
We provide a comparison between the utility of models trained with Poisson subsampling based DP-SGD, and the optimistic estimates of utility when using shuffling, via our new lower bounds on the privacy guarantee of ABLQ with shuffling.

Main paper: 
 introduction: 
 A common approach for private training of differentiable models, such as neural networks, is to apply first-order methods with noisy gradients. This general framework is known as DP-SGD (Differentially Private Stochastic Gradient Descent) [Abadi et al., 2016] ; the framework itself is compatible with any optimization sub-routine. Multiple open source implementations exist for applying DP-SGD in practice, namely, Tensorflow Privacy, JAX Privacy [Balle et al., 2022] and PyTorch Opacus [Yousefpour et al., 2021] ; and DP-SGD has been applied widely in various machine learning domains [e.g., Tramer and Boneh, 2020 , De et al., 2022 , Bu et al., 2022 , Chen et al., 2020 , Dockhorn et al., 2023 , Anil et al., 2022 , He et al., 2022 , Igamberdiev et al., 2024 , Tang et al., 2024] .DP-SGD (Algorithm 1) processes the training data in a sequence of steps, where at each step, a noisy estimate of the average gradient over a mini-batch is computed and used to perform a first-order update over the differentiable model. To obtain the noisy (average) gradient, the gradient g for each example in the mini-batch is clipped to have norm at most C (a pre-determined fixed bound), by setting [g] C := g • min{1, C/∥g∥ 2 }, and computing the sum over the batch; then independent zero-mean noise drawn from the Gaussian distribution of scale σC is added to each coordinate of the summed gradient. This could then be scaled by the "target" mini-batch size to obtain a noisy Algorithm 1 DP-SGD: Differentially Private Stochastic Gradient Descent [Abadi et al., 2016] Params: Batch sampler B (samples T batches, with "target" batch size b), differentiable loss ℓ : R d × X → R d , initial model state w 0 , clipping norm C, noise scale σ. Input: Dataset x = (x 1 , . . . , x n ). Output: Final model state w T ∈ R d .(S 1 , . . . , S T ) ← B(n) for t = 1, . . . , T do g t ←1 b N (0, σ 2 C 2 I d ) + x∈St [∇ w ℓ(w; x)] C w t ← w t-1 -η t g t ▷ Could also be some other optimization method. return w T average gradient. 1 The privacy guarantee of the mechanism depends on the following parameters: the noise scale σ, the number of examples in the training dataset, the size of mini-batches, the number of training steps, and the mini-batch generation process.In practice, almost all deep learning systems generate mini-batches of fixed-size by sequentially going over the dataset, possibly applying a global shuffling of all the examples in the dataset for each training epoch; each epoch corresponds to a single pass over the dataset, and the ordering of the examples may be kept the same or resampled between different epochs. However, performing the privacy analysis for such a mechanism has appeared to be technically difficult due to correlation between the different mini-batches. Abadi et al. [2016] instead consider a different mini-batch generation process of Poisson subsampling, wherein each mini-batch is generated independently by including each example with a fixed probability. This mini-batch generation process is however rarely implemented in practice, and consequently it has become common practice to use some form of shuffling in applications, but to report privacy parameters as if Poisson subsampling was used (see, e.g., the survey by Ponomareva et al. [2023, Section 4.3] ). A notable exception is the PyTorch Opacus library [Yousefpour et al., 2021] that supports the option of Poisson subsampling; however, this implementation only works well for datasets that allow efficient random access (for instance by loading it entirely into memory). To the best of knowledge, Poisson subsampling has not been used for training with DP-SGD on massive datasets.The privacy analysis of DP-SGD is usually performed by viewing it as a post-processing of an Adaptive Batch Linear Queries (ABLQ) mechanism that releases the estimates of a sequence of adaptively chosen linear queries on the mini-batches (formal definitions in Section 2.1). Chua et al. [2024] showed that the privacy loss of ABLQ with shuffling can be significantly higher than that with Poisson subsampling for small values of σ. Even though their analysis only applied to a single epoch mechanism, this has put under serious question the aforementioned common practice of implementing DP-SGD with some form of shuffling while reporting privacy parameters assuming Poisson subsampling. The motivating question for our work is:Which batch sampler provides the best utility for models trained with DP-SGD, when applied with the correct corresponding privacy accounting?1.1 Contributions Our contributions are summarized as follows.Privacy Analysis of Multi-Epoch ABLQ with Shuffling. We provide lower bounds on the privacy guarantees of shuffling-based ABLQ to handle multiple epochs. We consider the cases of both (i) Persistent Shuffling, wherein the examples are globally shuffled once and the order is kept the same between epochs, and (ii) Dynamic Shuffling, wherein the examples are globally shuffled independently for each epoch. Since our technique provides a lower bound on the privacy guarantee, the utility of the models obtained via shuffling-based DP-SGD with this privacy accounting is an optimistic estimate of the utility under the correct accounting. We choose B to be sufficiently larger than the expected batch size, so that the probability that the sampled batch size b exceeds the maximum allowed batch size B is small. We provide a modification to the analysis of ABLQ with Poisson subsampling in order to handle this difference.Generating these truncated Poisson subsampled batches can be difficult when the dataset is too large to fit in memory. We provide a scalable approach to the generation of batches with truncated Poisson subsampling using massively parallel computation [Dean and Ghemawat, 2004] . This can be easily specified using frameworks like beam [Apache Beam] and implemented on distributed platforms such as Apache Flink, Apache Spark, or Google Cloud Dataflow.Our detailed experimental results are presented in Section 4, and summarized below:• DP-SGD with Shuffle batch samplers performs similarly to Poisson subsampling for the same σ.• However, DP-SGD with Shuffle batch samplers, with our optimistic privacy accounting, perform worse than Poisson subsampling in high privacy regimes (small values of ε). Thus, our results suggest that Poisson subsampling is a viable option for implementing DP-SGD at scale, with almost no loss in utility compared to the traditional approach that uses shuffling with (incorrect) accounting assuming Poisson subsampling. et al. [2024] demonstrated gaps in the privacy analysis of ABLQ using shuffling and Poisson subsampling, by providing a lower bound on the privacy guarantee of ABLQ with shuffling; their technique, however, was specialized for one epoch. We extend their technique to the multi-epoch version of ABLQ with shuffling and provide lower bounds for both persistent and dynamic batching. Lebeda et al. [2024] also point out gaps in the privacy analysis of ABLQ with Poisson subsampling and with sampling batches of fixed size independently, showing that the latter has worse privacy guarantees than Poisson subsampling. We do not cover this sampling in our experimental study, since sampling independent batches of fixed size is not commonly implemented in practice, and DP-SGD using this sampling is only expected to be worse as compared to Poisson subsampling. Yousefpour et al. [2021] report the model utility (and computational cost overhead) under training with DP-SGD with Poisson subsampling. However, to the best of our knowledge, there is no prior work that has compared the model utility of DP-SGD under Poisson subsampling with that under shuffling, let alone compared it against DP-SGD under (Dynamic/Persistent) shuffling or studied the gaps between the privacy accounting of the two approaches.One possible gap between the privacy analysis of DP-SGD and ABLQ is that the former only releases the final iterate, whereas the latter releases the responses to all the queries. An interesting result by Annamalai [2024] shows that in general the privacy analysis of the last-iterate of DP-SGD cannot be improved over that of ABLQ, when using Poisson subsampling. This suggests that at least without any further assumptions, e.g., on the loss function, it is not possible to improve the privacy analysis of DP-SGD beyond that provided by ABLQ; this is in contrast to the techniques of privacy amplification by iteration for convex loss functions [e.g. Feldman et al., 2018, Altschuler and Talwar, 2022] . 
