Minimax Rates for Conditional Generative Learning under Wasserstein Loss
Read the original on arXiv Statistics ML →The Flow has not summarised this story yet — read it at arXiv Statistics ML.
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arXiv:2606. 19117v1 Announce Type: cross Abstract: Offline policy learning has received growing attention in causal inference.
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The paper investigates quantile distributional policy evaluation by approximating the return distribution under a policy with a quantile Bellman fixed point. It constructs an estimator from an empirical Markov decision process and derives non‑asymptotic error bounds in Wasserstein distances, achieving root‑n rates that are independent of the quantile resolution for a wide range of Wasserstein orders. The authors also characterize the exact local asymptotic minimax risk and show that their estimator attains this optimal rate, while examining how the risk depends on resolution in simple state‑space examples.
The paper develops a statistical framework showing that empirical score matching can achieve minimax optimal performance for diffusion generative models when data lie on a low‑dimensional manifold. It establishes convergence rates for empirical risk minimization over constrained Sobolev classes and demonstrates that the resulting score estimators yield minimax optimal Wasserstein‑1 rates for the generated distribution, up to logarithmic factors. The analysis reveals that the statistical rates depend on the intrinsic dimension of the data manifold rather than its ambient dimension, providing conditions under which score‑based generative modeling generalizes rather than overfits.