Simultaneous adaptivity of single-scale smoothed discrepancy stopping rules

Establish whether stopping times of the form defined by the single-smoothing-parameter smoothed discrepancy principle can be simultaneously adaptive over a wide range of target regularities, including varying source-condition smoothness and effective-dimension parameters.

Background

The paper reviews a smoothed discrepancy principle that premultiplies the residual by a matrix indexed by a smoothing parameter T and stops when the resulting residual falls below a variance threshold. The authors explain that the stopping rule is sensitive to the choice of T: choices that are too large or too small can cause premature stopping and suboptimal rates.

The proposed multi-scale rule is introduced specifically to address this unresolved limitation by imposing the residual criterion simultaneously over a logarithmic collection of smoothing parameters. The paper subsequently proves adaptivity for its own multi-scale random-feature procedure, but does not establish the corresponding result for the earlier single-scale stopping rules.

References

To the best of our knowledge, it has not been shown for any stopping time of the form~martstop that it can be simultaneously adaptive over a wide range of target regularities.

— A Smoothed Discrepancy Principle for Random Feature Methods and Neural Networks  (2609.21017 - Nguyen et al., 17 Sep 2026) in Section 2, subsection “Stopping Time Motivation,” immediately after equation (martstop)

We conjecture that our stopping rule can be adapted to achieve optimal rates for NOs within the operator-valued NTK regime.

— A Smoothed Discrepancy Principle for Random Feature Methods and Neural Networks  (2609.21017 - Nguyen et al., 17 Sep 2026) in Conclusion, subsection “Architectural Extensions”