preliminaries: 
 A differentially private (DP) mechanism M : X * → ∆ O can be viewed as a mapping from input datasets to distributions over an output space, namely, on input dataset x = (x 1 , . . . , x n ) where each example x i ∈ X , M(x) is a probability measure over the output space O; for ease of notation, we often refer to the corresponding random variable also as M(x). Two datasets x and x ′ are said to be adjacent, denoted x ∼ x ′ , if they "differ in one example"; in particular, we use the "zeroing-out" adjacency defined shortly.Algorithm 2 ABLQ B : Adaptive Batch Linear Queries (as formalized in Chua et al. [2024] ) Params: Batch sampler B (samples T batches), noise scale σ, and (adaptive) query method A : Definition 2.1 (DP). For ε, δ ≥ 0, a mechanism M satisfies (ε, δ)-DP if for all "adjacent" datasets x ∼ x ′ , and for any (measurable) event Γ it holds that Pr(R d ) * × X → B d . Input: Dataset x = (x 1 , . . . , x n ). Output: Query estimates g 1 , . . . , g T ∈ R d (S 1 , . . . , S T ) ← B(n) for t = 1, . . . , T do ψ t (•) := A(g 1 , . . . , g t-1 ; •) g t ← e t + i∈St ψ t (x i ) for e t ∼ N (0, σ 2 I d ) return (g 1 , . . . , g T ) Algorithm 3 Π b,T (n; ⃗ π):[M(x) ∈ Γ] ≤ e ε Pr[M(x ′ ) ∈ Γ] + δ.For any mechanism M, we use δ M : R ≥0 → [0, 1] to denote its privacy loss curve, namely δ M (ε) is the smallest δ such that M satisfies (ε, δ)-DP; ε M : [0, 1] → R ≥0 is defined similarly.Following the notation in Chua et al. [2024] , we study the adaptive batch linear queries mechanism ABLQ B (Algorithm 2) using a batch sampler B and an adaptive query method A, defined. The batch sampler B can be any algorithm that randomly samples a sequence S 1 , . . . , S T of batches. ABLQ B operates by processing the batches in a sequential order, and produces a sequence (g 1 , . . . , g T ), where the response g t ∈ R d is produced as the sum of ψ t (x) over the batch S t with added zero-mean Gaussian noise of scale σ to all coordinates, where the query ψ t : X → B d (for B d := {v ∈ R d : ∥v∥ 2 ≤ 1}) is produced by the adaptive query method A, based on the previous responses g 1 , . . . , g t-1 . DP-SGD can be viewed as a post-processing of an adaptive query method that maps examples to the clipped gradient at the last iterate, namely ψ t (x) := [∇ w ℓ(w t-1 , x)] 1 (we treat the clipping norm C = 1 for simplicity, as it is just a scaling term).In this work, we consider the following multi-epoch batch samplers: Deterministic D b,T , Persistent Shuffle S ⋄ b,T , and Dynamic Shuffle S ⟳ b,T batch sampler defined as instantiations of Algorithm 3 in Figure 1 and truncated Poisson P b,B,T (Algorithm 4); we drop the subscripts of each sampler whenever it is clear from context. Note that, while P b,B,T has no restriction on the value of n, the samplers D b,T , S ⋄ b,T , and S ⟳ b,T require that the number of examples n is such that E := bT /n and S := n/b are integers, where E corresponds to the number of epochs and S corresponds to the number of steps per epoch. We call the tuple (n, b, T ) as "valid" if that holds, and we will often implicitly assume that this holds. Also note that P b,B,T corresponds to the standard Poisson subsampling without truncation when B = ∞. We use δ B (ε) to denote the privacy loss curve of ABLQ B for any B ∈ {D, P, S ⋄ , S ⟳ }, where other parameters such as σ, T , etc. are implicit. Namely, for all ε > 0, let δ B (ε) be the smallest δ ≥ 0 such that ABLQ B satisfies (ε, δ)-DP for all choices of the underlying adaptive query method A. We define ε B (δ) similarly. Finally, we define σ B (ε, δ) as the smallest σ such that ABLQ B satisfies (ε, δ)-DP, with other parameters being implicit in B.Deterministic Batch Sampler D b,T (n): Realized as Π b,T (n; Id, Id, . . .). where Id is the identity permutation, i.e., the data is not permuted. Persistent Shuffle Batch Sampler S ⋄ b,T : Realized as Π b,T (n; π, π, . . .), where π is a random permutation over [n], i.e., the data is shuffled once and the order is persistent across epochs.Dynamic Shuffle Batch Sampler S ⟳ b,T : Realized as Π b,T (n; π 0 , π 1 , . . .), where π e 's are i.i.d. random permutations over [n], i.e., the data is reshuffled in each epoch. Adjacency notion. The common notion of Add-Remove adjacency is not applicable for mechanisms such as ABLQ D , ABLQ S ⋄ , ABLQ S ⟳ because these methods require that bT /n and n/b are integers, and changing n by ±1 does not respect this requirement. And while the other common notion of Substitution is applicable for all the mechanisms we consider, the standard analysis for ABLQ P is done w.r.t Add-Remove adjacency [Abadi et al., 2016 , Mironov, 2017] . Therefore, we use the "Zeroing-out" adjacency introduced by Kairouz et al. [2021] , namely we consider the augmented input space X ⊥ := X ∪ {⊥} where any adaptive query method A is extended as A(g 1 , . . . , g t ; ⊥) := 0 for all g 1 , . . . , g t ∈ R d . Datasets x, x ′ ∈ X n ⊥ are said to be zero-out adjacent if there exists i such that x -i = x ′ -i , and exactly one of {x i , x ′ i } is in X and the other is ⊥. We use x → z x ′ to specifically denote adjacent datasets with x i ∈ X andx ′ i = ⊥. Thus x ∼ x ′ if either x → z x ′ or x ′ → z x.For two probability density functions P and Q and α, β ∈ R ≥0 , we use αP + βQ to denote the weighted sum of the density functions. We use P ⊗ Q to denote the product distribution sampled as(ω 1 , ω 2 ) for ω 1 ∼ P , ω 2 ∼ Q, and, P ⊗T to denote the T -fold product distribution P ⊗ • • • ⊗ P . For all ε ∈ R, the e ε -hockey stick divergence between P and Q is D e ε (P ∥Q) := sup Γ P (Γ) -e ε Q(Γ). Thus, by definition a mechanism M satisfies (ε, δ)-DP iff for all adjacent x ∼ x ′ , it holds that D e ε (M(x)∥M(x ′ )) ≤ δ.Definition 2.2 (Dominating Pair [Zhu et al., 2022] ). The pair (P, Q) dominates the pair (A, B)(denoted (P, Q) ≽ (A, B)) if D e ε (P ∥Q) ≥ D e ε (A∥B) holds for all ε ∈ R. We say that (P, Q) dominates a mechanism M (denoted (P, Q) ≽ M) if (P, Q) ≽ (M(x), M(x ′ )) for all adjacent x → z x ′ .If (P, Q) ≽ M, then for all ε ≥ 0, it holds that δ M (ε) ≤ max{D e ε (P ∥Q), D e ε (Q∥P )}, and conversely, if there exists adjacent datasetsx → z x ′ such that (M(x), M(x ′ )) ≽ (P, Q), then δ M (ε) ≥ max{D e ε (P ∥Q), D e ε (Q∥P )}.When both of these hold, we say that (P, Q) tightly dominates the mechanism M (denoted (P, Q) ≡ M) and in this case it holds that δ M (ε) = max{D e ε (P ∥Q), D e ε (Q∥P )}. Thus, tightly dominating pairs completely characterize the privacy loss of a mechanism (although they are not guaranteed to exist for all mechanisms). Dominating pairs behave nicely under mechanism compositions: