Provable Differentially Private Computation of the Cross-Attention Mechanism

Technical Summary: Provable Differentially Private Computation of the Cross-Attention Mechanism

Problem Statement
Cross-attention mechanisms are foundational to modern Large Generative Models (LGMs), enabling critical capabilities such as Retrieval-Augmented Generation (RAG), system prompting, and guided stable diffusion. However, these mechanisms process sensitive data—including private user documents, proprietary business logic, and confidential system prompts—directly within the attention layers. This creates a significant privacy vulnerability where adversarial attacks (e.g., jailbreaking or prompt recovery) can extract this sensitive information. While Differential Privacy (DP) has been applied to the pre-training and fine-tuning stages of LGMs, and some empirical attempts exist for protecting prompts, there is a lack of theoretically provable methods to secure the cross-attention mechanism itself during inference. Existing approaches often lack rigorous accuracy bounds or fail to address adaptive query attacks.

Methodology
The paper proposes a novel framework that transforms the cross-attention computation into a weighted distance problem, enabling the application of differential privacy with provable guarantees. The core technical approach involves the following steps:

1. Mathematical Reformulation: The authors demonstrate that the Softmax cross-attention computation, defined as $Attn(Q, K, V) = D^{-1}AV$ where $A_{i,j} = \exp(\langle Q_i, K_j \rangle / d)$, is equivalent to a weighted distance problem. By utilizing polynomial kernel approximation methods (from Alman and Song, 2023), the exponential inner product $\exp(\langle x, y \rangle / d)$ is approximated by a polynomial kernel function $P(x)^\top P(y)$. Using the Law of Cosines, this inner product is further transformed into a weighted $\ell_2^2$ distance measure:
   $$2P(x)^\top P(y) = -\|P(x) - P(y)\|_2^2 + \|P(x)\|_2^2 + \|P(y)\|_2^2$$
   This transformation allows the complex Softmax query to be decomposed into weighted distance queries.

2. Data Structure Design (DPTree): To solve these weighted distance queries under DP, the authors design a hierarchy of data structures:

   • DPTree: A binary segment tree where truncated Laplace noise is added to nodes during initialization to ensure DP for summation queries.
   • DPTreeDistance: Extends DPTree to handle one-dimensional weighted $\ell_p^p$ distance queries by decomposing the distance calculation into polynomial terms.
   • DPTreeHighDim: Generalizes the one-dimensional structure to high dimensions by constructing independent trees for each coordinate.
   • DPTreeSoftmax: Combines the high-dimensional distance structures with the polynomial kernel approximation to answer weighted Softmax queries.
   • DPTreeSoftmaxAdaptive: Enhances the Softmax structure to be robust against adaptive queries (where an attacker chooses queries based on previous outputs). This is achieved by creating multiple independent copies of the data structure, taking the median of the outputs, and utilizing an $\epsilon_0$-net argument combined with the Lipschitz property of the weighted Softmax function to bound errors across the entire query space.

3. Cross-Attention Integration (Algorithm 1): The final algorithm, DPCrossAttention, initializes $d+1$ instances of DPTreeSoftmaxAdaptive. $d$ instances handle the weighted values ($V$), and one instance handles the normalization term ($D$, effectively $V=\mathbf{1}_n$). The query process computes the noisy numerator ($AV$) and denominator ($D$) separately and combines them to produce the final attention output.

Key Results and Complexity
The paper provides rigorous theoretical guarantees for the proposed mechanism:

• Privacy: The output process satisfies $(\epsilon, \delta)$-differential privacy.
• Accuracy: The mechanism achieves a relative error of $2\epsilon_s / (1 - \epsilon_s)$ and an additive error of $\tilde{O}((1-\epsilon_s)^{-1} n^{-1} \epsilon^{-1} R^2_s R_w r^2)$, where $n$ is the sequence length, $d$ is the feature dimension, $R$ and $R_w$ are magnitude bounds, and $r, s, \epsilon_s$ are polynomial kernel parameters. Crucially, the additive error decreases as the input token length $n$ increases, diminishing to zero.
• Complexity:
  • Space & Initialization: $\tilde{O}(ndr^2)$.
  • Query Time: $\tilde{O}(dr^2)$ per token.
• Robustness: The framework is proven to be robust against adaptive queries, ensuring that attackers cannot iteratively extract information about system prompts or RAG data.

Significance and Claims
The authors claim this work constitutes the first approach to provide provable differential privacy guarantees specifically for the cross-attention mechanism in large generative models. Unlike previous empirical studies or methods focused on training-stage privacy (e.g., DP-SGD), this work integrates DP directly into the inference-time attention mechanism. This offers a practical alternative for models trained with standard SGD that lack inherent privacy, avoiding the computational cost of retraining. By establishing a theoretical foundation for privacy-preserving cross-attention, the paper aims to enable the safe deployment of personalized AI agents and RAG systems that rely on sensitive user and proprietary data.