Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Noise-Aware Differentially Private Variational Inference (2410.19371v2)

Published 25 Oct 2024 in stat.ML, cs.CR, and cs.LG

Abstract: Differential privacy (DP) provides robust privacy guarantees for statistical inference, but this can lead to unreliable results and biases in downstream applications. While several noise-aware approaches have been proposed which integrate DP perturbation into the inference, they are limited to specific types of simple probabilistic models. In this work, we propose a novel method for noise-aware approximate Bayesian inference based on stochastic gradient variational inference which can also be applied to high-dimensional and non-conjugate models. We also propose a more accurate evaluation method for noise-aware posteriors. Empirically, our inference method has similar performance to existing methods in the domain where they are applicable. Outside this domain, we obtain accurate coverages on high-dimensional Bayesian linear regression and well-calibrated predictive probabilities on Bayesian logistic regression with the UCI Adult dataset.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (31)
  1. Deep learning with differential privacy. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, CCS’16. ACM, 2016.
  2. B. Becker and R. Kohavi. Adult. UCI Machine Learning Repository, 1996. DOI: https://doi.org/10.24432/C5XW20.
  3. G. Bernstein and D. Sheldon. Differentially private Bayesian inference for exponential families. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, pages 2924–2934, 2018.
  4. G. Bernstein and D. Sheldon. Differentially private Bayesian linear regression. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, pages 523–533, 2019.
  5. Variational inference: A review for statisticians. Journal of the American Statistical Association, 112(518):859–877, 2017.
  6. G. W. Brier. Verification of forecasts expressed in terms of probability. Monthly Weather Review, 78(1):1 – 3, 1950.
  7. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography, Third Theory of Cryptography Conference, TCC 2006, Proceedings, volume 3876 of Lecture Notes in Computer Science, pages 265–284. Springer, 2006.
  8. G. Folland. Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts. Wiley, 2013. ISBN 9781118626399.
  9. R. Gong. Exact inference with approximate computation for differentially private data via perturbations. J. Priv. Confidentiality, 12(2), 2022.
  10. Differentially private Markov chain Monte Carlo. In Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, pages 4115–4125, 2019.
  11. Probabilistic Numerics: Computation as Machine Learning. Cambridge University Press, 2022.
  12. M. D. Hoffman and A. Gelman. The No-U-Turn Sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(47):1593–1623, 2014.
  13. Differentially private variational inference for non-conjugate models. In Uncertainty in Artificial Intelligence 2017. The Association for Uncertainty in Artificial Intelligence, 2017. ISBN 978-1-5108-4779-8.
  14. DPVIm: Differentially private variational inference improved. Transactions on Machine Learning Research, 2023.
  15. An introduction to variational methods for graphical models. Mach. Learn., 37(2):183–233, 1999.
  16. Data augmentation MCMC for Bayesian inference from privatized data. In Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, NeurIPS 2022, 2022.
  17. D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations, ICLR 2015, Conference Track Proceedings, 2015.
  18. D. P. Kingma and M. Welling. Auto-encoding variational Bayes. In 2nd International Conference on Learning Representations, ICLR 2014, Conference Track Proceedings, 2014.
  19. Automatic differentiation variational inference. Journal of Machine Learning Research, 18(14):1–45, 2017.
  20. Differentially private Bayesian inference for generalized linear models. In Proceedings of the 38th International Conference on Machine Learning, ICML 2021, volume 139 of Proceedings of Machine Learning Research, pages 5838–5849. PMLR, 2021.
  21. Sampling-based accuracy testing of posterior estimators for general inference. In Proceedings of the 40th International Conference on Machine Learning, volume 202 of Proceedings of Machine Learning Research, pages 19256–19273. PMLR, 2023.
  22. Stochastic modified equations and dynamics of stochastic gradient algorithms I: mathematical foundations. J. Mach. Learn. Res., 20:40:1–40:47, 2019.
  23. On the validity of modeling SGD with stochastic differential equations (SDEs). In Advances in Neural Information Processing Systems 34: Annual Conference on Neural Information Processing Systems 2021, NeurIPS 2021, pages 12712–12725, 2021.
  24. A variational analysis of stochastic gradient algorithms. In Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, volume 48 of JMLR Workshop and Conference Proceedings, pages 354–363. JMLR.org, 2016.
  25. R. M. Neal. Probabilistic inference using Markov chain Monte Carlo methods. Technical report, Department of Computer Science, University of Toronto Toronto, ON, Canada, 1993.
  26. Noise-aware statistical inference with differentially private synthetic data. In International Conference on Artificial Intelligence and Statistics (AISTATS 2023), volume 206 of Proceedings of Machine Learning Research, pages 3620–3643. PMLR, 2023.
  27. A. Rajkumar and S. Agarwal. A differentially private stochastic gradient descent algorithm for multiparty classification. In Proceedings of the Fifteenth International Conference on Artificial Intelligence and Statistics, volume 22 of Proceedings of Machine Learning Research, pages 933–941. PMLR, 2012.
  28. L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales. Cambridge Mathematical Library. Cambridge University Press, 2 edition, 2000.
  29. Stochastic gradient descent with differentially private updates. In 2013 IEEE Global Conference on Signal and Information Processing, pages 245–248, 2013.
  30. Privacy for free: Posterior sampling and stochastic gradient Monte Carlo. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, volume 37 of JMLR Workshop and Conference Proceedings, pages 2493–2502. JMLR.org, 2015.
  31. Conditional density estimations from privacy-protected data. CoRR, abs/2310.12781, 2023.