if(P 1 , Q 1 ) ≽ M 1 and (P 2 , Q 2 ) ≽ M 2 , then (P 1 ⊗ P 2 , Q 1 ⊗ Q 2 ) ≽ M 1 • M 2 , where M 1 • M 2 denotes the (adaptively) composed mechanism.3 Privacy analysis of multi-epoch ABLQ BWe discuss the privacy analysis of ABLQ B for B ∈ {D, P, S ⋄ , S ⟳ } via dominating pairs.Privacy analysis for ABLQ D . A single epoch of the ABLQ D mechanism corresponds to a Gaussian mechanism with noise scale σ. And thus, E := bT /n epochs of the ABLQ D mechanism corresponds to an E-fold composition of the Gaussian mechanism, which is privacy-wise equivalent to a Gaussian mechanism with noise scale σ/ √ E [Dong et al., 2019, Corollary 3.3 ]. Thus, a closed-form expression for δ D (ε) exists via the dominating pair(P D = N (1, σ 2 E ), Q D = N (0, σ 2 E )). Theorem 3.1 (Balle and Wang [2018, Theorem 8] ). For all σ > 0, ε ≥ 0, and valid n, b, T , it holds thatδ D (ε) = Φ -σ ′ ε + 1 2σ ′ -e ε Φ -σ ′ ε -1 2σ ′ ,where σ ′ = σ √ E , and Φ(•) is the cumulative density function (CDF) of the standard normal random variable N (0, 1).Privacy analysis of ABLQ P . First, let us consider the case of Poisson subsampling without truncation, namely B = ∞. Zhu et al. [2022] showed 2 that the tightly dominating pair for a single step of ABLQ P , a Poisson sub-sampled Gaussian mechanism, is given by the pair(U = (1 -q)N (0, σ 2 ) + qN (1, σ 2 ), V = N (0, σ 2 )), where q is the sub-sampling probability of each example, namely q = b/n. Since ABLQ P is a T -fold composition of this Poisson subsampled Gaussian mechanism, it follows that (P P := U ⊗T , Q P := V ⊗T ) ≡ ABLQ P .A finite value of B however changes the mechanism slightly. In order to handle this, we use the following proposition, where d TV (P, P ′ ) denotes the statistical distance between P and P ′ . Proposition 3.2. For distributionsP, P ′ , Q, Q ′ such that d TV (P, P ′ ), d TV (Q, Q ′ ) ≤ η, and D e ε (P ′ ∥Q ′ ) ≤ δ, then D e ε (P ∥Q) ≤ δ + η(1 + e ε ).Proof. For any event Γ we have thatP (Γ) (i) ≤ P ′ (Γ) + η (ii) ≤ e ε Q ′ (Γ) + δ + η (iii) ≤ e ε (Q(Γ) + η) + δ + η = e ε Q(Γ) + δ + η(1 + e ε ),where (i) follows from d TV (P,P ′ ) ≤ η, (ii) follows from D e ε (P ′ ∥Q ′ ) ≤ δ and (iii) follows from d TV (Q, Q ′ ) ≤ η. Thus, we get that D e ε (P ∥Q) ≤ δ + η(1 + e ε ).The batch size |S t | before truncation in P b,B,T is distributed as the binomial distribution Bin(n, b/n), and thus, by a union bound over the events that the sampled batch size |S t | > B at any step, it follows that for any input dataset x,d TV (ABLQ P b,B,T (x), ABLQ P b,∞,T (x)) ≤ T • Ψ(n, b, B), where Ψ(n, b, B) := Pr r∼Bin(n,b/n) [r > B]. Applying Proposition 3.2 we get Theorem 3.3. For all σ > 0, ε ≥ 0, and integers b, n ≥ b, B ≥ b, T , it holds that δ P (ε) ≤ max{D e ε (P P ∥Q P ), D e ε (Q P ∥P P )} + T • (1 + e ε ) • Ψ(n, b, B).While the hockey stick divergences D e ε (P P ∥Q P ) and D e ε (Q P ∥P P ) do not have closed-form expressions, upper bounds on these can be obtained using privacy accountants based on the methods of Rényi DP (RDP) [Mironov, 2017] and privacy loss distributions (PLD) [Meiser and Mohammadi, 2018, Sommer et al., 2019] ; the latter admits numerically accurate algorithms [Koskela et al., 2020 , Gopi et al., 2021 , Ghazi et al., 2022 , Doroshenko et al., 2022] , with multiple open-source implementations [Prediger and Koskela, 2020 , Google's DP Library., 2020 , Microsoft., 2021] .Note that Ψ(n, b, B) can be made arbitrarily small by increasing B, which affects the computation cost. In particular, given a target privacy parameter (ε, δ), we can, for example, work backwards to first choose B such that Ψ(n, b, B) • T • (1 + e ε ) ≤ 10 -5 • δ, and then choose the noise scale σ such that max{D e ε (P P ∥Q P ), D e ε (Q P ∥P P )} ≤ (1 -10 -5 ) • δ, using aforementioned privacy accounting libraries. Notice that our use of Proposition 3.2 is likely not the optimal approach to account for the batch truncation. We do not optimize this further because we find that this approach already provides very minimal degradation to the choice of σ for a modest value of B relative to b. A more careful analysis could at best result in a slightly smaller B, which we do not consider as significant; see Figures 3 and 4 for more details.Privacy analysis of ABLQ S ⋄ . Obtaining the exact privacy guarantee for ABLQ S has been an open problem in the literature. Our starting point is the approach introduced by Chua et al. [2024] to prove a lower bound in the single epoch setting. Let the input space be X = [-1, 1], the (non-adaptive) query method A that produces the query ψ t (x) = x, and consider the adjacent datasets:x = (x 1 = -1, . . . , x n-1 = -1, x n = 1) and x ′ = (x 1 = -1, . . . , x n-1 = -1, x n = ⊥) .Recall that the number of epochs is E := bT /n and the number of steps per epoch is S := n/b. By considering the same setting, it is easy to see that the distributions U = ABLQ S ⋄ (x) and V = ABLQ S ⋄ (x ′ ) are given as:U = S s=1 1 S • N (-b • 1 + 2f s , σ 2 I T ), and V = S s=1 1 S • N (-b • 1 + f s , σ 2 I T ),2 Also implicit in prior work [Koskela et al., 2020 ].where f s ∈ R T is the sum of basis vectors E-1 ℓ=0 e ℓS+s , and 1 denotes the all-1's vector in R T . Basically, f s is the indicator vector encoding the batches that the differing example gets assigned to; in persistent shuffling, an example gets assigned to the sth batch within each epoch for a random s ∈ {1, . . . , S}. Shifting the distributions by b • 1 and projecting to the span of {f s : s ∈ [S]} does not change the hockey stick divergence D e ε (U ∥V ), hence we might as well consider the pairU ′ := S s=1 1 S • N (2 √ Ee s , σ 2 I S ) and V ′ := S s=1 1 S • N ( √ Ee s , σ 2 I S ).By scaling down the distributions by √ E on all coordinates we arrive at the following pair:EQUATIONThe pair (P S ⋄ , Q S ⋄ ) is essentially same as the pair obtained by Chua et al. [2024] , with σ replaced by σ/ √ E, and we get the following:Proposition 3.4. For all σ > 0, ε ≥ 0 and all valid n, b, T , it holds thatδ S ⋄ (ε) ≥ max{D e ε (P S ⋄ ∥Q S ⋄ ), D e ε (Q S ⋄ ∥P S ⋄ )}.Following Chua et al. [2024] , we can obtain a lower bound as δ S ⋄ (ε) ≥ P S ⋄ (Γ) -e ε Q S ⋄ (Γ) for any Γ, and in particular, we consider events of the form Γ C := {w ∈ R S : max s w s > C} for various values of C. P S ⋄ (Γ C ) and Q S ⋄ (Γ C ) are efficient to compute asP S ⋄ (Γ C ) = 1 -Φ C-2 σ/ √ E • Φ C σ/ √ E S-1 and Q S ⋄ (Γ C ) = 1 -Φ C-1 σ/ √ E • Φ C σ/ √ E S-1.Thus, using Proposition 3.4, we get that Theorem 3.5. For all σ > 0, ε ≥ 0, and all valid n, b, T , it holds thatδ S ⋄ (ε) ≥ sup C∈R P S ⋄ (Γ C ) -Q S ⋄ (Γ C ).Privacy analysis of ABLQ S ⟳ . Our starting point for providing a lower bound on δ S ⟳ (ε) is the pair (P S , Q S ) as defined below that provides a lower bound in the case of a single epoch.P S := S s=1 1 S • N (2e s , σ 2 I) and Q S := S s=11 S • N (e s , σ 2 I). ABLQ S ⟳ is an E-fold composition of the single-epoch mechanism. Hence by composition of dominating pairs, it follows that δ S ⟳ (ε) ≥ max{D e ε (P S ⟳ ∥Q S ⟳ ), D e ε (Q S ⟳ ∥P S ⟳ )}, where P S ⟳ := P ⊗E S and Q S ⟳ := Q ⊗E S . However, it is tricky to directly identify an event Γ for which the lower bound P S ⟳ (Γ) -e ε Q S ⟳ (Γ) is non-trivial and P S ⟳ (Γ), Q S ⟳ (Γ) are easy to compute. So in order to lower bound D e ε (P S ⟳ ∥Q S ⟳ ), below we construct a pair ( PS , QS ) of discrete distributions such that (P S , Q S ) ≽ ( PS , QS ) and thus (P S ⟳ , Q S ⟳ ) ≽ ( P ⊗E S , Q⊗E S ). For probability measures P and Q over a measurable space Ω, and a finite partition3 G = (G 1 , . . . , G m ) of Ω, we can consider the discrete distributions P G and Q G defined over {1, . . . , m}such that P G (i) = P (G i ) and Q G (i) = Q(G i ).The post-processing property of DP implies: Proposition 3.6 (DP Post-processing [Dwork and Roth, 2014] ). For all partitions E of Ω, it holds that (P, Q)≽ (P E , Q E ).We construct the pair ( PS , QS ) by instantiating Proposition 3.6 with the setG = {G 0 , G 1 , . . . , G m+1 } parameterized by a sequence C 1 < • • • < C m of values defined as follows: G 0 := {w ∈ R S : max s w s ≤ C 1 }, G i := {w ∈ R S : C i < max s w s ≤ C i+1 } for 1 ≤ i ≤ m and E m+1 := {w ∈ R S : C m < max s w s }; in other words, G 0 = R S ∖ Γ C1 , G i = Γ Ci ∖ Γ Ci+1 for 1 ≤ i ≤ m and G m+1 = Γ Cm .Theorem 3.7. For all σ > 0, ε ≥ 0, all valid n, b, T , and any finite sequence C 1 , . . . , C m of values used to define PS , QS as above, it holds thatδ S ⟳ (ε) ≥ max{D e ε ( P ⊗E S ∥ Q⊗E S ), D e ε ( Q⊗E S ∥ P ⊗E S )}.Add weight and sample batch indices We use the dp_accounting library [Google's DP Library., 2020] to numerically compute a lower bound on the quantity above, using PLD. In particular, we choose C 1 and C m such that P S (G 0 ) and P S (G m+1 ) are sufficiently small and choose other C i 's to get a sufficiently fine discretization of the interval between C 1 and C m . 