insight - Machine Learning - # Differential Privacy in Stochastic Gradient Descent

Privacy Analysis of Cyclically-Sampled DP-SGD with Last Iterate Release on Nonconvex Composite Losses

Q: Could the analysis be extended to handle adaptive gradient methods like Adam or Adagrad, which are widely used in practice but introduce additional challenges for privacy analysis?

Extending the analysis to adaptive gradient methods like Adam and Adagrad is a promising but non-trivial direction for future work. Here's why these methods pose challenges and potential ways to address them: Challenges: State-Dependent Updates: Adam and Adagrad maintain internal states (e.g., running averages of gradients and squared gradients) that are updated based on the observed data. This state-dependence introduces complexities in analyzing the sensitivity of the updates and requires careful accounting for how privacy leaks through these states. Non-Convexity and Non-Lipschitz Gradients: The analysis in the paper relies on properties like weak convexity and Lipschitz continuity of gradients. Adaptive methods often operate in settings where these assumptions might not hold or are difficult to verify. Potential Approaches for Extension: State Sensitivity Analysis: A key step would be to analyze the sensitivity of the internal states maintained by Adam and Adagrad. Techniques from composition theorems for differentially private mechanisms could be leveraged to bound the privacy loss accumulated through these states over multiple iterations. Modified Mechanisms: It might be necessary to modify the adaptive methods themselves to make them amenable to privacy analysis. For instance, one could explore clipping the updates to the internal states or using techniques like tree-based aggregation to limit the privacy leakage. Relaxed Assumptions: Investigating whether the analysis can be extended to handle weaker assumptions on the loss function (e.g., beyond weak convexity) would be crucial for practical applicability.

Q: How can these theoretical privacy guarantees be effectively communicated to practitioners and incorporated into the design of privacy-preserving machine learning systems deployed in real-world applications?

Bridging the gap between theoretical privacy guarantees and practical deployment is crucial for the responsible adoption of DP-SGD. Here are some ways to effectively communicate these results and incorporate them into real-world systems: Communication to Practitioners: Practical Guides and Tooling: Develop accessible guides and tutorials that explain the implications of these theoretical results in concrete terms. Providing easy-to-use software tools that automate the process of converting RDP bounds to $(\epsilon, \delta)$-DP guarantees and selecting appropriate noise parameters would be highly beneficial. Visualizations and Case Studies: Illustrate the impact of these tighter bounds on practical scenarios using visualizations and case studies. For example, showing how the required noise level changes for different privacy budgets and dataset sizes can provide intuitive understanding. Emphasis on Trade-offs: Clearly communicate the trade-offs between privacy, utility, and computational cost associated with different DP-SGD variants and parameter choices. This empowers practitioners to make informed decisions based on their specific application requirements. Incorporation into Real-World Systems: Integration with ML Libraries: Integrate these privacy-aware optimization algorithms into popular machine learning libraries (e.g., TensorFlow Privacy, PyTorch Opacus). This lowers the barrier to entry for practitioners and promotes wider adoption. Automated Privacy Auditing: Develop tools that can automatically audit the privacy guarantees of deployed machine learning models trained with DP-SGD. These tools can help ensure that the desired privacy levels are met in practice. Standardized Benchmarks: Establish standardized benchmarks and evaluation metrics for privacy-preserving machine learning. This allows for fair comparison of different approaches and tracks progress in the field.

Core Concepts

This paper presents a novel Rényi Differential Privacy (RDP) analysis for the last iterate of cyclically-traversed DP-SGD applied to nonconvex composite losses, addressing practical limitations of existing methods that rely on unrealistic assumptions like convexity and known sensitivity constants.

Abstract

Bibliographic Information: Kong, W., & Ribero, M. (2024). Privacy of the last iterate in cyclically-sampled DP-SGD on nonconvex composite losses. arXiv preprint arXiv:2407.05237v2.
Research Objective: This paper aims to develop tighter privacy analyses for the last iterate of DP-SGD under more realistic settings than existing works, particularly focusing on cyclically-traversed data, gradient clipping, and nonconvex loss functions.
Methodology: The authors leverage optimal transport techniques and analyze the Lipschitz properties of an SGD-like update to derive RDP bounds. They generalize the Privacy Amplification by Iteration (PABI) argument and combine it with analyses of weakly-convex functions and proximal operators.
Key Findings: The paper establishes new RDP upper bounds for the last iterate of cyclically-traversed DP-SGD under realistic assumptions of small step size and Lipschitz smoothness of the loss function. These bounds are parameterized by a weak convexity parameter and smoothly converge to convex bounds as the parameter approaches zero.
Main Conclusions: The proposed analysis provides tighter privacy guarantees for DP-SGD in practical settings, enabling implementations with lower noise variance and potentially higher model utility for the same privacy budget.
Significance: This work bridges a gap between theoretical DP analyses and practical DP-SGD implementations, offering valuable insights for privacy-preserving machine learning.
Limitations and Future Research: The paper focuses on RDP guarantees and does not explicitly address other DP definitions. Further research could explore extensions to different DP notions and investigate the trade-offs between privacy, utility, and computational efficiency in this context.

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Privacy of the last iterate in cyclically-sampled DP-SGD on nonconvex composite losses

by Weiw... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2407.05237.pdf

Privacy of the last iterate in cyclically-sampled DP-SGD on nonconvex composite losses

Deeper Inquiries

How do these RDP bounds translate to (ε, δ)-differential privacy guarantees, and how do they compare to existing bounds under this definition?

