Variational Bayesian Unlearning

The evolution of Bayesian inference methods, from early MCMC to modern variational inference (VI), has enabled scalable modeling of complex posteriors. Prior works like Black Box VI and normalizing flows have improved expressiveness, but model updates and data privacy remain challenges. Traditional unlearning techniques rely on retraining or model-specific inverses, which are costly and limited in scope. Recent interest in machine unlearning aims to address privacy regulations like GDPR, but effective solutions for Bayesian models with approximate posteriors are lacking. This gap motivates the development of scalable, theoretically grounded unlearning methods that can handle high-dimensional, non-conjugate models.

The key challenge is how to efficiently remove the influence of specific data points from a Bayesian model without full retraining. Exact unlearning is feasible only for simple models, while approximate methods risk catastrophic forgetting due to biases in the posterior approximation. The difficulty lies in designing a loss function that accurately measures the discrepancy between the unlearned and retrained posteriors, ensuring the model forgets targeted data while retaining overall performance. Addressing this in the context of variational approximations, which are inherently biased, complicates the task further.

The paper introduces three main innovations: 1) The use of the evidence upper bound (EUBO) as a loss function, providing a principled trade-off between unlearning and retention; 2) The adjustment of likelihood via a tunable parameter λ, focusing unlearning on high-confidence regions to mitigate bias; 3) The application of reverse KL divergence, which overestimates posterior variance, offering robustness against approximation errors. These innovations collectively enable scalable, theoretically justified Bayesian unlearning, extending applicability to complex models like normalizing flows.

• �� Define the unlearning goal as minimizing KL divergence between the approximate posterior after unlearning and the true posterior from retraining. • Use VI to maximize the ELBO for the full data, obtaining q(θ|D). • Derive the EUBO as a variational upper bound, balancing unlearning from erased data and retaining information from remaining data. • Introduce an adjusted likelihood function, padj(De|θ; λ), which downweights low-confidence regions based on q(θ|D). • Optimize the unlearning posterior by minimizing the EUBO with the adjusted likelihood, focusing on high-confidence θ regions. • Alternatively, minimize reverse KL divergence, which naturally emphasizes high-probability regions, enhancing robustness. • Incorporate generative models like normalizing flows to express complex posteriors, improving approximation fidelity.

Experiments involve synthetic datasets (e.g., moon-shaped Gaussian) and real datasets like banknote authentication and Fashion MNIST. The models include sparse Gaussian processes and logistic regression, with parameters trained via stochastic gradient ascent. The erased data De is selectively removed, and the remaining data Dr is used for comparison. Metrics include KL divergence between predictive distributions and accuracy. Hyperparameters such as λ are tuned to observe trade-offs. Baselines include full retraining and naive unlearning methods. Results demonstrate significant KL reduction and performance stability, validating the approach across different models and data distributions.

The proposed EUBO-based unlearning reduces KL divergence by over 20% compared to retraining, with minimal accuracy loss. Reverse KL techniques perform robustly on complex models, maintaining errors below 0.01. Adjusted likelihood parameter λ effectively balances unlearning strength and stability, with optimal values around 10^-9. Experiments confirm that combining these techniques prevents catastrophic forgetting and preserves model utility, especially in high-dimensional or non-conjugate settings. The methods scale well with data size and model complexity, outperforming existing approaches.

This framework applies to privacy-sensitive domains like healthcare, finance, and social media, where data deletion is mandated. It enables fast model updates without full retraining, reducing computational costs and downtime. The approach supports regulatory compliance and user data rights, facilitating industry adoption. Future applications include real-time unlearning in streaming data environments and integration with deep learning architectures, broadening the scope of privacy-preserving AI.

The effectiveness depends on the quality of variational approximation; biases may persist in highly complex models. Parameter λ tuning requires careful calibration, which may vary across datasets. Computational costs increase with model size and complexity, necessitating further optimization. The current approach assumes access to the approximate posterior and erased data, which may not always be feasible. Future work should focus on automating parameter selection and extending to deep neural networks.