Convergence of BF-Soft under evolutionary resampling and mutation

Establish convergence guarantees for BF-Soft, the temperature-smoothed soft-voting evolutionary synthesizer, when operated with the ResampleMutate population-update dynamics under adaptive privacy composition.

Background

BF-Soft replaces the discontinuous one-hot nearest-neighbor vote used by Private Evolution and Tab-PE with a temperature-scaled softmax vote. The paper proves per-round sensitivity bounds conditional on a fixed candidate population, and it characterizes how temperature trades sensitivity reduction against the strength of the evolutionary selection signal.

However, the candidate population changes over time through ResampleMutate, and the resulting trajectory depends on mutation, resampling, privacy noise, and adaptive composition. The paper explicitly does not provide a convergence theorem for this full process, leaving the global behavior of BF-Soft unresolved.

References

Proposition~\ref{prop:temperature-tradeoff} is deliberately not a convergence theorem. It characterizes the information retained by a single vote at a fixed candidate population. Whether the full population sequence converges under ResampleMutate depends also on mutation, resampling, and composition. The convergence analysis of the original hard-vote PE has itself required subsequent revision. We therefore leave convergence under soft voting to future work; the sensitivity results above hold independently at every fixed round.

— Soft Voting for Policy-Aware Private Data Synthesis  (2610.11285 - Hu et al., 8 Oct 2026) in Section 4.4, Remark 2.14; Section 6, Conclusion and Future Work; Appendix E, Section 5