Differentially Private Empirical Risk Minimization Revisited: Faster and More General (1802.05251v1)

Abstract: In this paper we study the differentially private Empirical Risk Minimization (ERM) problem in different settings. For smooth (strongly) convex loss function with or without (non)-smooth regularization, we give algorithms that achieve either optimal or near optimal utility bounds with less gradient complexity compared with previous work. For ERM with smooth convex loss function in high-dimensional ($p\gg n$) setting, we give an algorithm which achieves the upper bound with less gradient complexity than previous ones. At last, we generalize the expected excess empirical risk from convex loss functions to non-convex ones satisfying the Polyak-Lojasiewicz condition and give a tighter upper bound on the utility than the one in \cite{ijcai2017-548}.

• The paper introduces DP-SVRG and DP-SVRG++ algorithms that achieve near-optimal utility with significantly reduced gradient complexity.
• It efficiently handles both strongly convex and high-dimensional settings as well as non-convex loss functions under the Polyak-Lojasiewicz condition.
• The study broadens ERM applicability by incorporating non-smooth regularizers and using Gaussian width to refine theoretical utility bounds.

Overview of "Differentially Private Empirical Risk Minimization Revisited: Faster and More General"

The paper by Wang, Ye, and Xu revisits the topic of Differentiated Private (DP) Empirical Risk Minimization (ERM), presenting advancements in both computational efficiency and the breadth of applicable scenarios for ERM under differential privacy guarantees. This research builds upon a wealth of preceding studies and leverages contemporary algorithmic enhancements to make notable improvements.

In particular, the paper targets scenarios involving both convex and non-convex loss functions or regularizers—common circumstances in real-world machine learning tasks. ERM is a fundamental technique in machine learning where the objective is to minimize the empirical risk represented by a loss function over a dataset. Integrating differential privacy—a framework ensuring that no single data point overly influences the output—into ERM is a critical demand when dealing with sensitive information.

Key Contributions

1. Algorithmic Efficiency: The authors introduce DP-SVRG and DP-SVRG++, which are based on the Stochastic Variance Reduced Gradient (SVRG) method and its enhanced variant. These algorithms achieve near-optimal utility bounds with lower gradient complexity than previous solutions, specifically GD and SGD. This reduction in complexity is pivotal for processing large datasets efficiently.
2. Optimizing Strongly Convex and High-Dimension Settings: For strongly convex loss functions, the paper reports achieving a near-optimal excess empirical risk bound with significantly reduced computational complexity. The provision of a comprehensive analysis and novel algorithms for high-dimensional settings—where the number of parameters significantly exceeds the number of data points—is a pronounced contribution, given such conditions are prevalent in modern applications.
3. Incorporating Non-Convex Loss Functions: Extending beyond convex functions, the authors also address ERM problems with non-convex losses that satisfy the Polyak-Lojasiewicz condition. Their work clarifies that it is possible to provide tighter utility bounds than earlier studies for such cases, expanding the potential applicability of ERM under differential privacy.
4. Robustness to Non-Smooth Regularization: By embracing non-smooth regularizers within their solutions, the authors broaden the practical utility of their approaches. Many real-world problems entail non-smooth penalty terms, and the ability to handle such scenarios without compromising on privacy or utility is significant.
5. Gaussian Width Insights: The paper investigates high-dimensional cases using Gaussian width—a crucial geometric measure—to replace dimensionality terms in utility bounds, leading to more nuanced insights and theoretical robustness.

Implications and Future Directions

The research presented is influential for fields like data science, medicine, and any domain reliant on sensitive data. Differential privacy is pivotal for ensuring ethical data handling, particularly as data scales in volume and variety. The proposed algorithms, which require fewer computations for privacy-preserving training, could expedite the development and deployment of machine learning models in sensitive applications.

Looking forward, continued exploration into differential privacy's applications in more diverse classes of optimization problems (beyond those with smoothness or convexity assumptions) appears promising. Additionally, establishing lower bounds for non-convex problems under differential privacy and refining utility metrics in these regimes offer fertile grounds for future paper.

Overall, the paper represents a methodological enhancement in privacy-preserving learning, balancing computational demands against rigorous privacy guarantees—a balance crucial for broader adoption in industry and academia alike.