4An illustration of these accounting methods is presented in Figure 3 , which demonstrates the significant gap where the optimal σ for dynamic/persistent shuffling is significantly larger than compared to Poisson subsampling, even when using an optimistic estimate for shuffling as above. We provide the implementation of our privacy accounting methods described above in an iPython notebook5 hosted on Google Colab, executable using the freely available Python CPU runtime.Question: Does the paper discuss the limitations of the work performed by the authors? Answer: [Yes] Justification: We discuss limitations and future directions of this paper 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. 
theory assumptions and proofs: 
 Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes] Justification: Proofs of all statements are included either in the main body, or in the supplementary material.Guidelines:• The answer NA means that the paper does not include theoretical results.• All the theorems, formulas, and proofs in the paper should be numbered and cross-referenced.• All assumptions should be clearly stated or referenced in the statement of any theorems.• The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition. • Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material. • Theorems and Lemmas that the proof relies upon should be properly referenced. 
experiments: 
 We compare DP-SGD using the following batch sampling algorithms at corresponding noise scales:• Deterministic batches (using nearly exact value of σ D (ε, δ) via Theorem 3.1),• Truncated Poisson subsampled batches (using upper bound on σ P (ε, δ) via Theorem 3.3),• Persistent shuffled batches (using lower bound on σ S ⋄ (ε, δ) via Theorem 3.5), and • Dynamic shuffled batches (using lower bound on σ S ⟳ (ε, δ) via Theorem 3.7). As a comparison, we also evaluate DP-SGD with dynamic shuffled batches, but using noise that is an upper bound on σ P (ε, δ) (with no truncation, i.e. B = ∞), to capture the incorrect, but commonly employed approach in practice. Finally, in order to understand the impact of using different batch sampling to model training in isolation, we compare models trained with SGD under truncated Poisson subsampling, and dynamic and persistent shuffling without any clipping or noise. We use massively parallel computation (Map-Reduce operations [Dean and Ghemawat, 2004] ) to generate batches with truncated Poisson subsampling in a scalable manner as visualized in Figure 2 ; the details, with a beam pipeline implementation, is provided in Appendix A. We run experiments varying the (expected) batch size from 1 024 to 262 144, for both private training with (ε = 5, δ = 2.7 • 10 -8 ) and non-private training and with 1 or 5 epochs. We plot the results in Figure 3 . As mostly expected, we observe that the model utility generally improves with smaller σ. Truncated Poisson subsampling performs similarly to dynamic shuffling for the same value of σ, although it performs worse for non-private training. The latter could be attributed to the fact that when using truncated Poisson subsampling, a substantial fraction6 of examples are never seen in the training with high probability. However, this does not appear to significantly affect the private training model utility for the range of parameters that we consider, since we observe that truncated Poisson subsampling behaves similarly to dynamic shuffling with noise scale of σ P (ε, δ) (the values of σ for "P" and "S ⟳ (P accounting)" visually overlap in Figure 3 ; the values for P are only negligibly larger, since it accounts for truncation). Truncated Poisson subsampling performs better when compared to shuffling when the latter using our lower bound on σ S ⟳ (ε, δ), which suggests that shuffling with correct accounting (that is, with potentially even larger σ) would only perform worse.We also run experiments with a fixed batch size 65 536 and varying ε from 1 to 256, fixing δ = 2.7 • 10 -8 . We plot the results and the corresponding σ values in Figure 4 . We again observe that shuffling (with our lower bound accounting) performs worse than truncated Poisson subsampling in the high privacy (low ε) regime, but performs slightly better in the low privacy (high ε) regime (this is because at large ε the noise required under Poisson subsampling is in fact larger than that under shuffling). Moreover, we observe that shuffling performs similarly to truncated Poisson subsampling when we use similar value of σ, consistent with our observations from Figure 3 .Finally, as a comparison, we compute the upper bounds on σ S (ε, δ) via the privacy amplification by shuffling bounds by Feldman et al. [2021] . We find that these bounds tend to be vacuous in many regime of parameters, namely they are no better than σ D (ε, δ), which is clearly an upper bound on σ S (ε, δ). Details are provided in Appendix C. Question: Does the paper fully disclose all the information needed to reproduce the main experimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?Answer: [Yes] Justification: All experimental details are provided. 
conclusion: 
 We provide new lower bounds on the privacy analysis of Adaptive Batch Linear Query mechanisms, under persistent and dynamic shuffling batch samplers, extending the prior work of Chua et al. [2024] that analyzed the single epoch case. Our lower bound method continues to identify separations in the multi-epoch setting, showing that the amplification guarantees due to even dynamic shuffling can be significantly limited compared to the amplification due to Poisson subsampling in regimes of practical interest.We also provide evaluation of DP-SGD with various batch samplers with the corresponding privacy accounting, and propose an approach for implementing Poisson subsampling at scale using massively parallel computation. Our findings suggest that with provable privacy guarantees on model training, Poisson-subsampling-based DP-SGD has better privacy-utility trade-off than shuffling-based DP-SGD in many practical parameter regimes of interest, and in fact, essentially match the utility of shuffling-based DP-SGD at the same noise level. Thus, we consider Poisson-subsampling-based DP-SGD as a viable approach for implementing DP-SGD at scale, given the lower bound on the privacy analysis when using shuffling.Several interesting directions remain to be investigated. Firstly, our technique only provides a lower bound on the privacy guarantee when using persistent / dynamic shuffled batches. While some privacy amplification results are known [Feldman et al., 2021 [Feldman et al., , 2023]] , providing a tight (non-vacuous) upper bound on the privacy guarantee in these settings remains an open challenge. This can be important in regimes where shuffling does provide better privacy guarantees than Poisson subsampling.Another important point to note is that persistent and dynamic shuffling are not the only forms of shuffling used in practice. For example, methods such as tf.data.Dataset.shuffle or torchdata.datapipes.iter.Shuffler provide a uniformly random shuffle, only when the size of its "buffer" is larger than the dataset. Otherwise, for buffer size b, it returns a random record among the first b records, and immediately replaces it with the next record ((b + 1)th in this case), and repeats this process, which leads to an asymmetric form of shuffling. Such batch samplers merit more careful privacy analysis. 
