Universal finite-sample guarantees for SNIS beyond the studied estimator

Establish additional universal, distribution-free finite-sample guarantees for self-normalized importance sampling, including guarantees for functionals of normalized importance weights other than the entropy-based Kullback–Leibler estimator, and determine whether the additive slack in the existing bounds can be further tightened.

Background

The paper establishes a universal finite-sample upper bound for the expectation of an entropy-based estimator of the forward Kullback–Leibler divergence constructed from normalized importance weights. Specifically, the expectation exceeds the target–proposal divergence by less than an additive log 2, without requiring bounded importance weights or finite higher-order moments.

The authors identify a broader unresolved direction: extending universal, distribution-free finite-sample analysis to other functionals of normalized importance weights and improving the additive constant in their current bound. Their experiments suggest that the log 2 slack may be conservative, but do not establish a sharper optimal constant.

References

Several directions remain for future work. On the theoretical side, our results raise the broader question of what other universal, distribution-free finite-sample guarantees can be established for SNIS, including for other functionals of the normalized importance weights, and whether the additive slack in our bounds can be further tightened.

— Universal Bounds for Self-Normalized Kullback-Leibler Estimation  (2610.12851 - Phan et al., 9 Oct 2026) in Conclusion, final section

How $K$ interacts with $N$ is not immediately clear.

— Universal Bounds for Self-Normalized Kullback-Leibler Estimation  (2610.12851 - Phan et al., 9 Oct 2026) in Section 2.1, subsection “Experiments,” paragraph “How many Monte Carlo samples are needed?”