Universal Bounds for Self-Normalized Kullback-Leibler Estimation
This work studies an entropy-based estimator of Kullback-Leibler (KL) divergence constructed from normalized importance weights. We prove a universal finite-sample bound showing that its expectation can exceed the target-proposal KL divergence by at most $\log 2$ for any sample size.
This paper presents theoretical advancements in estimating Kullback-Leibler (KL) divergence using self-normalized importance sampling techniques, with applications to generative model alignment and entropy coding.
What happened
Researchers developed an entropy-based KL divergence estimator using normalized importance weights, proving theoretical bounds that apply without restrictions on importance weight moments. The method was applied to test-time alignment of generative models and entropy coding applications.
Why it matters
The results provide rigorous guarantees for importance-sampling-based techniques critical in generative modeling and probabilistic inference, enabling more reliable model alignment and coding efficiency analysis without distributional assumptions.
- The log 2 bound applies universally to KL divergence estimation regardless of sample size
- The method eliminates requirements for bounded importance weights or finite higher moments
- Applications include generative model alignment and coding cost analysis
- Theoretical results support importance-sampling-based techniques in practical ML systems
The bigger picture
The findings enable more robust analysis of generative models and information coding systems, particularly in scenarios where traditional moment assumptions may not hold, while providing theoretical foundations for importance-sampling-based scaling techniques.
This work advances the theoretical understanding of KL divergence estimation through self-normalized importance sampling, offering practical benefits for generative modeling and information coding applications with minimal distributional assumptions.
Technical details
- Self-normalized importance sampling estimators
- Kullback-Leibler divergence bounds
- Soft Best-of-N sampling technique
- Relative entropy coding frameworks
- Moment-free theoretical guarantees
FAQ
What is the significance of the log 2 bound in this context?
The log 2 bound represents a universal guarantee that the estimator's expectation cannot exceed the true KL divergence by more than this fixed value, regardless of sample size or distributional properties.
How does this research improve upon existing KL divergence estimation methods?
It removes constraints on importance weight moments and provides finite-sample guarantees without requiring bounded weights, expanding applicability to a wider range of distributions.
What are the key applications of these theoretical results?
The results apply to generative model alignment via Soft Best-of-N sampling and relative entropy coding, enabling more reliable analysis of these techniques without distributional assumptions.
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Originally published by arXiv - stat.ML on Oct 12, 2026.
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