experiment statistical significance: 
 Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments? Answer: [Yes] Justification: The training experiments have been repeated independently a few times and error bars are provided. However, due to significant cost of the experiments, we limit ourselves to only a small number of independent runs. These details are described in Section 4.Guidelines:• The answer NA means that the paper does not include experiments.• 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. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text. 
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 Appendix: We use massively parallel computation to generate batches with truncated Poisson subsampling in a scalable manner. Given the input parameters b, B, T , and n, we first compute the maximum batch size B such that Ψ(n, b, B) • T • (1 + e ε ) ≤ 10 -5 • δ. For each example in the input dataset, we generate a list of batches that the example would be in when sampled using Poisson subsampling. We include a code snippet for implementing truncated Poisson subsampling using massively parallel computation. This is written using Apache beam [Apache Beam] in Python, which can be implemented on distributed platforms such as Apache Flink, Apache Spark, Google Cloud Dataflow. """Returns the indices of the batches that an example is in.Assuming that an example is sampled in each batch using poisson subsampling, return the list of indices of the batches that the example is in. We use a neural network with five layers and ∼78M parameters as the model. The first layer consists of feature transforms for each of the categorical and integer features. Categorical features are mapped into dense feature vectors using an embedding layer, where the embedding dimensions are fixed at 48. We apply a log transform for the remaining integer features, and concatenate all the features together. The next three layers are fully connected layers with 598 hidden units each and a ReLU activation function. The last layer consists of a fully connected layer which gives a scalar logit prediction.We use the Adam or Adagrad optimizer with a base learning rate in {0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1}, which is scaled with a cosine decay, and we tune the norm bound C ∈ {1, 5, 10, 50}. For the experiments with varying batch sizes, we use batch sizes that are powers of 2 between 1 024 and 262 144, with corresponding maximum batch sizes B in {1 328, 2 469, 4 681, 9 007, 17 520, 34 355, 67 754, 134 172, 266 475}. For the experiments with varying ε, we vary ε as powers of 2 between 1 and 256, with batch size 65 536 and corresponding maximum batch sizes B in {67 642, 67 667, 67 725, 67 841, 68 059, 68 449, 69 106, 70 156, 71 760} . With these choices of the maximum batch sizes, the σ values for truncated Poisson subsampling are only slightly larger than without truncation, as we observe from the nearly overlapping curves in Figure 3 and Figure 4 . The training is done using NVIDIA Tesla P100 GPUs, where each epoch of training takes 1-2 hours on a single GPU.We evaluate the upper bounds on σ S (ε, δ) via privacy amplification by shuffling results of Feldman et al. [2021] , as applied in the context of our experiments in Figure 3 . In particular, their Proposition 5.3 states that if the unamplified (Gaussian) mechanism satisfies (ε 0 , δ 0 )-DP, then ABLQ S with T steps (in a single epoch setting) will satisfy (ε, η + O(e ε δ 0 n)-DP for any η ∈ [0, 1] andε = O (1 -e -ε0 ) √ e ε 0 log(1/η) √ T + e ε 0 T .They also obtain a tighter numerical bound on ε with an implementation provided in a GitHub repository.7 We evaluate their bounds in an optimistic manner. Namely, for a given value of δ, we compute an optimistic estimate on ε compared to the bound above by setting δ 0 = δ/n and setting ε 0 = ε D (δ 0 ), and set η = δ (note that this is optimistic because the above proposition requires setting δ = η + O(e ε δ 0 n), which is larger than the δ we are claiming). We use the numerical analysis method provided in the library by Feldman et al. [2021] to compute σ S (via a binary search on top of their method to compute an upper bound on ε S ), and plot it in Figure 5 , along with the lower bound on σ S as obtained by Chua et al. [2024] , as well as σ D . We find that the bounds by Feldman et al. [2021] are vacuous for batch sizes 2 048 and above, in that they are even larger than the bounds without any amplification. We include a comparison of the σ values for varying numbers of epochs, to show how the same trends hold beyond the 1 and 5 epoch regimes. 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.• The answer NA means that the paper does not include experiments.• If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not. • If the contribution is a dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable. • Depending on the contribution, reproducibility can be accomplished in various ways. 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For example (a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.(b) If the contribution is primarily a new model architecture, the paper should describe the architecture clearly and fully. (c) If the contribution is a new model (e.g., a large language model), then there should either be a way to access this model for reproducing the results or a way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset). (d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results. • The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them. • The full details can be provided either with the code, in appendix, or as supplemental material.Justification: The paper conforms to the NeurIPS Code of Ethics as it only uses experiments on publicly available datasets, and does not use any crowdsourcing or human subjects.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).As explained later, for Poisson batch sampler, the mini-batch size is not a constant, but the scaling has to be done with the "target" mini-batch size, and not the realized mini-batch size.Gi's are pairwise disjoint and ∪ m i=1 Gi = Ω.In our evaluation, we choose C1 and Cm to ensure PS (G0) + PS (Gm+1) ≤ e -40 . We chose other Ci's to be equally spaced in between C1 and Cm with a gap of ∆ • σ 2 , where ∆ is the desired discretization of the PLD. This heuristic choice is guided by the intuition that the privacy loss is approximately linear in maxt xt/σ 2 , and thus the chosen gap means that this approximate privacy loss varies by ∆ between buckets.https://colab.research.google.com/drive/1vCijMEQqRCm0x3EOUUKomcZnwx76sz64?usp=sharingThe expected number of examples that are never used even once during training is at least (1 -b n ) T , which approaches e -E in the limit as b/n → 0, for a fixed number of epochs E = bT /n.https://github.com/apple/ml-shuffling-amplification 


