# Universal Bounds for Self-Normalized Kullback-Leibler Estimation

arXiv: https://arxiv.org/abs/2610.12851 · HTML: https://mlsanity.com/paper/2610.12851/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 26 references that are in the map.
- First posted on arXiv 9 Oct 2026.

## Builds on (most cited first, 14 of 14)

- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Theoretical guarantees on the best-of-n alignment policy](https://mlsanity.com/paper/2401.01879/) (2024)
- [Is Best-of-N the Best of Them? Coverage, Scaling, and Optimality in Inference-Time Alignment](https://mlsanity.com/paper/2503.21878/) (2025)
- [Multi-Agent Verification: Scaling Test-Time Compute with Multiple Verifiers](https://mlsanity.com/paper/2502.20379/) (2025)
- [Reinterpreting Importance-Weighted Autoencoders](https://mlsanity.com/paper/1704.02916/) (2017)
- [Algorithms for the Communication of Samples](https://mlsanity.com/paper/2110.12805/) (2021)
- [Soft Best-of-n Sampling for Model Alignment](https://mlsanity.com/paper/2505.03156/) (2025)
- [Information Theoretic Guarantees For Policy Alignment In Large Language Models](https://mlsanity.com/paper/2406.05883/) (2024)
- [Best-of-N through the Smoothing Lens: KL Divergence and Regret Analysis](https://mlsanity.com/paper/2507.05913/) (2025)
- [Guided Speculative Inference for Efficient Test-Time Alignment of LLMs](https://mlsanity.com/paper/2506.04118/) (2025)
- [Improving Mutual Information Estimation with Annealed and Energy-Based Bounds](https://mlsanity.com/paper/2303.06992/) (2023)
- [Best-of-Majority: Minimax-Optimal Strategy for Pass@k Inference Scaling](https://mlsanity.com/paper/2510.03199/) (2025)
- [On importance sampling and independent Metropolis-Hastings with an unbounded weight function](https://mlsanity.com/paper/2411.09514/) (2024)
- [Requential Coding: Pushing the Limits of Model Compression with Self-Generated Training Data](https://mlsanity.com/paper/2607.11883/) (2026)

## Abstract (arXiv)

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.