Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

TL;DR

This paper explores the relaxed triangle inequality for KL divergence between multivariate Gaussian distributions, providing the supremum bounds and conditions for attainment, with applications in out-of-distribution detection and reinforcement learning.

Contribution

It derives the supremum of KL divergence between Gaussian distributions under relaxed triangle inequality constraints and identifies conditions for achieving this supremum.

Findings

Supremum of KL divergence is $oxed{ ext{epsilon}_1+ ext{epsilon}_2+2 oot{ ext{epsilon}_1 ext{epsilon}_2}$ for small epsilon values.

Provides conditions under which the supremum is attained.

Demonstrates applications in out-of-distribution detection and safe reinforcement learning.

Abstract

The Kullback-Leibler (KL) divergence is not a proper distance metric and does not satisfy the triangle inequality, posing theoretical challenges in certain practical applications. Existing work has demonstrated that KL divergence between multivariate Gaussian distributions follows a relaxed triangle inequality. Given any three multivariate Gaussian distributions $\mathcal{N}_1, \mathcal{N}_2$, and $\mathcal{N}_3$, if $KL(\mathcal{N}_1, \mathcal{N}_2)\leq \epsilon_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq \epsilon_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3\epsilon_1+3\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. However, the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ is still unknown. In this paper, we investigate the relaxed triangle inequality for the KL divergence between multivariate Gaussian distributions and give the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ as well as the conditions when the supremum can be attained. When $\epsilon_1$ and $\epsilon_2$ are small, the supremum is $\epsilon_1+\epsilon_2+2\sqrt{\epsilon_1\epsilon_2}+o(\epsilon_1)+o(\epsilon_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.