Decision: The main critiques of this paper were that the techniques may not be the most novel, and the contribution may be somewhat modest. Nonetheless reviewers agreed that this is an impactful advance on a very important problem, and thus worthy of acceptance.

In the discussion, Reviewers k7t2 and 9tRV brought up some points that they feel deserve mentioning in the final paper (highlighted in their reviews), and I anticipate the authors will take them into account.

Reviews: review: paper_summary: paper_summary: This paper investigates the utility of models trained with DP-SGD based on previous findings on the gap in privacy guarantee between shuffled batch sampling (commonly used in practice) and Poisson subsampling (used in theoretical analysis) for private training. A scalable implementation of Posson subsampling is proposed via truncation and parallelization, provided with lower bounds of the privacy guarantee via multi-epoch ABLQ with shuffling.
strength_weakness: strength_weakness: - The techniques for extending ABLQ from multi-batch to multi-epoch are straightforward. 
- It is difficult to distinguish from the existing results of Chua et al. [2024], given much of the material in Sec. 2 and Sec. 3 overlaps and is sometimes verbatim.
- Different types of batch sampling with different privacy accounting are evaluated. Still, all of them underestimate the privacy loss and cannot rigorously reflect the model utility under the claimed $(\epsilon, \delta)$.
questions: questions: - The authors mentioned that the Opacus library supports Possion subsampling but might not be used for training on massive datasets. Have the authors tested the implementation from Opacus on the Criteo dataset? If so, how large is a batch it can support (from 1000 to 200,000)? Is this the major obstacle to be included as a reference for Fig.2 and 3?
- Is the target batch size $b$ still used to average the summed noisy gradients for the truncated Poisson Batch sampler? Could the authors elaborate on how the size of truncated batches would affect the tracking of privacy loss, especially in Theorem 3.3?
- Line 254 -255, could the authors provide more details on selecting $C_i$ in practice? And what is the computation complexity regarding this part?
limitations: limitations: See W2,3 and Q1
paper_summary: paper_summary: The paper focuses on practical implementations of DP training of ML models at scale in the multi-epoch setting. The contribution is two-fold: the paper gives a rigorous analysis for a practical version of Poisson subsampled DP-SGD where the batch size is upper bounded, and proposes lower $(\varepsilon,\delta)$-bounds for the shuffled Gaussian mechanism. Both of these contributions require a careful theoretical analysis which the paper succeeds in carrying out such that the bounds are sharp: there is no big slack in the approximative Poisson subsampling upper bound and the lower bounds is able to illustrate that the shuffling mechanism generally leads to a worse privacy-utility trade-off than the Poisson subsampled Gaussian mechanism. So the message becomes clear: although the shuffling + disjoint batch training is practical, the practical version the paper proposes will likely lead to a better privacy-utility trade-off. The paper builds upon the paper (Chua et al., ICML 2024) and borrows several results from it.
strength_weakness: strength_weakness: - The paper heavily builds upon the previous work by Chua et al. (ICML 2024), and the novelty is relatively thin, although the paper succeeds in delivering a clear message.

