Universal Bounds for Self-Normalized Kullback-Leibler Estimation

Abstract: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. We then characterize the KL divergence between the distribution induced by importance-weighted resampling and the proposal, showing that it is upper bounded by both the expectation of the entropy-based estimator and the target-proposal KL divergence. Importantly, these bounds require neither bounded importance weights nor finite higher moments. We apply these results to test-time alignment of generative models via Soft Best-of-$N$ sampling, an importance-sampling-based scaling technique proposed by Verdun et al. (2025), and to relative entropy coding, obtaining finite-sample policy-divergence certificates and moment-free bounds on coding cost.

Machine Learning (stat.ML); Machine Learning (cs.LG)

arXiv:2610.12851 [stat.ML]

(or arXiv:2610.12851v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2610.12851 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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