Ordering of Divergences for Variational Inference with Factorized Gaussian Approximations

Основные понятия

Different divergences in variational inference yield varying estimates of uncertainty, leading to an ordering based on marginal variances.

Аннотация

The content delves into the analysis of different divergences in variational inference, focusing on factorized Gaussian approximations. It explores the impact of divergence choice on the estimation of uncertainty measures like variance, precision, and entropy. The structure is divided into sections discussing KL divergence, R´enyi divergence, score-based divergence, and an impossibility theorem for factorized VI. The implications of these analyses are then used to establish an ordering of divergences based on their performance in estimating uncertainty measures.
1. Introduction to Variational Inference

Problem: Approximating intractable distribution p by tractable q within family Q.
Importance of Variational Inference (VI) in Bayesian statistics and machine learning.
2. Divergence Definitions

Explanation of Kullback-Leibler (KL), R´enyi, and score-based divergences.
Comparison between reverse and forward KL divergences.
3. Analysis of Divergence Minimization

Detailed examination of minimizing KL and R´enyi divergences for factorized Gaussian VI.
Formulation and optimization using NQP for score-based divergences.
4. Ordering of Divergences

Definition: Dominance based on marginal variances estimation.
Propositions establishing relationships between precisions, entropies, and dominance.
Theorem presenting a comprehensive ordering of divergences for variational inference.

Статистика

"The α-divergence has also been studied in the context of VI (Li and Turner, 2016)."
"It has been widely observed that VI based on the reverse KL divergence tends to underestimate uncertainty."
"Proposition 3.1 (Precision matching). If q in eq. (4) has a diagonal covariance but p in eq. (3) does not..."

Цитаты

"The goal is to discover the best-matching distribution q within some larger parameterized family Q."
"VI often achieves a smaller error than Markov chain Monte Carlo methods with finite computational budgets."

Ключевые выводы из

An Ordering of Divergences for Variational Inference with Factorized Gaussian Approximations

by Charles C. M... в arxiv.org 03-21-2024

https://arxiv.org/pdf/2403.13748.pdf

Дополнительные вопросы

How does the choice of divergence impact the efficiency and accuracy of variational inference

The choice of divergence plays a crucial role in determining the efficiency and accuracy of variational inference. Different divergences lead to different solutions when approximating a target distribution with a more tractable one. The impact on efficiency stems from how well the chosen divergence aligns with the inferential goals of the specific application. For instance, if the goal is to estimate certain moments or measures of uncertainty accurately, choosing a divergence that prioritizes matching those specific aspects can lead to more efficient inference. On the other hand, inaccuracies in estimating these quantities due to mismatched divergences can result in inefficient and less reliable approximations.
In terms of accuracy, the choice of divergence directly influences how well the variational approximation captures key characteristics of the target distribution. A suitable divergence should guide FG-VI towards an approximation that closely resembles important features such as variance, precision, and entropy. When using an appropriate divergence that aligns with these critical aspects of uncertainty estimation, variational inference can be more accurate by providing better approximations that reflect essential properties of the true distribution.

What are potential drawbacks or limitations when using factorized Gaussian approximations

While factorized Gaussian approximations offer computational convenience and simplicity in variational inference settings like FG-VI, they come with potential drawbacks and limitations. One significant limitation is related to their inability to capture correlations between variables adequately. Factorizing distributions into independent components may oversimplify complex relationships present in real-world data or target distributions.
Another drawback is associated with variational collapse when using factorized Gaussian approximations based on certain divergences like score-based divergences where estimates for marginal variances might become zero or infinite under specific conditions leading to ill-defined solutions not belonging within Q -the family of proper distributions- which could hinder accurate modeling outcomes.
Additionally, factorized Gaussian approximations may struggle when dealing with high-dimensional data or complex models where capturing intricate dependencies among variables is crucial for accurate representation. In such cases, relying solely on diagonal covariance matrices may limit model flexibility and compromise overall performance.

How can insights from this analysis be applied to improve other probabilistic modeling techniques

Insights gained from analyzing different divergences for FG-VI can be applied beyond this specific context to improve various probabilistic modeling techniques:

Divergence Selection: Understanding how different divergences impact variational inference outcomes can help researchers choose appropriate metrics tailored to their specific inferential goals across diverse applications ranging from Bayesian statistics to machine learning tasks.

Model Flexibility: Recognizing limitations associated with factorized Gaussian approximations highlights opportunities for exploring alternative approaches that allow for capturing richer dependencies among variables while maintaining computational tractability.

Algorithmic Development: Leveraging insights from this analysis could inspire advancements in optimization algorithms tailored for handling non-Gaussian distributions efficiently within VI frameworks by considering multiple valid choices beyond KL divergence.

4Performance Evaluation: Applying similar analyses across various probabilistic models enables researchers to compare different methods' effectiveness based on their ability to estimate uncertainties accurately through appropriate selection and ordering criteria derived from understanding how each method performs under varying conditions.
These applications demonstrate how insights derived from studying divergences in FG-VI extend beyond this particular scenario's scope into broader probabilistic modeling contexts enhancing model performance and advancing research methodologies effectively..