- The privacy amplification by iteration analysis - type of analysis is not mentioned. Although the current analyses in that line of work are only applicable to convex problems, they give $(\varepsilon,\delta)$-DP bounds for e.g. noisy cyclic GD (Bok et al., ICML 2024) that lead to models whose privacy-utility trade-off is competitive with the DP-SGD trained models (see e.g. experiments by Bok et al.).
questions: questions: - You mention that the analysis for the shuffled Gaussian mechanism is an open problem. Why do you think this is an important open problem if the lower bounds for the shuffled Gaussian mechanism already indicate that DP-SGD give much better privacy-utility trade-off? In which scenarios would the upper bounds be useful?

- Why do you only consider the Poisson subsampling? Is there something particular in the subsampling without replacement that makes it less suitable to the large scale setting? In that case you would not need to have those approximations that you carry out to limit the batch size.
limitations: limitations: Yes.
paper_summary: paper_summary: This paper provides theoretical and empirical analyses of three different DP-SGD minibatch sampling schemes, and also implement an efficient beam pipeline for using Poisson sampling in practice via a truncation-based approach.  Theoretically, they derive new lower bounds for the privacy accounting / noise calibration of shuffling-based DP-SGD, and truncated Poisson-sampling-based DP-SGD.  Empirically, they show (1) for the same noise multiplier, shuffling does better (in terms of AUC) than Poisson sampling and (2) for the same privacy budget, Poisson sampling does better, mainly due to the more favorable privacy accounting.  In one experiment (Fig 3 upper left) they show that shuffling can sometimes be better than Poisson sampling for small epsilon.
strength_weakness: strength_weakness: There are a few missing pieces that I think would strengthen the paper and hopefully not require too much extra work:

1) Thm 3.3 uses the bound from Prop 3.2, and may not be the tightest possible result.  Would be good to quantify the gap between Poisson sampling and Truncated Poisson sampling in terms of the noise multiplier you obtain for each, fixing p for Poisson sampling and B for the Truncated Poisson sampling so that the expected batch sizes of Poisson sampling matches the physical batch sizes of truncated Poisson.
2) There is not enough discussion or analysis on how to configure the Truncated Poisson parameters.  For a fixed physical batch size, privacy budget, and other relevant parameters (number of iterations), can you give a simple procedure to find the best sampling probability to use in terms of the expected signal to noise ratio?
3) Do the same findings hold up beyond the 1- and 5-epoch regimes? Would be nice to include a comparison where Epochs is varied on the x-axis and sigma is varied on the y-axis (no need to do new experiments on real data which might be very expensive).  

Minor:

4) In plots you should evenly space out the independent variables.  If plotting in log space, evenly space them in log space (e.g., Batch Size = 1024, 2048, 4096, ..., 262144 and epsilon = 1, 2, 4, 8, ..., 16).  Also the legend is a bit small, and shared across plots.  Consider making 1-row 5-column legend spanning or otherwise increasing the font size.
questions: questions: 1) In Fig 2, it seems like P corresponds to  truncated Poisson sampling, but green line corresponds to non-truncated Poisson accounting.  Why do these lines line up in the middle plots then?
limitations: limitations: N/A
paper_summary: paper_summary: DP-SGD is one of the must common algorithms currently deployed for performing machine learning tasks while maintaining differential privacy. As the "S" in its name reminds us, the gradient is not computed over the full dataset at each turn, but instead over a subset sampled from it. The commonly used privacy analysis typically relies on Poisson sub-sampling, whereby each element is independently added to each batch with some constant probability. In contrast, due to technical challenges stemming from the fact the Poisson sub-sampled batch size is not constant, this is typically implemented by first shuffling the full dataset, then batching it into subsequent subsets.

Unfortunately, there is a gap between existing upper bounds on the privacy loss of the shuffle version and those of the sub-sampling. Recently Lynn et al. [1] provided a lower bound on the privacy of the shuffled version, which shows a clear gap between the two settings in some parameter regimes (mainly low local privacy).

This paper continues and extends [1] work in two ways:
1. It extends the analysis to multi-epoch setting
2. It proposes a variant of Poisson sub-sampling with effective constant batch size, by paying a small price in the privacy failure probability $\delta$
The authors also perform several empirical experiments to compare the privacy and accuracy of various versions of private training.

[1] Lynn Chua, Badih Ghazi, Pritish Kamath, Ravi Kumar, Pasin Manurangsi, Amer Sinha, and Chiyuan Zhang. How Private is DP-SGD? In International Conference on Machine Learning, ICML (to appear), 2024.
strength_weakness: strength_weakness: While this work is clearly written and provides useful empirical and theoretical tools, its novelty is somewhat limited.
As mentioned, this is a follow-up work to the one by Lynn et al. The extension of their analysis to multi epochs is relatively straight forward, and so is the analysis of the truncated Poisson sub-sampling.
questions: questions: Can the authors add a comparison of the the various privacy analyses to the upper bounds provided by existing literature on general shuffle model, such as the one provided by Feldman et al. [2]?

Additionally, to the best of my understanding, amplification by sub-sampling a constant size batches without replacement (per sampling)  is will understood, and enjoys privacy guarantees that are comparable to those of Poisson sub-sampling, up to a factor 2, depending on the exact neighboring notion. What is the advantage of the proposed method relative to the WOR sampling option?

