Characterize universal learning rates for agnostic squared-loss regression

Characterize which hypothesis spaces are learnable at which universal rates for agnostic regression with squared loss, beyond the countably infinite settings analyzed in the paper.

Background

The paper studies universal learning for agnostic regression under squared loss but does not provide a general characterization of the universal rates attainable by different hypothesis spaces. Its results establish several structural facts for finite and countably infinite dictionaries, including arbitrarily slow rates and nearly exponential lower bounds, but leave the broader rate-classification problem unresolved.

References

Several questions remain open and suggest directions for future work. For one, we do not characterize which hypothesis spaces are learnable at which universal rates for squared loss in the agnostic setting (beyond \cref{sec:countably-infinite-hyp-spaces}).

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

Our focus on exponential rates is motivated by the finite case, in which exponential rates are always possible but the problem is already nontrivial. It remains open to investigate the cost of general universal rates for uniform guarantees.

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