# Minimax Estimation of Neural Net Distance

arXiv: https://arxiv.org/abs/1811.01054 · HTML: https://mlsanity.com/paper/1811.01054/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2018.

## Built on by (most cited first, 7 of 7)

- [Lower Bounds and Accelerated Algorithms for Bilevel Optimization](https://mlsanity.com/paper/2102.03926/) (2021)
- [Learning to Generalize Provably in Learning to Optimize](https://mlsanity.com/paper/2302.11085/) (2023)
- [α-GAN: Convergence and Estimation Guarantees](https://mlsanity.com/paper/2205.06393/) (2022)
- [Addressing GAN Training Instabilities via Tunable Classification Losses](https://mlsanity.com/paper/2310.18291/) (2023)
- [A Statistical Learning Perspective on Semi-dual Adversarial Neural Optimal Transport Solvers](https://mlsanity.com/paper/2502.01310/) (2025)
- [On the Optimization Landscape of Maximum Mean Discrepancy](https://mlsanity.com/paper/2110.13452/) (2021)
- [Theoretical Analysis of Measure Consistency Regularization for Partially Observed Data](https://mlsanity.com/paper/2602.01437/) (2026)

## Abstract (arXiv)

An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance based on samples drawn from the distributions. We develop the first known minimax lower bound on the estimation error of the neural net distance, and an upper bound tighter than an existing bound on the estimator error for the empirical neural net distance. Our lower and upper bounds match not only in the order of the sample size but also in terms of the norm of the parameter matrices of neural networks, which justifies the empirical neural net distance as a good approximation of the true neural net distance for training GANs in practice.