[2] Feldman, V., McMillan, A., and Talwar, K. Hiding among the clones: A simple and nearly optimal analysis of privacy amplification by shuffling. In FOCS, pp. 954–964, 2021
limitations: limitations: N/A

Comments: title: Response to Official Comment by Reviewer 9tRV
comment: > better clarify in the notations which lines correspond to upper bounds and which ones to lower bounds

Just to clarify, the noise multiplier for $\cal D$ is essentially exact, the noise multipliers for $\cal P$ (as well as ${\cal S}^{\circlearrowright}$ with $\cal P$ accounting) correspond to upper bounds, and the noise multipliers for ${\cal S}^{\circlearrowright}$ and ${\cal S}^\diamond$ are lower bounds. Thanks for the feedback, we will incorporate this in the revision.
I see, so only a ~1.7% "cost" -- perhaps there is not much to be gained by optimizing the parameters.  Thanks for clarification, would be good to include a version of this table in the paper or appendix.
> Opacus supports (distributed) Poisson sampling without the need to load the entire dataset (e.g., simply by indexing). I suggest the authors do their due diligence and check the implementation at https://opacus.ai/api/_modules/opacus/utils/uniform_sampler.html

Thanks to the reviewer for the pointer. We apologize that our original text in lines 47-49 could be misinterpreted as “Opacus solution *requires* loading the dataset into the memory”. We meant to say that “loading into memory” is one way to facilitate this feature for small datasets. We will revise the text to make it more precise.

To avoid any further misunderstanding between us and the reviewer on this matter, we summarize our view of the Opacus solution here: Opacus Poisson Subsampling works by using a unique identifier to index each example, and sampling the indices. They also provide a distributed sampler where each worker samples from a random subset of indices. This approach relies on a DataLoader that can provide efficient random access --- given an arbitrary index, read the corresponding data point efficiently. If random access is slow, the data pipeline will be slow, even though the (index) sampler is not the bottleneck.

For small datasets, this can be easily supported by loading the data into memory. For large datasets that do not fit in the memory, various technical challenges need to be addressed to make it work. For example, many formats for raw data (e.g. `csv`, typically used for tabular or text data) or serialized data (e.g. `tfrecords` files, commonly used by the Tensorflow Datasets Catalog) do _not_ naturally support random indexing; moreover, random access can be much slower than sequential bulk reading in cases such as when the dataset is stored in a distributed file system. Various workarounds do exist depending on the resources and constraints, for example, by storing each example as an individual file in a fast SSD attached to the trainer machine. But those solutions usually need to be tailored to specific hardware and cluster infrastructures. On the contrary, we provide a **generic solution** that works for a large range of scales of dataset sizes and with **minimal assumption / requirement** on the underlying IO infrastructure. 

We agree that an empirical comparison of different technical solutions could be interesting but it is quite difficult to get right and useful as everyone has different configurations of (distributed) file systems, file format of preference, cluster configuration with communication channels of different properties, etc. Therefore it is quite far from the scope of our paper.

Finally, regardless of the underlying technical solution, please note that ours is the _first large-scale_ DP training experiment with Poisson subsampling, comparing its feasibility with other solutions from multiple angles, including accounting, efficiency, and model utility.

> truncated Poisson sampling distorts the probability of each sample being included in a batch

It is true that due to truncation in Poisson subsampling, the probability that an example lands in a batch is reduced slightly. This is handled in the privacy analysis (Theorem 3.3).

Please let us know if there are any further questions.
For truncated Poisson subsampling, we use a maximum batch size as specified below, which includes the dummy examples added to ensure that all batches have the same size. The normalization is still by expected batch size (before truncation), which is $b = 200,000$.

| Epochs | 1 | 3 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|---|
| **Max Batch size** | 203288 | 203353 | 203383 | 203423 | 203446 | 203463 |
I see, I would have thought the plot would be normalized for physical batch size.  

For Truncated Poisson sampling what is the physical batch size you use then?
We would like to clarify that for the table we shared, the expected batch sizes for Poisson subsampling are the same as the physical batch size for Shuffle and Deterministic; by "${\cal S}^{\circlearrowright}$ (with $\cal P$ accounting)", we mean the same method as ${\cal S}^{\circlearrowright}$, but using the noise multiplier assuming Poisson subsampling (without any truncation). While truncation in Poisson subsampling does reduce the expected batch size by a small amount, this is quite negligible and hence the normalization would be by approximately the same quantity. In particular, we use the values of $\varepsilon = 5$, $\delta = 2.7 \cdot 10^{-8}$ and (expected) batch size $b = 200000$. Please let us know if there are still any further questions.
Apologies I am just looking at this carefully now.  For the table you shared, those noise multipliers appear to not account for the differing expected batch sizes, is that correct?

For P, I would like to see noise multiplier / (expected batch size)

For S (with P accounting) I would like to see noise multipliers / (physical batch size)

Please update and send back ASAP, I would like to follow-up after seeing those numbers.
Thank the authors for the response and clarification. There are still some issues to be addressed:

- W1 The authors highlight the difference between single-epoch and multi-epoch analysis of ABLQ. However, from my point of view, as reviewer k7t2 pointed out, the theoretical contribution of this part is limited.

- W3 Thanks for the clarification. I wanted to emphasize that even with the new lower bounds for multi-epoch, the gap between Poisson subsampling and shuffle-based sampling is not addressed, especially the amplification effect in the low $\sigma$ region.

- Q1 The statement in lines 47-49 is not supported by empirical results or further investigation, which may be inaccurate and misleading to other researchers/practitioners in the field. As far as I know, Opacus supports (distributed) Poisson sampling without the need to load the entire dataset (e.g., simply by indexing). I suggest the authors do their due diligence and check the implementation at https://opacus.ai/api/_modules/opacus/utils/uniform_sampler.html

- Q2 The concern here is on the effect of batch size value used in averaging the sum of noisy gradients for privacy accounting. The batch size essentially reflects the sampling rate. However, truncated Poisson sampling distorts the probability of each sample being included in a batch. Has this been considered?

- W2 \& Q3 Thank the authors for the response. I hope the authors will include these discussions in the manuscript. 

Given the amount of revisions required and the questions that remain, I am inclined to keep my initial assessment.
I thank the authors for their response.

As previously mentioned, the comparison of lower bounds for the shuffle model to upper bounds for the Poisson sub-sampling model were already performed by Chia et al. One of the main contributions of this current work as stated by the authors, is the proposal of the truncated Poisson sub-sampling model, to remedy the existing gap in private SGD based training. I find it hard to asses the importance of this contribution, where the upper bounds for two existing alternatives (shuffle and constant size sub-sampling) where not considered, and might very well provide accuracy results that are comparable to the new proposed technique.

Additionally, I would recommend the authors better clarify in the notations which lines correspond to upper bounds and which ones to lower bounds. This seemed to cause confusion for some of the reviewers, and might affect the readers as well. In fact, the comparison discussed in line 284 creates the impression that the two compared lines represent the same quantity while - to the best of my understanding - they do not.
Thank you for the replies! I will keep my score. Please do add the discussion about subsampling without replacement. Please take into account also the conjecture by [Lebeda et al.](https://arxiv.org/pdf/2405.20769) that subsampling without replacement behaves similarly both for add/remove and substitute neighborhood relations.