Quantitative convergence rates for approximate optimization weights

Derive quantitative convergence rates for the uniform convergence of approximate minimizers of the empirical source-to-target weighting criterion to the oracle source-to-target weights.

Background

The paper proves that approximate minimizers of the empirical covariate-shifted distance-covariance and energy-distance criterion converge uniformly to the oracle source-to-target weight under Sobolev regularity and other assumptions. However, the result is qualitative: it establishes convergence without specifying an explicit rate in terms of the source and target sample sizes, optimization tolerance, or function-class complexity. The authors identify quantitative epigraphical-distance techniques as a possible route to obtaining such rates.

References

First, the uniform convergence theorem is stated for approximate minimizers of the empirical criterion, but obtaining quantitative rates for this convergence remains open. Such rates may be possible by adapting quantitative epigraphical-distance arguments such as those in \citet{attouch1991quantitative}.

— Causal Generalization of Continuous Treatment Effects under Covariate Shift  (2608.19383 - Cheng et al., 19 Aug 2026) in Discussion, paragraph beginning “Several limitations suggest directions for future work.”