The paper focuses on deriving tighter Rényi Differential Privacy (RDP) bounds for the last iterate of DP-SGD under more realistic assumptions. While RDP offers a convenient framework for analyzing privacy, practitioners often work with the more widely known $(\epsilon, \delta)$-differential privacy.
Here's how the RDP bounds can be translated to $(\epsilon, \delta)$-DP and a comparison with existing bounds:
Conversion from RDP to $(\epsilon, \delta)$-DP:

Theorem: For any $(\alpha, \epsilon_\alpha)$-RDP mechanism, it also satisfies $(\epsilon, \delta)$-DP for
$\epsilon = \epsilon_\alpha + \frac{log(1/\delta)}{\alpha - 1}$.

Application:  Given an RDP bound from the paper (e.g., those in Theorem 4.3 or 4.4), we can obtain an $(\epsilon, \delta)$-DP guarantee by optimizing over the choice of $\alpha$. This typically involves finding the $\alpha$ that minimizes the right-hand side of the above equation for a fixed $\delta$.
Comparison with Existing $(\epsilon, \delta)$-DP Bounds:

Direct Comparison Challenging:  Directly comparing the $(\epsilon, \delta)$-DP guarantees derived from the paper's RDP bounds to existing bounds is not straightforward. This is because existing $(\epsilon, \delta)$-DP analyses often rely on different assumptions (e.g., random sampling of batches, convexity) that are not made in this work.

Qualitative Advantages: The tighter RDP bounds derived in the paper suggest that, when converted to $(\epsilon, \delta)$-DP, they would lead to improved guarantees compared to approaches based on loose RDP bounds (e.g., those relying on advanced composition theorems). This means that for a fixed privacy budget $\epsilon$ and failure probability $\delta$, the noise added to the gradients can be reduced while maintaining the same level of privacy.

Empirical Validation Needed:  To concretely demonstrate the improvements in $(\epsilon, \delta)$-DP, empirical evaluations comparing the noise levels required for a fixed $(\epsilon, \delta)$ guarantee under the different analyses would be necessary.

Could the analysis be extended to handle adaptive gradient methods like Adam or Adagrad, which are widely used in practice but introduce additional challenges for privacy analysis?

Extending the analysis to adaptive gradient methods like Adam and Adagrad is a promising but non-trivial direction for future work. Here's why these methods pose challenges and potential ways to address them:
Challenges:

State-Dependent Updates: Adam and Adagrad maintain internal states (e.g., running averages of gradients and squared gradients) that are updated based on the observed data. This state-dependence introduces complexities in analyzing the sensitivity of the updates and requires careful accounting for how privacy leaks through these states.

Non-Convexity and Non-Lipschitz Gradients: The analysis in the paper relies on properties like weak convexity and Lipschitz continuity of gradients. Adaptive methods often operate in settings where these assumptions might not hold or are difficult to verify.
Potential Approaches for Extension:

State Sensitivity Analysis:  A key step would be to analyze the sensitivity of the internal states maintained by Adam and Adagrad. Techniques from composition theorems for differentially private mechanisms could be leveraged to bound the privacy loss accumulated through these states over multiple iterations.

Modified Mechanisms:  It might be necessary to modify the adaptive methods themselves to make them amenable to privacy analysis. For instance, one could explore clipping the updates to the internal states or using techniques like tree-based aggregation to limit the privacy leakage.

Relaxed Assumptions:  Investigating whether the analysis can be extended to handle weaker assumptions on the loss function (e.g., beyond weak convexity) would be crucial for practical applicability.

How can these theoretical privacy guarantees be effectively communicated to practitioners and incorporated into the design of privacy-preserving machine learning systems deployed in real-world applications?

Bridging the gap between theoretical privacy guarantees and practical deployment is crucial for the responsible adoption of DP-SGD. Here are some ways to effectively communicate these results and incorporate them into real-world systems:
Communication to Practitioners:

Practical Guides and Tooling: Develop accessible guides and tutorials that explain the implications of these theoretical results in concrete terms. Providing easy-to-use software tools that automate the process of converting RDP bounds to $(\epsilon, \delta)$-DP guarantees and selecting appropriate noise parameters would be highly beneficial.

Visualizations and Case Studies:  Illustrate the impact of these tighter bounds on practical scenarios using visualizations and case studies. For example, showing how the required noise level changes for different privacy budgets and dataset sizes can provide intuitive understanding.

Emphasis on Trade-offs: Clearly communicate the trade-offs between privacy, utility, and computational cost associated with different DP-SGD variants and parameter choices. This empowers practitioners to make informed decisions based on their specific application requirements.
Incorporation into Real-World Systems:

Integration with ML Libraries: Integrate these privacy-aware optimization algorithms into popular machine learning libraries (e.g., TensorFlow Privacy, PyTorch Opacus). This lowers the barrier to entry for practitioners and promotes wider adoption.

Automated Privacy Auditing: Develop tools that can automatically audit the privacy guarantees of deployed machine learning models trained with DP-SGD. These tools can help ensure that the desired privacy levels are met in practice.

Standardized Benchmarks: Establish standardized benchmarks and evaluation metrics for privacy-preserving machine learning. This allows for fair comparison of different approaches and tracks progress in the field.