# Reliable Estimation of KL Divergence using a Discriminator in Reproducing Kernel Hilbert Space

arXiv: https://arxiv.org/abs/2109.14688 · HTML: https://mlsanity.com/paper/2109.14688/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 29 Sep 2021.

## Builds on (most cited first, 18 of 18)

- [Representation Learning with Contrastive Predictive Coding](https://mlsanity.com/paper/1807.03748/) (2018)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [MINE: Mutual Information Neural Estimation](https://mlsanity.com/paper/1801.04062/) (2018)
- [Which Training Methods for GANs do actually Converge?](https://mlsanity.com/paper/1801.04406/) (2018)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Isolating Sources of Disentanglement in Variational Autoencoders](https://mlsanity.com/paper/1802.04942/) (2018)
- [Understanding the Limitations of Variational Mutual Information Estimators](https://mlsanity.com/paper/1910.06222/) (2019)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [Amortised MAP Inference for Image Super-resolution](https://mlsanity.com/paper/1610.04490/) (2016)
- [Maximum Margin Multiclass Nearest Neighbors](https://mlsanity.com/paper/1401.7898/) (2014)
- [Estimating Kullback-Leibler Divergence Using Kernel Machines](https://mlsanity.com/paper/1905.00586/) (2019)

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

- [The Representation Jensen-Shannon Divergence](https://mlsanity.com/paper/2305.16446/) (2023)
- [Automated Discovery of Pairwise Interactions from Unstructured Data](https://mlsanity.com/paper/2409.07594/) (2024)
- [Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances](https://mlsanity.com/paper/2206.03230/) (2022)
- [Optimizing EEG Graph Structure for Seizure Detection: An Information Bottleneck and Self-Supervised Learning Approach](https://mlsanity.com/paper/2604.01595/) (2026)
- [Feature-aligned N-BEATS with Sinkhorn divergence](https://mlsanity.com/paper/2305.15196/) (2023)
- [OBSER: Object-Based Sub-Environment Recognition for Zero-Shot Environmental Inference](https://mlsanity.com/paper/2507.02929/) (2025)
- [A transfer learning approach for automatic conflicts detection in software requirement sentence pairs based on dual encoders](https://mlsanity.com/paper/2511.23007/) (2025)

## Abstract (arXiv)

Estimating Kullback Leibler (KL) divergence from samples of two distributions is essential in many machine learning problems. Variational methods using neural network discriminator have been proposed to achieve this task in a scalable manner. However, we noted that most of these methods using neural network discriminators suffer from high fluctuations (variance) in estimates and instability in training. In this paper, we look at this issue from statistical learning theory and function space complexity perspective to understand why this happens and how to solve it. We argue that the cause of these pathologies is lack of control over the complexity of the neural network discriminator function and could be mitigated by controlling it. To achieve this objective, we 1) present a novel construction of the discriminator in the Reproducing Kernel Hilbert Space (RKHS), 2) theoretically relate the error probability bound of the KL estimates to the complexity of the discriminator in the RKHS space, 3) present a scalable way to control the complexity (RKHS norm) of the discriminator for a reliable estimation of KL divergence, and 4) prove the consistency of the proposed estimator. In three different applications of KL divergence : estimation of KL, estimation of mutual information and Variational Bayes, we show that by controlling the complexity as developed in the theory, we are able to reduce the variance of KL estimates and stabilize the training