Minimizing Regularized R'enyi Divergence Using Bregman Proximal Gradient Algorithms: A Novel Approach to Variational Inference

Conceitos Básicos

This paper introduces a novel variational inference algorithm based on Bregman proximal gradient descent that minimizes the regularized R'enyi divergence between a target distribution and an approximating distribution from an exponential family, offering theoretical convergence guarantees and practical advantages over existing methods.

Resumo

• Bibliographic Information: Guilmeau, T., Chouzenoux, E., & Elvira, V. (2024). Regularized R'enyi divergence minimization through Bregman proximal gradient algorithms. arXiv preprint arXiv:2211.04776v5.

• Research Objective: This paper proposes a novel algorithm for variational inference (VI) that leverages the geometry of exponential families through Bregman proximal gradient descent to minimize a regularized R'enyi divergence, aiming to address limitations of existing VI methods in handling divergences beyond KL and providing strong convergence guarantees.

• Methodology: The authors develop a Bregman proximal gradient algorithm tailored for minimizing a regularized R'enyi divergence between a target distribution and an approximating distribution from an exponential family. They utilize the Bregman divergence induced by the KL divergence to exploit the geometry of the approximating family. A sampling-based stochastic implementation is also proposed to handle the black-box setting. The convergence analysis leverages existing and novel techniques for studying Bregman proximal gradient methods.

• Key Findings: The proposed algorithm is shown to be interpretable as a relaxed moment-matching algorithm with an additional proximal step. The authors establish strong convergence guarantees for both deterministic and stochastic versions of the algorithm, including monotonic decrease of the objective, convergence to a stationary point or the minimizer, and geometric convergence rates under certain conditions. Numerical experiments demonstrate the algorithm's efficiency, robustness, and advantages over Euclidean geometry-based methods, particularly for Gaussian approximations and sparse solution enforcement.

• Main Conclusions: The research introduces a versatile, robust, and competitive method for variational inference by combining the strengths of Bregman proximal gradient descent, R'enyi divergence minimization, and regularization within the framework of exponential families. The theoretical analysis provides strong convergence guarantees, and numerical experiments confirm the practical benefits of the proposed approach.

• Significance: This work contributes significantly to the field of variational inference by expanding the scope of tractable divergences beyond the commonly used KL divergence, providing a principled framework for incorporating regularization, and offering strong theoretical guarantees for both deterministic and stochastic implementations.

• Limitations and Future Research: The paper primarily focuses on exponential families as approximating distributions. Future research could explore extensions to broader distribution families. Additionally, investigating the impact of different Bregman divergences and regularization choices on the algorithm's performance could be of interest.

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Regularized R\'enyi divergence minimization through Bregman proximal gradient algorithms

by Thom... às arxiv.org 10-17-2024

https://arxiv.org/pdf/2211.04776.pdf

Perguntas Mais Profundas

How might this Bregman proximal gradient approach be extended to handle variational inference in deep generative models, where the approximating distributions are typically not in the exponential family?

Extending the Bregman proximal gradient approach to deep generative models, where the approximating distributions are often not in the exponential family, presents several challenges and opportunities:
Challenges:

Non-Tractable KL Divergence: The key advantage of using Bregman divergence based on KL divergence in the exponential family setting is its tractability. In deep generative models, the KL divergence between the true posterior and the variational approximation is usually intractable.
Non-Conjugate Models: Deep generative models rarely exhibit conjugacy, making it difficult to derive closed-form updates for the Bregman proximal step.
High Dimensionality: The parameter space of deep generative models is typically very high-dimensional, which can make optimization difficult.
Potential Extensions and Strategies:

Alternative Bregman Divergences: Instead of relying on the KL divergence, explore alternative Bregman divergences that are more suitable for the specific distributions used in deep generative models. For instance:

Stein Divergences: These divergences can be computed using only samples from the distributions and have been successfully used in variational inference [Ranganath et al., 2016].
Other Tractable Divergences: Identify divergences that admit tractable bounds or approximations within the context of the specific deep generative model.

Black-Box Variational Inference: Embrace black-box variational inference techniques that rely on stochastic gradient estimates of the objective function and its gradients. This allows for flexibility in the choice of divergence and approximating family.

Amortized Variational Inference: Utilize inference networks to learn a mapping from data points to the parameters of the variational distribution. This amortization can significantly improve efficiency in high-dimensional settings.

Hybrid Methods: Combine the strengths of Bregman proximal methods with other optimization techniques commonly used in deep learning, such as:

Stochastic Variance Reduction: Techniques like SVRG or SAGA can help accelerate convergence in the stochastic optimization setting.
Adaptive Learning Rates: Methods like Adam or RMSprop can dynamically adjust learning rates to improve optimization efficiency.

Exploiting Structure: If possible, exploit any specific structure present in the deep generative model or the approximating distribution to simplify computations or derive more efficient updates.

Overall, extending the Bregman proximal gradient approach to deep generative models requires careful consideration of the specific challenges posed by these models. Exploring alternative divergences, leveraging black-box techniques, and combining with other optimization methods are promising directions for future research.

