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Privacy-Aware Randomized Quantization via Linear Programming

Zhongteng Cai, Xueru Zhang, Mohammad Mahdi Khalili

TL;DR

This work addresses the challenge of achieving differential privacy with discrete outputs by introducing a family of unbiased discrete quantization mechanisms and formulating an optimization framework. The core idea is to quantize a scalar input into a finite set of symmetric bins via two-bin sampling around the input, with output probabilities tuned to minimize mean absolute error under a DP budget. The Exponential Randomized Mechanism (ERM) is a special case, and the optimal randomized quantization mechanism (OPTM) is found by solving a linear program that upper-bounds MAE and imposes linearized DP constraints. Empirical results on scalar and vector inputs, as well as DP-SGD tasks, show that OPTM generally achieves a superior privacy-accuracy trade-off compared to baselines like MVU and RQM, with ERM offering competitive performance. The framework also enables extensions to high-dimensional, biased, and dynamic quantization, making discrete DP outputs more practical for bandwidth-limited and secure-aggregation settings.

Abstract

Differential privacy mechanisms such as the Gaussian or Laplace mechanism have been widely used in data analytics for preserving individual privacy. However, they are mostly designed for continuous outputs and are unsuitable for scenarios where discrete values are necessary. Although various quantization mechanisms were proposed recently to generate discrete outputs under differential privacy, the outcomes are either biased or have an inferior accuracy-privacy trade-off. In this paper, we propose a family of quantization mechanisms that is unbiased and differentially private. It has a high degree of freedom and we show that some existing mechanisms can be considered as special cases of ours. To find the optimal mechanism, we formulate a linear optimization that can be solved efficiently using linear programming tools. Experiments show that our proposed mechanism can attain a better privacy-accuracy trade-off compared to baselines.

Privacy-Aware Randomized Quantization via Linear Programming

TL;DR

This work addresses the challenge of achieving differential privacy with discrete outputs by introducing a family of unbiased discrete quantization mechanisms and formulating an optimization framework. The core idea is to quantize a scalar input into a finite set of symmetric bins via two-bin sampling around the input, with output probabilities tuned to minimize mean absolute error under a DP budget. The Exponential Randomized Mechanism (ERM) is a special case, and the optimal randomized quantization mechanism (OPTM) is found by solving a linear program that upper-bounds MAE and imposes linearized DP constraints. Empirical results on scalar and vector inputs, as well as DP-SGD tasks, show that OPTM generally achieves a superior privacy-accuracy trade-off compared to baselines like MVU and RQM, with ERM offering competitive performance. The framework also enables extensions to high-dimensional, biased, and dynamic quantization, making discrete DP outputs more practical for bandwidth-limited and secure-aggregation settings.

Abstract

Differential privacy mechanisms such as the Gaussian or Laplace mechanism have been widely used in data analytics for preserving individual privacy. However, they are mostly designed for continuous outputs and are unsuitable for scenarios where discrete values are necessary. Although various quantization mechanisms were proposed recently to generate discrete outputs under differential privacy, the outcomes are either biased or have an inferior accuracy-privacy trade-off. In this paper, we propose a family of quantization mechanisms that is unbiased and differentially private. It has a high degree of freedom and we show that some existing mechanisms can be considered as special cases of ours. To find the optimal mechanism, we formulate a linear optimization that can be solved efficiently using linear programming tools. Experiments show that our proposed mechanism can attain a better privacy-accuracy trade-off compared to baselines.
Paper Structure (30 sections, 8 theorems, 66 equations, 4 figures, 9 tables, 2 algorithms)

This paper contains 30 sections, 8 theorems, 66 equations, 4 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Discrete differential privacy.
  4. Private quantization.
  5. Problem Formulation
  6. Background: differential privacy
  7. Proposed quantization mechanism
  8. Selection distribution.
  9. Special cases
  10. Exponential Randomized Mechanism (ERM).
  11. Optimal Mechanism
  12. Performance measure
  13. Performance measure.
  14. OPTM as a linear program
  15. Linear upper bound for MAE.
  16. ...and 15 more sections

Key Result

Theorem 1

Assume the interval $[-c-\Delta, c+\Delta]$ is divided uniformly into $m$ bins, i.e., Then ERM satisfies DP with privacy loss

Figures (4)

  • Figure 1: An example of selection distribution
  • Figure 2: An example of output distribution
  • Figure 3: Comparison of mean absolute error under the same privacy on scalar inputs
  • Figure 4: a) Average error of $L_1$ bounded vectors , b) Average error of $L_2$ bounded vectors, c) Training accuracy on breast cancer dataset, d) Training accuracy on MNIST dataset

Theorems & Definitions (17)

  • Definition 1: Exponential Mechanism
  • Theorem 1: Privacy loss of ERM
  • Theorem 2: Error of ERM
  • Lemma 1
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • Example 1
  • Example 2
  • Theorem 5
  • ...and 7 more