Private Learning and Sanitization: Pure vs. Approximate Differential Privacy (1407.2674v1)

Abstract: We compare the sample complexity of private learning [Kasiviswanathan et al. 2008] and sanitization~[Blum et al. 2008] under pure $\epsilon$-differential privacy [Dwork et al. TCC 2006] and approximate $(\epsilon,\delta)$-differential privacy [Dwork et al. Eurocrypt 2006]. We show that the sample complexity of these tasks under approximate differential privacy can be significantly lower than that under pure differential privacy. We define a family of optimization problems, which we call Quasi-Concave Promise Problems, that generalizes some of our considered tasks. We observe that a quasi-concave promise problem can be privately approximated using a solution to a smaller instance of a quasi-concave promise problem. This allows us to construct an efficient recursive algorithm solving such problems privately. Specifically, we construct private learners for point functions, threshold functions, and axis-aligned rectangles in high dimension. Similarly, we construct sanitizers for point functions and threshold functions. We also examine the sample complexity of label-private learners, a relaxation of private learning where the learner is required to only protect the privacy of the labels in the sample. We show that the VC dimension completely characterizes the sample complexity of such learners, that is, the sample complexity of learning with label privacy is equal (up to constants) to learning without privacy.

• The paper shows approximate differential privacy can significantly reduce sample complexity for learning and sanitization tasks compared to pure DP.
• The authors introduce Quasi-Concave Promise Problems and use them to build recursive algorithms for privately learning and sanitizing functions.
• An analysis of label-private learners reveals their sample complexity is characterized by VC dimension, similar to non-private learners.

Differential Privacy in Learning and Sanitization: Comparative Analysis

This paper conducts a rigorous analysis of the sample complexity involved in private learning and sanitization, comparing two paradigms: pure differential privacy (DP) and approximate differential privacy. The authors examine the intricate balance between privacy guarantees and the efficiency of machine learning algorithms operating on sensitive datasets.

The fundamental premise of differential privacy is to preserve the information of individuals by ensuring that any single data point does not significantly influence the outcome. Pure ε-differential privacy provides stronger privacy assurances, whereas (ε, δ)-differential privacy relaxes this guarantee, potentially allowing lower sample complexity and more efficient algorithms.

Key Findings

1. Sample Complexity Reduction: The work underscores how (ε, δ)-differential privacy can lead to substantial reductions in sample complexity for both learning and sanitization tasks compared to pure differential privacy. This result has significant implications for the feasibility and efficiency of privacy-preserving algorithms.
2. Quasi-Concave Promise Problems: The authors introduce and leverage a concept termed Quasi-Concave Promise Problems to solve optimization problems privately and efficiently. They employ this abstraction to construct recursive algorithms for learning and sanitizing various function classes, including point functions and threshold functions.
3. Label-Private Learners: An exploration into label-private learners—where privacy is only required for the labels—reveals that the VC dimension characterizes their sample complexity precisely like non-private learners. This finding bridges the conceptual gap between privacy-preserving and traditional learning models.
4. Sanitization Implications: The paper distinguishes between the requirements for sanitizers under pure and approximate DP, revealing lower sample complexity under the approximate framework. However, pure-private tasks often demand considerably larger datasets.
5. Reductions to Private Learning: The authors establish that the sanitization of a concept class can yield private learning algorithms for the same class. Such reductions illustrate deep connections between different privacy-preserving tasks in learning.

Theoretical and Practical Implications

The theoretical ramifications of this work are manifold. The paper enhances our understanding of the relationships between privacy guarantees and computational constraints in the context of machine learning. The introduction of Quasi-Concave Promise Problems showcases a novel approach to handling complex optimization scenarios under privacy constraints.

Practically, the findings provide concrete guidelines for designing efficient privacy-preserving algorithms in domains where data sensitivity is paramount, such as in healthcare and finance. By highlighting the trade-offs between privacy and computational feasibility, the research directs future endeavors towards constructing versatile algorithms that judiciously balance these aspects.

Speculation on Future Developments

As artificial intelligence and data privacy continue to intersect robustly, several pathways for further research emerge:

• Enhanced Mechanisms for Complex Concept Classes: Future work could extend these concepts to learning more sophisticated structures like hyperplanes and neural networks, potentially yielding efficient privacy-preserving learners across diverse applications.
• Empirical Validation: While this paper focuses on theoretical constructs, comprehensive empirical studies could further elucidate the practical performance and limitations of these privacy mechanisms.
• Hybrid Privacy Frameworks: Developing frameworks that mix pure and approximate differential privacy dynamically based on contextual requirements could offer adaptive privacy-compliant solutions with an optimal balance of accuracy and privacy.

In summary, this paper represents a substantial step forward in the domain of privacy-preserving machine learning, laying foundational insights that promise to catalyze further innovation and application across sectors demanding rigorous data privacy.