给出高斯分布KL散度的松弛三角不等式上界，可提升生成模型与强化学习安全性。
Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions
- 推导出多变量高斯分布KL散度的紧致松弛三角不等式上界
- 小误差下上界为ε₁+ε₂+2√(ε₁ε₂)+o(ε₁)+o(ε₂)
- 适用于流模型异常检测与安全强化学习场景
KL散度不是正规距离度量，不满足三角不等式，在某些实际应用中带来理论挑战。已有研究证明，多变量高斯分布间的KL散度满足松弛三角不等式：对任意三个多变量高斯分布 𝒩₁, 𝒩₂, 𝒩₃，若 KL(𝒩₁, 𝒩₂)≤ε₁ 且 KL(𝒩₂, 𝒩₃)≤ε₂，则 KL(𝒩₁, 𝒩₃)<3ε₁+3ε₂+2√(ε₁ε₂)+o(ε₁)+o(ε₂)。然而，KL(𝒩₁, 𝒩₃)的上确界仍未知。本文研究多变量高斯分布间KL散度的松弛三角不等式，给出其上确界及达到条件。当ε₁和ε₂较小时，上确界为ε₁+ε₂+2√(ε₁ε₂)+o(ε₁)+o(ε₂)。最后，我们展示了结果在基于流的生成模型异常检测与安全强化学习中的应用。
原文摘要 · Abstract (English)
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 $ε_1+ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.
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