The Right Complexity Measure in Locally Private Estimation: It is not the Fisher Information (1806.05756v3)

Abstract: We identify fundamental tradeoffs between statistical utility and privacy under local models of privacy in which data is kept private even from the statistician, providing instance-specific bounds for private estimation and learning problems by developing the \emph{local minimax risk}. In contrast to approaches based on worst-case (minimax) error, which are conservative, this allows us to evaluate the difficulty of individual problem instances and delineate the possibilities for adaptation in private estimation and inference. Our main results show that the local modulus of continuity of the estimand with respect to the variation distance---as opposed to the Hellinger distance central to classical statistics---characterizes rates of convergence under locally private estimation for many notions of privacy, including differential privacy and its relaxations. As consequences of these results, we identify an alternative to the Fisher information for private estimation, giving a more nuanced understanding of the challenges of adaptivity and optimality.

• The paper demonstrates that classical Fisher information fails under local privacy, proposing local minimax risk as a more appropriate complexity measure.
• It introduces the local modulus of continuity with respect to variation distance, which improves the understanding of convergence rates in private estimation.
• The analysis reveals that problems considered easy by traditional metrics become challenging when stringent privacy constraints are enforced.

The paper "The Right Complexity Measure in Locally Private Estimation: It is not the Fisher Information" by John C. Duchi and Feng Ruan addresses fundamental challenges in locally private estimation by exploring the tradeoffs between privacy and statistical utility. The authors argue that traditional complexity measures, like the Fisher information, are inadequate under local privacy constraints, proposing an alternative framework based on the local minimax risk.

Key Contributions

1. Local Minimax Risk:
   • The paper introduces a framework to evaluate private estimation difficulties by focusing on the local minimax risk, which considers instance-specific complexity rather than the traditional minimax viewpoint.
   • The minimax risk is contextualized in locally private settings such as differential privacy, highlighting the limitations of classical statistical measures.
2. Local Modulus of Continuity:
   • The authors propose the use of the local modulus of continuity with respect to variation distance, as opposed to the Hellinger distance used in classical statistics.
   • This choice significantly impacts the rates of convergence in private estimation problems, offering a better understanding of adaptivity and optimality under privacy constraints.
3. Alternative to Fisher Information:
   • They identify an alternative measure to the Fisher information for locally private estimation, suggesting that complexity in such contexts should be understood through different lenses.
   • This metric reflects the nuances and challenges found in adapting classical statistical theory to private settings.
4. Numerical and Theoretical Analyses:
   • Through theoretical constructs and numerical experiments, the paper demonstrates that the variance-constrained approach of classical statistical methods may not hold under local privacy conditions.
   • They showcase that easy problems, as per classical metrics, are not necessarily easy when privacy is considered, demonstrating cases where local privacy influences estimation complexity significantly.

Practical Implications

• The paper has significant implications for the design of statistical procedures in environments where strong privacy protections are needed, such as technology companies implementing privacy-preserving data collection.
• The results suggest that careful consideration of privacy constraints and adaptation to problem-specific complexities can lead to more accurate and efficient estimations in practice.

Limitations and Future Work

• While this paper provides a foundational shift in conceptualizing complexity under privacy, the practical implementation of such concepts in real-world systems may require further refinement.
• Future research could expand these ideas to multi-dimensional and more complex parametric models to verify the framework's applicability across broader statistical and machine learning applications.

Overall, the authors argue convincingly for a departure from classical measures like the Fisher information in favor of metrics better suited to the requirements of local privacy, paving the way for more robust and privacy-conscious statistical methodologies. The proposed framework could potentially revolutionize how complexity is measured in privacy-sensitive environments, leading to better-aligned theoretical and practical tools for estimation tasks.