EigenVI: score-based variational inference with orthogonal function expansions

TL;DR

EigenVI introduces an eigenvalue-based variational inference method using orthogonal function expansions, enabling flexible, non-Gaussian approximations that are computationally efficient and outperform existing Gaussian BBVI methods.

Contribution

EigenVI presents a novel eigenvalue-based approach for variational inference that avoids gradient-based optimization by solving a minimum eigenvalue problem, enhancing accuracy and stability.

Findings

EigenVI accurately approximates complex target distributions.

It outperforms existing Gaussian BBVI methods on benchmark models.

The method efficiently models non-Gaussian, multimodal, and asymmetric distributions.

Abstract

We develop EigenVI, an eigenvalue-based approach for black-box variational inference (BBVI). EigenVI constructs its variational approximations from orthogonal function expansions. For distributions over $\mathbb{R}^D$, the lowest order term in these expansions provides a Gaussian variational approximation, while higher-order terms provide a systematic way to model non-Gaussianity. These approximations are flexible enough to model complex distributions (multimodal, asymmetric), but they are simple enough that one can calculate their low-order moments and draw samples from them. EigenVI can also model other types of random variables (e.g., nonnegative, bounded) by constructing variational approximations from different families of orthogonal functions. Within these families, EigenVI computes the variational approximation that best matches the score function of the target distribution by minimizing a stochastic estimate of the Fisher divergence. Notably, this optimization reduces to solving a minimum eigenvalue problem, so that EigenVI effectively sidesteps the iterative gradient-based optimizations that are required for many other BBVI algorithms. (Gradient-based methods can be sensitive to learning rates, termination criteria, and other tunable hyperparameters.) We use EigenVI to approximate a variety of target distributions, including a benchmark suite of Bayesian models from posteriordb. On these distributions, we find that EigenVI is more accurate than existing methods for Gaussian BBVI.