Optimal Stability of KL Divergence under Gaussian Perturbations

2026-04-16 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

This paper establishes a generalized relaxed triangle inequality for Kullback-Leibler (KL) divergence, extending its applicability beyond purely Gaussian distributions. Previous work on KL divergence stability critically relied on the assumption that all involved distributions were Gaussian, limiting its use in modern applications like out-of-distribution (OOD) detection with flow-based generative models. The new theorem demonstrates that if an arbitrary distribution P with a finite second moment is far from a Gaussian distribution $\mathcal{N}_{1}$ (i.e., $KL(P||\mathcal{N}_{1})>C$), and $\mathcal{N}_{1}$ is close to another Gaussian $\mathcal{N}_{2}$ (i.e., $KL(\mathcal{N}_{1}||\mathcal{N}_{2})\leq\epsilon$), then P remains far from $\mathcal{N}_{2}$, specifically $KL(P||\mathcal{N}_{2})\geq C-O(\sqrt{\epsilon})$. The authors prove this $\sqrt{\epsilon}$ rate is optimal, even within Gaussian families, revealing an intrinsic stability property. This theoretical advancement provides a rigorous foundation for KL-based OOD analysis in flow-based models and enables KL-based reasoning in non-Gaussian deep learning and reinforcement learning contexts.

Key takeaway

Research Scientists working on out-of-distribution detection or reinforcement learning with KL regularization should recognize that this generalized KL stability bound removes restrictive Gaussian assumptions. You can now rigorously justify KL-based OOD detection in flow-based models even when latent representations are non-Gaussian, and ensure robust policy separation in RL, preventing accidental drift into unsafe regions. This expands the theoretical foundation for applying KL divergence in complex, non-Gaussian deep learning scenarios.

Key insights

KL divergence stability under Gaussian perturbations extends to arbitrary distributions with optimal $\sqrt{\epsilon}$ degradation.

Principles

• KL divergence exhibits intrinsic $\sqrt{\epsilon}$ stability under Gaussian perturbations.
• Flow-based models preserve KL divergence between data and latent spaces.

Method

The method involves decomposing KL terms, bounding components using matrix norms and eigenvalue properties, and applying Taylor expansions to derive the $O(\sqrt{\epsilon})$ bound.

In practice

• Justifies KL-based OOD detection in flow-based models without Gaussian assumptions.
• Guarantees robust policy separation in reinforcement learning under KL regularization.

Topics

• KL Divergence Stability
• Gaussian Perturbations
• Relaxed Triangle Inequality
• Out-of-Distribution Detection
• Flow-Based Generative Models

Best for: Research Scientist, AI Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.