On the Growth of Mistakes in Differentially Private Online Learning: A Lower Bound Perspective (2402.16778v3)

Abstract: In this paper, we provide lower bounds for Differentially Private (DP) Online Learning algorithms. Our result shows that, for a broad class of $(\varepsilon,\delta)$-DP online algorithms, for number of rounds $T$ such that $\log T\leq O(1 / \delta)$, the expected number of mistakes incurred by the algorithm grows as $\Omega(\log \frac{T}{\delta})$. This matches the upper bound obtained by Golowich and Livni (2021) and is in contrast to non-private online learning where the number of mistakes is independent of $T$. To the best of our knowledge, our work is the first result towards settling lower bounds for DP-Online learning and partially addresses the open question in Sanyal and Ramponi (2022).

• The paper establishes a theoretical lower bound on expected errors, demonstrating that mistake counts grow as Ω(log(T/δ)/ε).
• It introduces a novel analytical approach by modeling concentrated learners and uniform firing algorithms under differential privacy constraints.
• The findings highlight a trade-off between privacy and accuracy, emphasizing the inherent cost of embedding differential privacy in online learning.

Lower Bounds for Differentially Private Online Learning

Introduction to Differentially Private Online Learning

Differential Privacy (DP) serves as a cornerstone in the privacy-preserving data analysis domain, aimed at safeguarding individual privacy while still allowing for meaningful statistical analysis. Despite its advantages, the incorporation of DP in online learning models introduces a trade-off between privacy and accuracy. Specifically, this paper explores the cost of privacy in the context of DP online learning algorithms, extending the investigation to the domain where learner’s mistakes gradually increase with the processing of more data points. This phenomenon starkly contrasts with non-private online learning models, where the mistake count typically remains independent of the data size.

Theoretical Insights and Main Contributions

This research delineates a theoretical framework establishing lower bounds on the number of errors incurred by differentially private online learning algorithms across a broad class of (ε, δ)-DP online algorithms. It demonstrates that the expected mistake count grows logarithmically with the time horizon T, given by Ω(log(T/δ)/ε), indicating a compulsory increase in error rates as the learning process prolongs. This outcome not only provides a foundational understanding of the cost of embedding differential privacy into online learning but also aligns with previously obtained upper bounds, essentially highlighting a trade-off between privacy guarantees and learning performance.

Novelty in Analytical Approach

The novelty of the paper lies in its analytical approach towards establishing these lower bounds, which are validated under specific assumptions about the learning algorithm’s behavior and concentration properties. By conceptualizing a "concentrated" learning model, the research showcases how certain algorithms that exhibit high confidence in their output on non-distinguishing input sequences inevitably lead to a predictable mistake count lower bound. Moreover, the paper extends its analysis to a subclass of algorithms termed "uniform firing algorithms," offering insights into their performance under differentially private constraints.

Practical Implications and Future Directions

The theoretical findings of this paper have profound implications for the design and implementation of DP online learning models, urging a careful consideration of the inherent trade-offs between privacy and accuracy. The established lower bounds not only benchmark the inevitable cost of privacy in online learning scenarios but also push the envelope for developing more efficient algorithms that can navigate these trade-offs more effectively.

Looking ahead, this paper sets the stage for a plethora of research opportunities, inviting exploration into the realms of pure DP online learning algorithms and beyond. Investigating potential upper and lower bounds for a wider variety of hypothesis classes and learning models could further enhance the understanding of DP’s impact on online learning. Additionally, the question of whether generic learners with minimal dependency on class size or more effective continuous observation to online learning transformations can be designed remains open, promising exciting avenues for future research.

Acknowledgements

The authors extend their gratitude to several individuals and institutions for their support and intellectual contributions to this research, signifying the collaborative spirit that drives advancements in the field of differentially private online learning.