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Naturally Private Recommendations with Determinantal Point Processes

Jack Fitzsimons, Agustín Freitas Pasqualini, Robert Pisarczyk, Dmitrii Usynin

TL;DR

The work investigates whether determinantal point processes (DPPs), which inherently balance popularity and diversity, can provide natural privacy protection without explicit noise. It casts DPP sampling as an instance of the exponential mechanism by relating eigenvalue-based scores to $u(\lambda)=\log(\lambda)$ and derives a bound $\Delta_u = n \log\left(1 + \dfrac{\Delta_L}{\sigma \sqrt{n}}\right)$ under a jitter $\sigma$, leading to $\epsilon$-privacy for the eigenvalue step via $\epsilon = 2\Delta_u$. It further analyzes eigenvector-based sampling, invoking the Davis–Kahan theorem to relate perturbations to eigengap, and notes that the total privacy budget also depends on the subset-construction step, which remains to be quantified. The study highlights practical limitations, such as dependence on eigenvalue separation and potentially large privacy budgets, and proposes integrating DPPs with standard DP techniques (e.g., SVT) to improve the privacy-utility balance in future work.

Abstract

Often we consider machine learning models or statistical analysis methods which we endeavour to alter, by introducing a randomized mechanism, to make the model conform to a differential privacy constraint. However, certain models can often be implicitly differentially private or require significantly fewer alterations. In this work, we discuss Determinantal Point Processes (DPPs) which are dispersion models that balance recommendations based on both the popularity and the diversity of the content. We introduce DPPs, derive and discuss the alternations required for them to satisfy epsilon-Differential Privacy and provide an analysis of their sensitivity. We conclude by proposing simple alternatives to DPPs which would make them more efficient with respect to their privacy-utility trade-off.

Naturally Private Recommendations with Determinantal Point Processes

TL;DR

The work investigates whether determinantal point processes (DPPs), which inherently balance popularity and diversity, can provide natural privacy protection without explicit noise. It casts DPP sampling as an instance of the exponential mechanism by relating eigenvalue-based scores to and derives a bound under a jitter , leading to -privacy for the eigenvalue step via . It further analyzes eigenvector-based sampling, invoking the Davis–Kahan theorem to relate perturbations to eigengap, and notes that the total privacy budget also depends on the subset-construction step, which remains to be quantified. The study highlights practical limitations, such as dependence on eigenvalue separation and potentially large privacy budgets, and proposes integrating DPPs with standard DP techniques (e.g., SVT) to improve the privacy-utility balance in future work.

Abstract

Often we consider machine learning models or statistical analysis methods which we endeavour to alter, by introducing a randomized mechanism, to make the model conform to a differential privacy constraint. However, certain models can often be implicitly differentially private or require significantly fewer alterations. In this work, we discuss Determinantal Point Processes (DPPs) which are dispersion models that balance recommendations based on both the popularity and the diversity of the content. We introduce DPPs, derive and discuss the alternations required for them to satisfy epsilon-Differential Privacy and provide an analysis of their sensitivity. We conclude by proposing simple alternatives to DPPs which would make them more efficient with respect to their privacy-utility trade-off.
Paper Structure (11 sections, 1 theorem, 6 equations, 1 algorithm)

This paper contains 11 sections, 1 theorem, 6 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Determinantal Point Processes (DPPs) and L-ensembles
  4. Sampling from a DPP Using Eigenvalue Decomposition
  5. Kernel Construction for DPPs
  6. Privacy Analysis of DPPs
  7. Sensitivity Analysis of DPPs
  8. Eigenvalue Sampling (with Preconditioning)
  9. Eigenvector Sampling (with Preconditioning)
  10. Implicit Privacy Budget Consumption
  11. Shortcomings and Future Work

Key Result

Theorem 3.1

An L-ensemble-based Determinantal Point Process (DPP) with kernel matrix $\mathbf{L}$ achieves $\epsilon$-differential privacy only if $\mathbf{L}$ is strictly positive definite, thereby preventing any eigenvalue $\lambda_i$ from approaching zero and ensuring the defined scoring function $u(\lambda)

Theorems & Definitions (1)

  • Theorem 3.1