Relaxing boundedness assumptions in ensembling stability guarantees

Establish ensembling-based stability guarantees without requiring the base algorithm to produce outputs in the unit interval or, for Hilbert-space-valued outputs, in a bounded subset of a Hilbert space, including settings with data-dependent bounds.

Background

The paper’s main stability theorem assumes that the base algorithm takes values in [0,1]. Its Hilbert-space extension similarly requires the algorithm’s outputs to lie in a bounded subset of a Hilbert space. These conditions make the operator-norm framework directly applicable but may be restrictive for applications involving unbounded or data-dependent outputs.

The authors explicitly identify relaxing these output-boundedness assumptions, or permitting bounds that depend on the data, as an open question and an important direction for future work. Such an extension would broaden the framework beyond uniformly bounded algorithms and generalize the stability guarantees currently established under fixed boundedness conditions.

References

These results naturally lead to some important open questions. First, our theoretical guarantees rely on a boundedness assumption: we assume in Theorem~\ref{thm:main} that $A$ returns outputs lying in $[0,1]$ (or, in the extension to Hilbert space-valued output in Appendix~\ref{sec:HSExtension}, that the output lies in a bounded subset of a Hilbert space). While this is reasonable for certain applications, relaxing these conditions or allowing for data-dependent bounds is an important direction for future work (see \citet{soloff2024bagging} for some extensions of this type in the specific setting of bagging for stability with respect to data deletion).

— Algorithmic stability via ensembling  (2609.10428 - Barber et al., 9 Sep 2026) in Section Discussion