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arXiv Machine Learning By Buu Phan, Lester Mackey, David Alvarez-Melis

Universal Bounds for Self-Normalized Kullback-Leibler Estimation

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The paper introduces an entropy-based estimator for Kullback-Leibler divergence that uses normalized importance weights. It establishes a universal finite-sample bound, proving that the estimator’s expectation can exceed the true target‑proposal KL divergence by no more than λ2, regardless of sample size. Additionally, it shows that the KL divergence between the distribution produced by importance‑weighted resampling and the proposal is bounded by both the estimator’s expectation and the target‑proposal KL divergence, without requiring bounded weights or finite higher moments. The authors apply these theoretical results to test‑time alignment of generative models via Soft Best‑of‑N sampling and to relative entropy coding, yielding finite‑sample policy‑divergence guarantees and moment‑free bounds on coding cost.

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