Model-class approximation error and regularity-controlled capacity scaling

Quantify how the model-class approximation error for improved MeanFlow transport maps decreases with model capacity while controlling the associated regularity constants.

Background

The paper derives posterior-stability and end-to-end sampling-error bounds in terms of the joint improved MeanFlow risk, which decomposes into population training suboptimality and model-class approximation error. The resulting transport and posterior guarantees depend not only on this approximation error but also on regularity quantities such as Lipschitz constants and related bounds. Although the theory shows that the error bounds improve as the approximation error decreases, it does not characterize how that error scales with neural-network capacity or how the regularity constants behave as capacity increases. The authors therefore identify establishing this capacity-versus-approximation relationship under controlled regularity as an unresolved problem.