Summary

  • The paper introduces NA-DPVI, a novel method that integrates DP noise into Bayesian inference to improve uncertainty quantification.
  • It establishes a formal framework using Bayesian linear models to post-process gradient traces and analyze hyperparameter effects.
  • Empirical evaluations in high-dimensional Bayesian linear and logistic regression show improved calibration on real-world datasets.

Noise-Aware Differentially Private Variational Inference

The paper presents a novel approach to incorporating differential privacy (DP) into Bayesian inference by proposing Noise-Aware Differentially Private Variational Inference (NA-DPVI). Traditional methods suffer from poor uncertainty quantification due to disregarding the noise added for privacy purposes. The proposed NA-DPVI method effectively integrates DP noise into high-dimensional and non-conjugate probabilistic models, offering a significant enhancement over existing techniques limited to simpler models.

Key Contributions

The authors make several important contributions:

  1. Theoretical Framework: A formal framework for noise-aware inference is established, expanding upon previous work to allow for more comprehensive analysis of approximate noise-aware posteriors.
  2. NA-DPVI Method: A new methodology is introduced for performing approximate noise-aware inference by post-processing gradient traces using a Bayesian linear model. This captures uncertainty from DP while utilizing VI posterior approximations for data modeling.
  3. Theoretical Analysis: The paper provides a rigorous examination of conditions under which the proposed method is effective, focusing on the influence of hyperparameters, such as learning rates, on posterior approximation.
  4. Evaluation Method: An advanced evaluation technique is developed for estimating noise-aware posteriors, adapted from the TARP method. This is used to gauge the effectiveness of the NA-DPVI method in empirical tests.

Numerical Results and Experiments

The empirical evaluation demonstrates that NA-DPVI performs comparably to traditional methods when they are applicable, with enhanced accuracy in scenarios where they are not. Specifically, the paper reports strong results in high-dimensional Bayesian linear regression and Bayesian logistic regression with the UCI Adult dataset.

  • Exponential Families: NA-DPVI achieves competitive coverage errors compared to existing noise-aware techniques, proving its efficacy across various conjugate models.
  • High-Dimensional Models: In scenarios such as 10D Bayesian linear regression, NA-DPVI shows substantially improved results over naive baselines, particularly when using more robust inference techniques like NUTS.
  • Real-World Data: The model's real-world applicability is illustrated through Bayesian logistic regression on the UCI Adult dataset, where NA-DPVI achieves better calibrated predictive distributions.

Implications and Future Directions

This paper's contributions are two-fold, impacting both theoretical understanding and practical execution of DP Bayesian inference. By incorporating noise-awareness into the inference process, NA-DPVI enables more precise uncertainty quantification in private data analysis. Future research could focus on further refining these techniques and extending their applicability to broader classes of models and datasets.

Moreover, the framework and methodology pave the way for future exploration in noise-aware inference algorithms that could offer even more robust solutions for incorporating DP in various fields. Continued investigation into the implications of privacy noise on statistical inference accuracy remains a critical area of development.

In conclusion, the framework and results presented constitute a substantial step forward in noise-aware Bayesian inference under differential privacy, promising more reliable applications in data-sensitive environments.

File Document Download Save Streamline Icon: https://streamlinehq.com PDF File Document Download Save Streamline Icon: https://streamlinehq.com Markdown
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets