Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds (1405.7085v2)

Abstract: In this paper, we initiate a systematic investigation of differentially private algorithms for convex empirical risk minimization. Various instantiations of this problem have been studied before. We provide new algorithms and matching lower bounds for private ERM assuming only that each data point's contribution to the loss function is Lipschitz bounded and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex. Our algorithms run in polynomial time, and in some cases even match the optimal non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for $(\epsilon,0)$- and $(\epsilon,\delta)$-differential privacy; perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median.

• The paper introduces novel differentially private ERM algorithms that operate efficiently and sometimes match the performance of non-private methods.
• It establishes tight theoretical lower bounds for excess risk in both (ε,0) and (ε,δ) privacy models, confirming near-optimality.
• The approaches apply broadly to convex and strongly convex functions, offering practical insights for secure machine learning applications.

Overview of Differentially Private Empirical Risk Minimization

This paper presents a detailed paper of algorithms for differentially private empirical risk minimization (ERM) within the realms of convex optimization, an essential component in statistics and machine learning. The authors focus on creating algorithms that maintain the privacy of sensitive data while providing accurate solutions to ERM problems. They explore the settings where each data point's contribution to the loss function is Lipschitz and propose separate frameworks for general convex functions and those that are strongly convex.

Key Contributions

The main contributions of this research include:

1. Algorithm Development: The paper introduces new algorithms for differentially private ERM that operate efficiently in polynomial time and, in some cases, match the runtime of optimal non-private methods.
2. Lower Bounds and Optimality: The authors establish tight lower bounds for these algorithms, demonstrating their optimality in both the $(\epsilon,0)$ and $(\epsilon,\delta)$-differential privacy settings.
3. Diverse Techniques: The paper outlines distinct techniques required for optimal algorithm design depending on the differential privacy settings, highlighting that $(\epsilon,0)$ and $(\epsilon,\delta)$ frameworks necessitate fundamentally different approaches.
4. Applicability to Smooth Functions: Importantly, the lower bounds apply even to simple, smooth function families, ensuring the generality of the results and their applicability to scenarios like training support vector machines and solving high-dimensional median problems.

Numerical Results & Practical Implications

The paper provides rigorous numerical evaluations, establishing strong performance guarantees for the proposed algorithms. Key numerical results include:

• For general convex functions, the achieved excess empirical risk under $(\epsilon,0)$-differential privacy is $\Theta\left(\frac{p}{\epsilon n}\right)$, while under the $(\epsilon,\delta)$ model, it becomes $$\Theta\left(\frac{\sqrt{p\log(1/\delta)}}{\epsilon n}\right)$$.
• When the loss functions are also strongly convex, the excess risk improves to $\Theta\left(\frac{p^2}{\epsilon^2 n^2}\right)$ for $(\epsilon,0)$ and $$\Theta\left(\frac{p \log(1/\delta)}{\epsilon^2 n^2}\right)$$ in the $(\epsilon,\delta)$ setting.

These bounds are almost tight, aligning with their respective lower bounds, suggesting that the designed algorithms are near-optimal.

Future Directions

The findings of this paper open several avenues for further exploration:

1. Algorithmic Refinements: Future work may focus on refining these algorithms to reduce constants and logarithmic factors in the bounds or extending them to more complex or structured data types.
2. Alternative Techniques: Investigating alternative paradigms or incorporating additional privacy mechanisms could lead to even more efficient private learning algorithms.
3. Application to Other Domains: Translating these methods to emerging fields where privacy concerns intersect with data-intensive applications, like genomics or social networks, could offer new insights and practical solutions.

Conclusion

This paper provides a significant contribution to the field of private empirical risk minimization by presenting algorithms that marry efficiency with privacy, backed by robust theoretical guarantees. It establishes a foundational understanding of the limitations and capabilities of such algorithms, shaping future research in differentially private optimization techniques. The insights into the differential privacy landscape, especially the nuanced handling of convex functions, set a high standard for ongoing and future research in this critical area of data privacy.