Differentially Private Empirical Risk Minimization (0912.0071v5)

Abstract: Privacy-preserving machine learning algorithms are crucial for the increasingly common setting in which personal data, such as medical or financial records, are analyzed. We provide general techniques to produce privacy-preserving approximations of classifiers learned via (regularized) empirical risk minimization (ERM). These algorithms are private under the $\epsilon$-differential privacy definition due to Dwork et al. (2006). First we apply the output perturbation ideas of Dwork et al. (2006), to ERM classification. Then we propose a new method, objective perturbation, for privacy-preserving machine learning algorithm design. This method entails perturbing the objective function before optimizing over classifiers. If the loss and regularizer satisfy certain convexity and differentiability criteria, we prove theoretical results showing that our algorithms preserve privacy, and provide generalization bounds for linear and nonlinear kernels. We further present a privacy-preserving technique for tuning the parameters in general machine learning algorithms, thereby providing end-to-end privacy guarantees for the training process. We apply these results to produce privacy-preserving analogues of regularized logistic regression and support vector machines. We obtain encouraging results from evaluating their performance on real demographic and benchmark data sets. Our results show that both theoretically and empirically, objective perturbation is superior to the previous state-of-the-art, output perturbation, in managing the inherent tradeoff between privacy and learning performance.

• The paper introduces objective perturbation as a novel method to balance privacy preservation and learning performance in empirical risk minimization.
• It proves differential privacy guarantees for both output and objective perturbation under convex loss functions and strong regularizers.
• Empirical validation on real datasets demonstrates that objective perturbation achieves lower error rates and enables private parameter tuning compared to output perturbation.

An Overview of Differentially Private Empirical Risk Minimization

The paper "Differentially Private Empirical Risk Minimization" by Kamalika Chaudhuri, Claire Monteleoni, and Anand D. Sarwate addresses the challenges of designing machine learning algorithms that maintain the privacy of sensitive data. The primary focus is on Empirical Risk Minimization (ERM) within the context of differential privacy, a rigorously defined privacy model. This overview will cover the key methodologies, theoretical results, and practical implications presented in the paper.

Key Contributions

1. Output Perturbation Method: This method adapts the standard output perturbation technique for differentially private ERM by adding Laplace noise scaled to the sensitivity of the ERM solution. The authors prove that if the loss function and the regularizer are convex and differentiable, this method provides differential privacy guarantees.
2. Objective Perturbation Method: A novel contribution where noise is added to the ERM objective function itself rather than its output. This perturbation method ensures differential privacy, provided the regularizer is strongly convex and the loss function satisfies certain convexity and differentiability conditions. Objective perturbation is shown to be theoretically more efficient in trading off between privacy and learning performance compared to output perturbation.
3. Privacy-preserving Parameter Tuning: The authors develop a method to privately tune hyperparameters, addressing the often overlooked problem in end-to-end privacy-preserving learning. This is achieved by selecting hyperparameters through differentially private mechanisms on holdout data.
4. Privacy for Kernel Methods: They extend the approach to kernel methods, which are especially challenging due to their inherent data dependence. Using random projections, the authors convert kernel methods into a finite-dimensional linear representation, allowing the usage of differentially private ERM techniques.
5. Empirical Validation: Extensive experiments validate the theoretical results, demonstrating significant improvements in learning performance when using objective perturbation over output perturbation. Experiments were conducted on the Adult and KDDCup99 datasets, showcasing the practical relevance of their privacy-preserving methods.

Theoretical Insights and Numerical Results

The theoretical results presented in the paper include rigorous proofs of differential privacy for both output perturbation and objective perturbation methods. The sensitivity of the loss function and the strongly convex nature of the regularizer are critical to ensuring the privacy guarantees. Furthermore, the authors provide generalization error bounds, highlighting the sample complexity required to achieve a specific generalization error given the privacy constraints.

Numerically, the paper shows that the objective perturbation method generally outperforms output perturbation in balancing privacy and learning performance. For instance, in the case of logistic regression and support vector machines on the Adult and KDDCup99 datasets, the error rates of classifiers trained with objective perturbation were significantly lower than those trained with output perturbation under the same privacy budget.

Future Directions and Practical Implications

The implications of this research are both practical and theoretical. Practically, the techniques proposed can be readily integrated into existing machine learning pipelines where sensitive data such as medical records or financial transactions are analyzed. Theoretically, the paper opens several avenues for future research:

1. Relaxing Assumptions on Loss and Regularization: One direction is to explore privacy-preserving learning algorithms that do not require the strong convexity of the regularizer or differentiability of the loss function.
2. More Efficient Kernel Methods: While the paper provides a solution using random projections, more efficient and statistically robust methods for privacy-preserving kernel methods could be developed.
3. Broader Class of Learning Problems: Extending these privacy-preserving techniques to a broader class of learning problems, such as reinforcement learning or unsupervised learning, would be valuable.
4. Tighter Privacy-Utility Trade-offs: Further work is needed to minimize the loss of utility further while maintaining robust privacy guarantees. This would include exploring alternative noise distribution models or adaptive privacy budgets.

Conclusion

This paper makes substantial contributions to the field of privacy-preserving machine learning by addressing fundamental problems in differentially private ERM. Through innovative methods like objective perturbation and privacy-preserving parameter tuning, the authors offer practical solutions validated by strong theoretical guarantees and empirical results. The implications of this work are far-reaching, enabling the deployment of machine learning algorithms in sensitive data environments without compromising individual privacy.