Batch Clipping with Shuffling: A DP Method

• Batch clipping with shuffling is a method that aggregates all gradients in a batch, clips the combined sum once, and shuffles data to bolster privacy guarantees.
• It refines DP analysis by leveraging f-DP and information-theoretic bounds, improving group privacy scaling from O(g) to O(√g).
• The approach highlights trade-offs with per-example clipping and Poisson subsampling, influencing noise scaling and model utility in SGD implementations.

Batch clipping with shuffling is a procedure within differentially private stochastic gradient methods that alters both the mechanism by which gradients are clipped and the sampling design for minibatches, with the goal of optimizing privacy–utility trade-offs. Unlike standard differential privacy (DP) approaches using per-example clipping with Poisson or uniform subsampling, batch clipping aggregates all gradients in a batch, clips the sum, and applies shuffling to form disjoint, randomly permuted batches per epoch. This design impacts both algorithmic workflow and formal privacy guarantees, as captured in recent analyses via $f$-DP as well as information-theoretic lower bounds.

1. Algorithmic Framework for Batch Clipping with Shuffling

Given a dataset $D$ of size $N$, the standard generalized framework for batch clipping with shuffling proceeds as follows. For each of $E$ epochs:

• Draw a fresh random permutation of the dataset indices.
• Partition the permuted dataset into $B = N / s$ contiguous, disjoint batches of size $s$.
• For each batch:
  • Compute gradients $g_i = \nabla f(w, x_i)$ for all $i$ in the batch $S$ at the current model $w$.
  • Aggregate the batch gradient $a = \sum_{i\in S} g_i$.
  • Clip the aggregate: $U = a / \max(1, \|a\|/C)$ for norm bound $C$.
  • Add Gaussian noise: output $\widetilde U = U + \mathcal N(0, (2C\sigma)^2 I)$.
  • Update model: $w \leftarrow w - \eta_t (\widetilde U / s)$, with $\eta_t$ the step size.

The process is repeated for all epochs, with models $w_1, \ldots, w_T$ forming the trajectory. Importantly, this shuffling regime ensures each example appears exactly once per epoch, unlike i.i.d. subsampling (Dijk et al., 2022).

2. Batch Clipping vs. Individual Clipping

Standard DP-SGD applies clipping at the per-example level: each gradient is individually norm-limited, and then the sum is formed, which is followed by noise addition. Batch clipping, in contrast, accumulates all gradients in the batch without per-example clipping and clips only the aggregate once. The resulting difference in algorithmic sensitivity changes the DP analysis:

Method	Operation	Noise Addition
Individual Clipping	Clip each $\nabla f(w, x_i)$ to $\|g_i\| \le C$	$\mathcal N}(0, (2C\sigma)^2I)$
Batch Clipping	Sum raw gradients in batch, clip $\sum g_i$	$\mathcal N}(0, (2C\sigma)^2I)$

For both cases, the same overall noise scale applies to the clipped value (unless using probabilistic filtering), but the mechanism's sensitivity to changes in data differs in critical ways, especially in the group privacy regime (Dijk et al., 2022).

3. Privacy Amplification via Shuffling

Shuffling is implemented by randomly permuting the dataset at the start of each epoch before partitioning into batches. This “shuffling” amplifies privacy as each data point's exposure is limited: every record appears in exactly one batch per epoch. Unlike Poisson or i.i.d. subsampling, which can lead to overlapping sample inclusion, shuffling reduces the event likelihood for adversarial exposure in multiple rounds.

In the context of batch clipping, shuffling enables a new group privacy behavior: for a group of $g$ affected records, high-probability shuffling ensures these land in $g$ distinct batches. This structure permits tightening the DP bounds from $O(g)$ to $O(\sqrt{g})$ dependence, providing improved group privacy (Dijk et al., 2022).

