Typical improvement from inverse-leverage weighting

Determine whether the factor comparing the weighted best-approximation error under the inverse-leverage weight with the unweighted best-approximation error is typically smaller than one, thereby establishing whether inverse-leverage weighting generally improves unweighted sup-norm regression bounds.

Background

For exact quadrature, the paper compares unweighted sup-norm error bounds with bounds obtained using the weight function given by the inverse of the leverage function, scaled by the square root of the model dimension. The comparison involves the factor between the corresponding best-approximation errors.

The authors conjecture that this factor is typically less than one, motivated by polynomial approximation on unbounded domains: unweighted polynomial approximation of bounded functions is ineffective, whereas the inverse-leverage weight can compensate for polynomial growth. The conjecture is presented as unresolved and is intended to clarify whether inverse-leverage weighting is naturally advantageous for unweighted sup-norm guarantees.

References

We conjecture that this factor is typically smaller than one based on this polynomial example: As polynomials of degree greater zero always diverge to infinity, they cannot uniformly approximate bounded functions on unbounded domains.

— User-friendly approximation theory for statistical regression  (2610.01342 - Benning, 1 Oct 2026) in Remark following Example ‘Uniform bounds for exact quadrature’ in Section 3, subsection ‘Exact quadrature/cubature’

It is unclear how to bound \error_\modelSpace\infty[f_x] in general however.

— User-friendly approximation theory for statistical regression  (2610.01342 - Benning, 1 Oct 2026) in Remark ‘Towards tighter bounds on the Lebesgue constant’ in Section 3, subsection ‘Exact quadrature/cubature’