Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shiji Xiao, Yufeng Zhang, Chubo Liu, Yan Ding, Keqin Li, Kenli Li

TL;DR

This paper establishes a tight, dimension-free supremum for the Kullback-Leibler divergence between Gaussian distributions under fixed intermediate divergences: KL$(\mathcal{N}_1 \| \mathcal{N}_2)=\Delta_1$ and KL$(\mathcal{N}_2 \| \mathcal{N}_3)=\Delta_2$. The main result shows KL$(\mathcal{N}_1 \| \mathcal{N}_3) \le \tfrac{1}{2}[ w_2(2\Delta_1)-1 ][ w_2(2\Delta_2)-1 ] + \Delta_1 + \Delta_2$, where $w_2(t)$ is defined via Lambert W function, and equality characterizes exact Gaussian parameters achieving the bound. For small divergences, the supremum simplifies to $\Delta_1 + \Delta_2 + 2\sqrt{\Delta_1\Delta_2} + o(\Delta_1) + o(\Delta_2)$, improving prior looser bounds. The paper also provides explicit necessary and sufficient conditions for attainment, and validates the theory with numerical experiments. Finally, it discusses practical applications in out-of-distribution detection with flow-based models and in safe reinforcement learning, where tighter KL triangle-inequality bounds strengthen theoretical guarantees and guiding principles.

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 $KL(\mathcal{N}_1, \mathcal{N}_2)\leq ε_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq ε_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3ε_1+3ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. 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 $ε_1$ and $ε_2$ are small, the supremum is $ε_1+ε_2+\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.

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

TL;DR

This paper establishes a tight, dimension-free supremum for the Kullback-Leibler divergence between Gaussian distributions under fixed intermediate divergences: KL and KL. The main result shows KL, where is defined via Lambert W function, and equality characterizes exact Gaussian parameters achieving the bound. For small divergences, the supremum simplifies to , improving prior looser bounds. The paper also provides explicit necessary and sufficient conditions for attainment, and validates the theory with numerical experiments. Finally, it discusses practical applications in out-of-distribution detection with flow-based models and in safe reinforcement learning, where tighter KL triangle-inequality bounds strengthen theoretical guarantees and guiding principles.

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 , and , if and , then . However, the supremum of 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 as well as the conditions when the supremum can be attained. When and are small, the supremum is . Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.
Paper Structure (30 sections, 14 theorems, 155 equations, 5 figures, 2 tables)

This paper contains 30 sections, 14 theorems, 155 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notations
  4. Dimension-free Supremum
  5. Core Lemma \ref{['lem_KLSupremumForNormal']}
  6. Core Dimension-free Supremum
  7. Experiments
  8. The Properties of Supremum
  9. The Condition for Supremum
  10. The Experiments on Lemma \ref{['lem_MaxH']}
  11. Discussions and Applications
  12. Comparison with Existing Results
  13. Applications
  14. Application in Out-of-distribution Detection with Flow-based Models
  15. Application in Safe Reinforcement Learning
  16. ...and 15 more sections

Key Result

Lemma 3.2

zhang2023PropertiesKullbackleiblerDivergence For $t \geq 0$, the equation $f(x) = 1 + t$ has two solutions. The smaller solution is given by and the larger solution is given by where $W_{0}$ and $W_{-1}$ denote the principal and $-1$ branches of the Lambert $W$ function, respectively.

Figures (5)

  • Figure 1: Heatmap and surface of the supremum of $\mathop{\mathrm{KL}}\nolimits\left(\mathcal{N}_1 \,\|\, \mathcal{N}_3 \right)$, which is equal to $\frac{1}{2}\left[ w_2(2\Delta_1) - 1 \right] \left[ w_2(2\Delta_2) - 1 \right] + \Delta_1 + \Delta_2$.
  • Figure 2: Probability density functions of $\mathcal{N}_1({\mu}_1, {\sigma}_1^{2})$ and $\mathcal{N}_2({\mu}_2, {\sigma}_2^{2})$ for varying means $\mu_1$ and $\mu_2$, respectively, where $\mathcal{N}_1$ satisfies ${\sigma}_1^{2} = w_2\left( 2 \Delta_1 - {\mu}_1^{2} \right)$ and $\mathcal{N}_2$ satisfies $\dfrac{1}{{\sigma}_2^{2}} = \dfrac{ w_2\left[ 2 \Delta_2 - \log \left( {1 + {\mu}_2^{2}} \right) \right] }{1 + {\mu}_2^{2}}$, with $\Delta_1 = \Delta_2 = 0.1$.
  • Figure 3: Heatmap of $\mathop{\mathrm{KL}}\nolimits(\mathcal{N}_1 \| \mathcal{N}_2)$ with respect to $({\mu}_1, {\mu}_2)$, where $\mathcal{N}_1$ satisfies ${\sigma}_1^{2} = w_2\left( 2 \Delta_1 - {\mu}_1^{2} \right)$ and $\mathcal{N}_2$ satisfies $\dfrac{1}{{\sigma}_2^{2}} = \dfrac{ w_2\left[ 2 \Delta_2 - \log \left( {1 + {\mu}_2^{2}} \right) \right] }{1 + {\mu}_2^{2}}$, with $\Delta_1 = \Delta_2 = 0.1$.
  • Figure 4: Heatmaps and surfaces of the two-dimensional Gaussian probability density functions of $\mathcal{N}_1$ and $\mathcal{N}_2$ when $\mathop{\mathrm{KL}}\nolimits\left( \mathcal{N}_1 \, \| \, \mathcal{N}_2 \right)$ attains its supremum and $\bm{Q} = \bm{I}$. $\mathcal{N}_1$ and $\mathcal{N}_2$ satisfy the constraints $\mathop{\mathrm{KL}}\nolimits\left(\mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1) \, \| \, \mathcal{N}(\bm{0}, \bm{I}) \right) = \Delta_1$ and $\mathop{\mathrm{KL}}\nolimits\left( \mathcal{N}(\bm{0}, \bm{I}) \,\|\, \mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2) \right) = \Delta_2$, respectively, with $\Delta_1 = \Delta_2 = 0.1$.
  • Figure 5: Heatmaps of $\bar{H}(\bar{x}, \bar{y}; \Delta_1, \Delta_2)$ over $(\bar{x}, \bar{y}) \in [0,2]^2$ for various parameter pairs $(\Delta_1, \Delta_2)$. On each subplot, the blue curve $(\bar{x}, \bar{y}^*{\left( \bar{x} \right)})$ and the red curve $(\bar{x}^*{(\bar{y})}, \bar{y})$ trace the maximizers of $\bar{H}$ along vertical lines $\bar{x} = \text{const}$ and horizontal lines $\bar{y} = \text{const}$, respectively, satisfying $\frac{\partial \bar{H}(\bar{x}, \bar{y}; \Delta_1, \Delta_2)}{\partial \bar{y}} = 0$ and $\frac{\partial \bar{H}(\bar{x}, \bar{y}; \Delta_1, \Delta_2)}{\partial \bar{x}} = 0$, respectively. The red point in the upper-right corner of each subplot denotes the global maximizer $(\bar{x}^*, \bar{y}^*)$.

Theorems & Definitions (17)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Theorem 4.2
  • Remark 4.3
  • Theorem 4.4
  • Lemma A.1
  • Lemma A.2
  • Lemma B.1
  • Lemma B.2
  • ...and 7 more