Gaussian-Smoothed Sliced Divergences: Theoretical Properties and Applications in Privacy-Preserving Domain Adaptation

Core Concepts

This work investigates the theoretical properties of Gaussian-smoothed sliced divergences, including their topological and statistical properties. It establishes that under mild conditions, the smoothing and slicing operations preserve the metric property. The paper also focuses on the sample complexity of such divergences, particularly the Gaussian-smoothed sliced Wasserstein distance, and proves that it converges at a rate of O(n^(-1/2)). Additionally, the authors derive continuity properties of the divergences with respect to the smoothing parameter, which is crucial for the privacy-utility trade-off.

Abstract

The paper starts by introducing the concept of Gaussian-smoothed sliced divergences, which are a generalization of the Gaussian-smoothed sliced Wasserstein distance. It first establishes the topological properties of these divergences, showing that under mild conditions on the base divergence, the smoothing and slicing operations preserve the metric property and the weak topology. The paper then focuses on the statistical properties of these divergences. It introduces the concept of a "double empirical distribution" for the smoothed-projected origin distribution, which is a result of a double sampling process: one from sampling according to the origin distribution and the second according to the convolution of the projection of the origin distribution on the unit sphere and the Gaussian smoothing. The authors particularly focus on the Gaussian-smoothed sliced Wasserstein distance and prove that it converges at a rate of O(n^(-1/2)). The paper also derives other properties, including the continuity of the divergences with respect to the smoothing parameter. This property is crucial for the privacy-utility trade-off, as the smoothing parameter impacts the level of privacy achieved by the divergence. The authors show that the divergences satisfy an order relation with respect to the noise level and are also continuous with respect to this parameter. The theoretical findings are supported by empirical studies in the context of privacy-preserving domain adaptation, where the authors demonstrate that the Gaussian-smoothed sliced divergences can achieve similar performance to their non-smoothed counterparts while preserving privacy.

Stats

The condition R ∞ 0 e^(2ξ^2 / (σ^2 ϑ^2)) P(||X|| > ξ) dξ < ∞ needs P(||X|| > ξ) to go to 0 faster than e^(-κ ξ^2) for κ < 2/σ^2 ϑ^2. This can be satisfied when ||X|| is a ω-sub-Gaussian (ω ≥ 0). The sample complexity depends on the amount of smoothing through the moment of the Gaussian noise: the larger the amount of smoothing (and thus the privacy), the worse is the constant of the complexity.

Quotes

"The novelty of the present paper consists in the theoretical properties derived from the definition of the empirical measure ˆ ˆ μn. This latter is derived from a double process sampling, which is inspired from the implementation part." "We emphasize that the novelty of the present paper consists in the theoretical properties derived from the definition of the empirical measure ˆ ˆ μn. This latter is derived from a double process sampling, which is inspired from the implementation part."

Key Insights Distilled From

Gaussian-Smoothed Sliced Probability Divergences

by Mokhtar Z. A... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03273.pdf

Deeper Inquiries

What are the implications of the double sampling process in the theoretical analysis, and how does it differ from previous works that only considered a single sampling process

The double sampling process in the theoretical analysis plays a crucial role in understanding the sample complexity and statistical properties of the Gaussian-smoothed sliced divergences. By incorporating two levels of randomness - one from sampling according to the original distribution and the second from sampling according to the convolution of the projection of the distribution on the unit sphere and the Gaussian smoothing - the analysis captures the intricacies of the approximation process. This double sampling process allows for a more comprehensive evaluation of the convergence rates and the impact of the smoothing parameter on the divergence metrics. In contrast to previous works that only considered a single sampling process, the double sampling process provides a more nuanced understanding of how the divergence metrics behave under different conditions. It allows for a more accurate assessment of the sample complexity and projection complexity, leading to more robust theoretical results. By considering both levels of randomness, the theoretical analysis can better capture the underlying dynamics of the Gaussian-smoothed sliced divergences and their properties.

Can the theoretical results be extended to other types of smoothing distributions beyond the Gaussian, and what would be the key considerations in such an extension

The theoretical results regarding the Gaussian-smoothed sliced divergences can potentially be extended to other types of smoothing distributions beyond the Gaussian. However, such an extension would require careful consideration of the properties and characteristics of the alternative smoothing distributions. Key considerations for extending the theoretical results to other smoothing distributions include: Properties of the Smoothing Distribution: The new smoothing distribution should have properties that allow for similar mathematical manipulations and analyses as the Gaussian distribution. This includes properties related to convolution, moments, and continuity. Impact on Metric Properties: The new smoothing distribution should not significantly alter the metric properties of the divergence measures. It should still preserve the metric property, weak topology, and other key properties established in the theoretical analysis. Sample Complexity: The extension should consider how the new smoothing distribution affects the sample complexity of the divergence metrics. Understanding how the convergence rates change with the new distribution is essential for practical applications. Continuity and Stability: The extension should maintain the continuity and stability of the divergence metrics with respect to the smoothing parameter. Any new distribution should not introduce discontinuities or instabilities in the analysis. By carefully addressing these considerations and ensuring that the properties of the new smoothing distribution align with the requirements of the theoretical framework, the results can be extended to a broader range of smoothing distributions.

How can the insights from the privacy-utility trade-off analysis be leveraged to develop practical guidelines for selecting the optimal smoothing parameter in real-world applications

The insights from the privacy-utility trade-off analysis can be leveraged to develop practical guidelines for selecting the optimal smoothing parameter in real-world applications. By understanding the relationship between the smoothing parameter, privacy guarantees, and utility in the context of Gaussian-smoothed sliced divergences, practitioners can make informed decisions when implementing these divergence metrics. Practical guidelines for selecting the optimal smoothing parameter may include: Privacy Requirements: Determine the level of privacy required for the specific application. A higher smoothing parameter (larger noise level) provides increased privacy guarantees but may impact the utility of the divergence metric. Utility Considerations: Evaluate the trade-off between privacy and utility. Conduct experiments to assess the performance of the divergence metric at different smoothing parameters and choose the parameter that balances privacy and utility effectively. Fine-Tuning Strategy: Consider a fine-tuning strategy where models are initially trained with a high smoothing parameter for privacy and then fine-tuned with lower smoothing parameters for improved utility. This approach can help achieve a balance between privacy and performance. Empirical Validation: Validate the performance of the divergence metric at different smoothing parameters on real-world datasets. This empirical analysis can provide insights into the practical implications of the privacy-utility trade-off. By following these guidelines and considering the insights from the privacy-utility trade-off analysis, practitioners can effectively select the optimal smoothing parameter for Gaussian-smoothed sliced divergences in real-world applications.