Extending Mean-Field Variational Inference via Entropic Regularization: Theoretical and Computational Guarantees

Core Concepts

Ξ-variational inference (Ξ-VI) is a novel variational inference method that extends the naive mean-field approach by incorporating an entropic regularization term. This allows Ξ-VI to effectively recover the true posterior dependency structure while providing computational efficiency.

Abstract

The paper proposes a new variational inference method called Ξ-variational inference (Ξ-VI) that extends the traditional mean-field variational inference (MFVI) approach. Ξ-VI optimizes the variational objective over the entire space of distributions, but includes an entropic regularization term that encourages the variational posterior to resemble a factorized distribution.
The key insights are:

Ξ-VI has a close connection to the entropic optimal transport (EOT) problem, and the solution can be computed efficiently using the Sinkhorn algorithm.

Ξ-VI provides a smooth interpolation between the exact posterior and the mean-field approximation by varying the regularization parameter λ. When λ = 0, Ξ-VI recovers the exact posterior, while when λ = ∞, it reduces to the mean-field solution.

The authors provide theoretical guarantees for Ξ-VI, including posterior consistency, asymptotic normality (Bernstein-von Mises theorem), high-dimensional bounds, and algorithmic stability. They characterize regimes where Ξ-VI effectively approximates either the mean-field or the exact posterior.

Empirically, Ξ-VI is shown to outperform traditional MFVI and other VI methods on simulated and real-world datasets. The results exhibit a phase transition behavior as the regularization parameter λ is varied.

Overall, Ξ-VI is a principled variational inference framework that can adaptively balance statistical accuracy and computational efficiency by leveraging the connection to entropic optimal transport.

Stats

The log-likelihood function ℓ(x; θ) is uniformly bounded over the parameter space Θ.
The gradient ∇ℓ(x; θ) and Hessian ∇²ℓ(x; θ) of the log-likelihood are uniformly bounded over Θ.
The prior π(θ) has a Lebesgue density that is twice continuously differentiable and bounded in a neighborhood of the true parameter θ₀.

Quotes

"Ξ-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm."
"We show that Ξ-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter."
"We also investigate the frequentist properties of Ξ-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability."

Key Insights Distilled From

Extending Mean-Field Variational Inference via Entropic Regularization: Theory and Computation

by Bohan Wu,Dav... at arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09113.pdf

Deeper Inquiries

What are the potential applications of Ξ-VI beyond Bayesian modeling, such as in reinforcement learning or generative modeling

Ξ-Variational Inference (Ξ-VI) has the potential for various applications beyond Bayesian modeling. One such application is in reinforcement learning, where approximate inference methods are crucial for handling complex decision-making processes. In reinforcement learning, Ξ-VI can be used to approximate the posterior distribution over the latent variables, enabling more efficient and accurate decision-making under uncertainty. By incorporating Ξ-VI into reinforcement learning algorithms, we can improve the exploration-exploitation trade-off, enhance policy optimization, and enable more robust and adaptive learning in dynamic environments.
Another potential application of Ξ-VI is in generative modeling, particularly in the context of deep generative models such as variational autoencoders (VAEs) and generative adversarial networks (GANs). By leveraging the flexibility and accuracy of Ξ-VI, we can improve the training and inference processes in generative models. Ξ-VI can help in capturing complex dependencies in the data distribution, leading to better sample generation, improved model generalization, and enhanced model interpretability. Additionally, Ξ-VI can aid in addressing challenges such as mode collapse and posterior collapse in generative modeling, thereby advancing the state-of-the-art in this field.

How can the theoretical analysis of Ξ-VI be extended to handle non-compact parameter spaces or non-smooth likelihoods

Extending the theoretical analysis of Ξ-VI to handle non-compact parameter spaces or non-smooth likelihoods is a challenging but rewarding endeavor. To address non-compact parameter spaces, one approach could involve adapting the optimization framework of Ξ-VI to incorporate constraints that ensure the variational distributions remain within a feasible region. This could involve introducing regularization terms or penalty functions that penalize variational distributions that stray too far from the feasible parameter space. Additionally, techniques from convex optimization and constrained optimization could be employed to handle non-compact spaces effectively.
For non-smooth likelihoods, the theoretical analysis of Ξ-VI could be extended by considering subgradients or generalized derivatives of the likelihood function. By incorporating subgradients into the optimization framework of Ξ-VI, it becomes possible to handle non-smooth likelihoods and still derive meaningful theoretical results. Techniques from nonsmooth analysis and variational calculus can be leveraged to develop convergence guarantees and asymptotic properties for Ξ-VI in the presence of non-smooth likelihood functions.

Can the connection between Ξ-VI and entropic optimal transport be further exploited to develop new variational inference algorithms or to provide tighter bounds on the approximation error

The connection between Ξ-VI and entropic optimal transport offers exciting opportunities for the development of new variational inference algorithms and the derivation of tighter bounds on the approximation error. One potential direction is to explore the use of optimal transport metrics as divergence measures in variational inference. By incorporating entropic optimal transport principles into the variational framework, it may be possible to design variational objectives that better capture the underlying structure of the data distribution and improve the quality of the approximations.
Furthermore, the entropic optimal transport framework can be utilized to develop novel variational inference algorithms that leverage the efficient computation of optimal transport plans. By integrating the Sinkhorn algorithm or other optimal transport algorithms into the variational optimization process, it may be possible to enhance the scalability and convergence properties of variational inference methods. This could lead to faster and more accurate approximations of complex posterior distributions in high-dimensional settings.
Additionally, the connection to entropic optimal transport can provide insights into the geometry of the variational approximation space and enable the derivation of tighter bounds on the approximation error. By studying the geometric properties of the variational distributions in the context of optimal transport, it may be possible to quantify the quality of the approximation and establish more rigorous guarantees on the accuracy of the variational inference results. This could lead to a deeper understanding of the trade-offs between computational efficiency and statistical accuracy in variational inference.