Efficient and Near-Optimal Noise Generation for Streaming Differential Privacy (2404.16706v3)

Abstract: In the task of differentially private (DP) continual counting, we receive a stream of increments and our goal is to output an approximate running total of these increments, without revealing too much about any specific increment. Despite its simplicity, differentially private continual counting has attracted significant attention both in theory and in practice. Existing algorithms for differentially private continual counting are either inefficient in terms of their space usage or add an excessive amount of noise, inducing suboptimal utility. The most practical DP continual counting algorithms add carefully correlated Gaussian noise to the values. The task of choosing the covariance for this noise can be expressed in terms of factoring the lower-triangular matrix of ones (which computes prefix sums). We present two approaches from this class (for different parameter regimes) that achieve near-optimal utility for DP continual counting and only require logarithmic or polylogarithmic space (and time). Our first approach is based on a space-efficient streaming matrix multiplication algorithm for a class of Toeplitz matrices. We show that to instantiate this algorithm for DP continual counting, it is sufficient to find a low-degree rational function that approximates the square root on a circle in the complex plane. We then apply and extend tools from approximation theory to achieve this. We also derive efficient closed-forms for the objective function for arbitrarily many steps, and show direct numerical optimization yields a highly practical solution to the problem. Our second approach combines our first approach with a recursive construction similar to the binary tree mechanism.

• The paper presents a near-optimal matrix factorization technique that uses structured matrices and rational approximations to generate correlated noise for differential privacy in data streams.
• The paper leverages a novel streaming algorithm that achieves polylogarithmic space and time complexity by computing M(x) = B(Cx+z) efficiently on sequential data.
• The paper provides detailed bound analysis on the root mean squared error, ensuring that utility is maintained while meeting stringent differential privacy constraints.

Efficient Streaming Matrix Factorization for Differential Privacy

Introduction

Efficient matrix factorizations for differentially private continual counting are crucial in practical data analysis and machine learning applications. The core challenge lies in balancing computational and space complexity with differential privacy guarantees. Optimal matrix factorizations ensure noise addition maintains adequate utility while complying with privacy constraints, emphasizing the importance of strategic noise correlation across processing steps.

Matrix Factorization and Utility

The matrix $A \in \{0,1\}^{n \times n}$, a lower-triangular all-ones matrix, plays a central role in differentially private continual counting. Factorizing $A$ into products of simpler matrices $B$ and $C$, i.e., $A = BC$, where $B,C^T \in \mathbb{R}^{n \times n'}$, facilitates the generation of correlated noise essential for differential privacy. This noise correlation is crucial since each output depends only on inputs up to that point, not future ones due to the lower triangular invariance of matrices involved.

Algorithm Efficiency

The mechanism's efficiency is governed by the matrix multiplication $M(x) = B(Cx+z)$, with $z \gets \mathcal{N}(0,\sigma^2 I)$ denoting the Gaussian noise vector. The critical operational metrics are the space and time complexity, ideally both being polylogarithmic in $n$. The computational challenge pivots around generating the noise sequence $Bz$ without storing excessive data or noise values, which could breach either memory constraints or privacy by requiring information from extensive processing history.

Theoretical Contributions

1. Near-Optimal Matrix Factorization: The introduction of methods based on precise matrix factorizations involving Toeplitz structure and rational approximations to crucial functions allows highly efficient noise generation and effective differential privacy controls.
2. Algorithmic Advancements: A novel streaming algorithm leverages structured matrices to maintain continual updates over streaming data efficiently. This provides a crucial computational advantage in practical applications where data points arrive sequentially over time.
3. Bound Analysis and Performance Guarantees: Detailed theoretical analysis provides bounds on the root mean squared error, with the maximum error analyzed under various matrix compositions, ensuring that utility is maintained within acceptable limits defined by differential privacy requirements.

Practical Implications

Explicit delineation of methods facilitating near-optimal matrix factorizations offers a pathway to implement differentially private algorithms in environments with stringent memory and computational restrictions, such as mobile devices or in federated learning scenarios where data decentralization is crucial.

Future Directions

Future work may focus on extending these theoretical constructs to broader classes of matrices and exploring the implications of different noise distribution models on utility and privacy. Additionally, empirical validations of proposed methods in real-world datasets could substantiate theoretical claims, closing the gap between theoretical differential privacy and its practical implementations.

This paper progresses the discourse in differential privacy by detailing constructively how matrix factorizations can be strategically harnessed to enhance performance and privacy in continual data release frameworks, laying groundwork for future explorations in efficient data privacy mechanisms.