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Pré-Publication, Document De Travail Année : 2023

Wasserstein Generative Adversarial Networks are Minimax Optimal Distribution Estimators

Arthur Stéphanovitch
  • Fonction : Auteur
  • PersonId : 1318416
Clément Levrard
  • Fonction : Auteur
  • PersonId : 1318418

Résumé

We provide non asymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax optimal rate for estimating a certain probability measure $\mu$ with support in $\mathbb{R}^p$. The probability $\mu$ is considered to be the push forward of the Lebesgue measure on the $d$-dimensional torus $\mathbb{T}^d$ by a map $g^\star:\mathbb{T}^d\rightarrow \mathbb{R}^p$ of smoothness $\beta+1$. Measuring the error with the $\gamma$-Hölder Integral Probability Metric (IPM), we obtain up to logarithmic factors, the minimax optimal rate $O(n^{-\frac{\beta+\gamma}{2\beta +d}}\vee n^{-\frac{1}{2}})$ where $n$ is the sample size, $\beta$ determines the smoothness of the target measure $\mu$, $\gamma$ is the smoothness of the IPM ($\gamma=1$ is the Wasserstein case) and $d\leq p$ is the intrinsic dimension of $\mu$. In the process, we derive a sharp interpolation inequality between Hölder IPMs. This novel result of theory of functions spaces generalizes classical interpolation inequalities to the case where the measures involved have densities on different manifolds.
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Dates et versions

hal-04315916 , version 1 (30-11-2023)

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  • HAL Id : hal-04315916 , version 1

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Arthur Stéphanovitch, Eddie Aamari, Clément Levrard. Wasserstein Generative Adversarial Networks are Minimax Optimal Distribution Estimators. 2023. ⟨hal-04315916⟩
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