On the convergence rate of two-stage sampling distribution regression

The upper bound estimate for the convergence rate of two-stage sampling distribution regression is investigated. The excess risk is decomposed as the two-stage sample error and the approximation error, the approximation error is bounded by a [Formula: see text]-functional. The two-stage sample error is bounded with a convex analysis method whose advantage lies separating the two-stage optimal solution and the one-stage optimal solution from the mean optimal solution. On this basis, the two-stage sample error is attributed to the decay of a modulus of smoothness whose convergence rate has been investigated in approximation theory. Finally, a kind of explicit capacity independent convergence rate for the excess risk is provided. The feature of the investigation lies that the usual regularity condition is replaced by a decay assumption related to the modulus of smoothness.