4. f-DP and Information-Theoretic Privacy Bounds

Batch clipping with shuffling is analyzed using the hypothesis-testing framework of $f$-DP. In this setting:

• The sensitivity for each batch is proportional to the number $k_b$ of altered examples in batch $b$ between neighboring datasets $(D, D')$, yielding Gaussian DP $G_{k_b/\sigma}$ per batch.
• Aggregating across batches, the total sensitivity per epoch $c = \sqrt{k_1^2 + \cdots + k_B^2}$ becomes random, with its distribution tightly concentrated at $c \approx \sqrt{g}$ for $g$ group changes.
• The overall epoch-wise trade-off function is a mixture $$f(\alpha) = \inf_{(\alpha_c)} \sum_c q_E(c) G_{c/\sigma}(\alpha_c)$$, for $q_E(c)$ the probability mass for each $c$; composition across $E$ epochs gives $G_{\sqrt{gE}/\sigma}$ (Dijk et al., 2022).

A critical distinction is that, under shuffling and batch clipping, the amplification is only $O(\sqrt{E})$, rather than linear in $E$ as for i.i.d. composition. This has major consequences for multi-epoch DP-SGD analysis, especially regarding group privacy and the conversion from $f$-DP to $(\epsilon, \delta)$-DP.

Recent work shows that the true lower bounds on $(\epsilon, \delta)$ for multi-epoch batch clipping under shuffling (both persistent and dynamic) are strictly weaker than those obtained under Poisson subsampling. Analytical expressions are provided via dominating pairs of mixture Gaussians, yielding numerically quantifiable privacy degradations (Chua et al., 6 Nov 2024).

5. Comparison to Poisson Subsampling

Poisson subsampling, where each data point appears in a batch independently with fixed probability $q = b/n$, achieves privacy amplification scaling like $O(\sqrt{Tq})$ (with $T$ steps). This is strictly stronger than the $O(\sqrt{E})$ amplification under shuffling.

Empirical and theoretical analyses have demonstrated that, for fixed noise scale $\sigma$, shuffling and Poisson subsampling yield comparable model utility. However, correctly accounting for privacy under shuffling requires using significantly larger noise to meet the same $(\epsilon, \delta)$ target, especially in the high-privacy regime (small $\epsilon$): the gap in privacy loss can be orders of magnitude for small $\sigma$, making shuffling strictly inferior to Poisson subsampling for privacy amplification (Chua et al., 6 Nov 2024).

A summary table from experimental evaluations:

batch size $b$	$\sigma_{Poisson}$	$\sigma_{persist}$ (LB)	$\sigma_{dyn}$ (LB)	AUC$_{Poisson}$	AUC$_{shuffle}$
1,024	1.25	1.42	1.48	0.785	0.783
16,384	0.82	1.05	1.12	0.796	0.793
65,536	0.68	0.89	0.96	0.801	0.798

6. Implementation Considerations and Best Practices

Implementing shuffled-batch sampling and batch clipping is straightforward in standard SGD pipelines. For Poisson subsampling, variable batch sizes require careful engineering to fit into fixed-batch compilers (e.g., JAX/TPU), motivating the use of truncated Poisson subsampling with a padding and downsampling protocol scalable via Map-Reduce (Chua et al., 6 Nov 2024).

Best practices emerging from the literature:

• Privacy accounting must match the sampler: reporting $(\epsilon, \delta)$ based on Poisson subsampling when using shuffling overstates privacy and is considered optimistic.
• If feasible, Poisson subsampling should be preferred for privacy amplification and utility, especially when high privacy is paramount.
• Truncation for Poisson batches must be budgeted carefully; in practical parameters, truncation effects are negligible for $B \gtrsim 1.05 b$.
• Buffer-based shuffling methods, such as those in deep learning frameworks using limited-size buffers, require fresh privacy analysis and cannot be assumed to adhere to the established bounds (Chua et al., 6 Nov 2024).

7. Extensions and Applicability Beyond Standard SGD

The batch clipping with shuffling scheme generalizes to any first-order optimizer generating a per-batch update vector $a = A(w; S)$ with $\|a\| \leq C$. As long as a single clipped vector per batch is produced and appropriate Gaussian noise is added, the same $f$-DP analysis applies. This extends to momentum-SGD, Adam, RMSProp, AdaGrad, and multi-step local recursion. The general DP recipe remains: batch-clip once per batch, shuffle the dataset, add Gaussian noise, and update the model trajectory accordingly (Dijk et al., 2022).

A plausible implication is that the batch clipping with shuffling design forms a modular primitive for privacy in a wide range of federated and distributed optimization settings. However, its limitations in privacy amplification relative to Poisson subsampling must be accounted for to avoid overstatement of privacy guarantees.