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When the reward density is uniformly bounded away from zero, the paper establishes a root-n high-probability bound that holds simultaneously for all resolutions m and all Wasserstein orders p (1 ≤ p ≤ ∞), with the estimation error converging at the parametric rate n^{-1/2}.
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10/22/2026
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UnverifiedWhen the reward density vanishes at the support endpoints, for 1 ≤ p ≤ 2, the resolution-independent root-n rate is still maintained as long as the reward quantile function satisfies continuity and endpoint monotonicity.78% similarUnverifiedThe Stein identity states that for a function φ that decays sufficiently fast on the boundary, when the probability density p(x) is sufficiently smooth on R^d, E_p[∇_x log p(x)·φ(x) + ∇_x·φ(x)] = 0.73% similarUnverifiedThe error bound in this paper holds for all Wasserstein-p distances with p ≥ 1, rather than being limited to a specific metric.69% similarUnverifiedThe finite-sample bound compares the network with its Gaussian limit on a fixed finite sample set, using the quadratic Wasserstein distance W2, with a convergence rate of m^{-1/2} and explicit constants, but the constants grow with the number of samples, and the bound becomes uninformative as the sample set is progressively refined.68% similarUnverifiedFor polynomial activation functions and arbitrary compact input sets, this lifting gives a rate of m^{-1/2} under the supremum norm, with an explicit constant.67% similar
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