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Process."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>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. 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