Could the use of alternative divergences, such as the Wasserstein distance, within this Bregman proximal framework offer further advantages or insights for specific variational inference problems?

Yes, using alternative divergences like the Wasserstein distance within the Bregman proximal framework could offer significant advantages and insights for specific variational inference problems. Here's a breakdown:
Advantages of Wasserstein Distance:

Well-Defined for Non-Overlapping Supports: Unlike KL divergence, which becomes infinite when distributions have non-overlapping supports, the Wasserstein distance remains well-defined and provides meaningful gradients even in such cases. This is crucial when dealing with multimodal target distributions or when the variational approximation struggles to cover all modes initially.
Geometry Awareness: The Wasserstein distance captures the geometric structure of the underlying space, making it suitable for problems where preserving this structure is important. For example, in image generation, the Wasserstein distance can help generate more realistic and coherent images compared to KL-based methods.
Robustness to Noise: Wasserstein distance is known to be more robust to noise and outliers compared to KL divergence, leading to more stable and reliable variational inference.
Bregman Proximal Framework with Wasserstein Distance:
While directly using the Wasserstein distance as a Bregman divergence is not straightforward (it's not a Bregman divergence in the strict sense), there are ways to incorporate it into the framework:

Primal-Dual Methods: Formulate the variational inference problem as a primal-dual optimization problem, where the Wasserstein distance appears in the dual objective. This allows leveraging powerful primal-dual algorithms for optimization.

Entropic Regularization: Add an entropic regularization term to the Wasserstein distance, resulting in the Sinkhorn distance. This regularized distance is more computationally tractable and can still retain some of the desirable properties of the Wasserstein distance.

Approximations and Bounds: Utilize approximations or upper bounds of the Wasserstein distance that are easier to compute and differentiate. For instance, the sliced Wasserstein distance or adversarial learning-based approximations can be employed.

Specific Advantages and Insights:

Multimodal Target Distributions: For targets with multiple modes, Wasserstein-based methods can effectively explore different modes and avoid collapsing to a single mode, which is a common issue with KL-based methods.
Generative Models of Images and Text: In domains like image and text generation, where preserving the geometric structure of the data is crucial, Wasserstein-based variational inference can lead to more realistic and coherent samples.
Robust Variational Inference: When dealing with noisy data or model misspecification, the robustness of the Wasserstein distance can result in more reliable and stable inference compared to KL-based approaches.
In conclusion, incorporating the Wasserstein distance into the Bregman proximal framework holds great promise for specific variational inference problems. By leveraging its unique properties and employing suitable optimization techniques, we can achieve more accurate, robust, and geometrically meaningful inference in various challenging scenarios.

Given the connection between moment-matching and geometric averages as barycenters, what other geometric interpretations or generalizations could be explored for understanding and designing efficient variational inference algorithms?

The connection between moment-matching, geometric averages, and barycenters offers a rich geometric perspective on variational inference, opening doors to several intriguing interpretations and generalizations:
1. Exploring Other Barycenters:

Beyond KL Divergence: Moment-matching and geometric averages correspond to barycenters under the KL divergence. Investigate barycenters defined using other divergences, such as alpha-divergences, f-divergences, or optimal transport distances. These alternative barycenters might offer different trade-offs between computational complexity and approximation quality.
Weighted Barycenters: Introduce weights to the barycenter computation to emphasize certain regions or features of the target distribution. This could be particularly useful for focusing on regions of high probability mass or capturing specific characteristics of the target.
2. Geometric Insights for Algorithm Design:

Geodesic Flows: Interpret variational inference updates as moving along geodesics in the space of probability distributions equipped with a suitable metric. This perspective could lead to novel algorithms that efficiently traverse this space to reach the optimal approximation.
Projection and Retraction Operators: Formalize the moment-matching and geometric averaging steps as projection or retraction operations onto the manifold of approximating distributions. This geometric viewpoint can guide the design of more principled and efficient update rules.
3. Generalizations and Extensions:

Manifold-Valued Variational Inference: Extend variational inference to settings where the parameters of the approximating distribution lie on a Riemannian manifold. This framework can handle more complex and structured distributions beyond the exponential family.
Information Geometric Approaches: Leverage tools from information geometry, such as natural gradients and Fisher information metrics, to design algorithms that exploit the intrinsic geometry of the statistical model and the approximating family.
4. Connections to Optimal Transport:

Wasserstein Barycenters: Explore the use of Wasserstein barycenters, which offer a powerful way to define averages of distributions while taking into account the underlying geometry of the data space.
Optimal Transport Plans: Instead of directly minimizing a divergence, consider optimizing over the space of transport plans between the target and approximating distributions. This approach can provide a more flexible and expressive way to capture complex relationships between distributions.
By delving deeper into these geometric interpretations and generalizations, we can gain a more profound understanding of variational inference algorithms. This understanding can pave the way for developing novel and efficient methods that effectively tackle the challenges posed by complex and high-dimensional inference problems.