Characterize hypothesis-space and distribution-family trade-offs

Characterize exactly which pairs consisting of a hypothesis space and a family of distributions exhibit a trade-off between universal exponential rates and uniform learning guarantees, and determine whether such examples exist outside the class constructed in Theorem stated in the paper.

Background

For countably infinite dictionaries, the paper constructs families of distributions for which universal exponential learning and uniform learning are separately possible but cannot be achieved simultaneously by one algorithm. The authors do not establish whether their construction captures all such trade-off instances or whether fundamentally different examples exist.

References

We also do not provide an exact characterization of which pairs consisting of a hypothesis space and a family of distributions exhibit a trade-off between universal exponential rates and uniform rates. \cref{thm:bob-general} already provides a class of such instances, but whether there are examples outside this class remains open.

— Reconciling Universal and Uniform Learning with $Q$-Aggregation  (2609.05041 - Høgsgaard et al., 4 Sep 2026) in Conclusion

And finally, similar investigations into the trade-off between universal and uniform rates beyond regression with squared loss, for example for classification and other loss functions, could generalize this work.

— Reconciling Universal and Uniform Learning with $Q$-Aggregation  (2609.05041 - Høgsgaard et al., 4 Sep 2026) in Conclusion