insight - Algorithms and Data Structures - # Convergence of Coordinate Ascent Variational Inference for Log-Concave Measures

Core Concepts

The Coordinate Ascent Variational Inference (CAVI) algorithm converges to the minimizer of the mean field variational inference problem when the target probability measure has a log-concave density. The convergence rate is linear if the log-density has a Lipschitz gradient, and exponential if it is also strongly convex.

Abstract

The paper analyzes the convergence of the Coordinate Ascent Variational Inference (CAVI) algorithm for approximating a high-dimensional probability measure ρ with a product (factorized) measure.
Key highlights:
CAVI iteratively optimizes over one coordinate (factor) at a time, which can be done explicitly.
The authors prove convergence of CAVI for log-concave densities ρ.
If log ρ has a Lipschitz gradient, a linear rate of convergence is obtained.
If log ρ is also strongly log-concave, an exponential rate of convergence is obtained.
The analysis views MFVI as a geodesically convex optimization problem in Wasserstein space when ρ is log-concave.
CAVI is analyzed as an equivalent of the Block Coordinate Descent algorithm in Wasserstein space.
The authors leverage tools from optimal transport and convex optimization to derive the convergence results.
The assumptions of log-concavity and Lipschitz/strong convexity of log ρ are shown to cover important applications like Bayesian linear regression.

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Key Insights Distilled From

by Manuel Arnes... at **arxiv.org** 04-16-2024

Deeper Inquiries

The analysis presented in the context can be extended to more general classes of target measures beyond log-concave densities by considering different convexity assumptions and integrability conditions. For example, instead of assuming strict convexity or strong convexity of the target measure, one could explore the convergence properties under weaker convexity assumptions or even non-convex settings. By relaxing the convexity requirements, the analysis could be extended to a broader range of probability measures, allowing for a more comprehensive understanding of the convergence behavior of the Coordinate Ascent Variational Inference (CAVI) algorithm.

The implications of the Wasserstein geometry perspective for the design and analysis of other variational inference algorithms are significant. By viewing the Mean Field Variational Inference (MFVI) problem as a geodesically convex optimization problem on Wasserstein space, one can leverage the rich geometric structure of Wasserstein space to develop and analyze new variational inference algorithms. This perspective allows for the application of tools and techniques from optimal transport theory and convex optimization to improve the efficiency and convergence properties of variational inference algorithms. Additionally, the Wasserstein geometry perspective provides insights into the geometry of the space of probability measures, leading to a deeper understanding of the optimization landscape and convergence behavior of variational inference methods.

There are connections between the convergence rates obtained in the analysis of the Coordinate Ascent Variational Inference (CAVI) algorithm and the statistical accuracy of the CAVI approximation to the target measure. The convergence rates provide insights into how quickly the CAVI algorithm can approximate the true posterior distribution or target measure. Faster convergence rates imply that the algorithm reaches a more accurate approximation in fewer iterations, leading to a more precise estimation of the target measure. On the other hand, slower convergence rates may indicate that the algorithm requires more iterations to achieve a comparable level of accuracy. Understanding the relationship between convergence rates and statistical accuracy is crucial for assessing the performance and reliability of variational inference algorithms in practical applications.

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