# Regularized f-Divergence Kernel Tests

arXiv: https://arxiv.org/abs/2601.19755 · HTML: https://mlsanity.com/paper/2601.19755/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 35 of its 79 references that are in the map.
- First posted on arXiv 27 Jan 2026.

## Placed in

- LLM safety and security / [Machine unlearning](https://mlsanity.com/safety/machine-unlearning/)

## Builds on (most cited first, 30 of 35)

- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Remember What You Want to Forget: Algorithms for Machine Unlearning](https://mlsanity.com/paper/2103.03279/) (2021)
- [Fast Machine Unlearning Without Retraining Through Selective Synaptic Dampening](https://mlsanity.com/paper/2308.07707/) (2023)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Optimal Accounting of Differential Privacy via Characteristic Function](https://mlsanity.com/paper/2106.08567/) (2021)
- [MMD Aggregated Two-Sample Test](https://mlsanity.com/paper/2110.15073/) (2021)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)
- [Are we making progress in unlearning? Findings from the first NeurIPS unlearning competition](https://mlsanity.com/paper/2406.09073/) (2024)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [MAUVE Scores for Generative Models: Theory and Practice](https://mlsanity.com/paper/2212.14578/) (2022)
- [MMD-FUSE: Learning and Combining Kernels for Two-Sample Testing Without Data Splitting](https://mlsanity.com/paper/2306.08777/) (2023)
- [A Wild Bootstrap for Degenerate Kernel Tests](https://mlsanity.com/paper/1408.5404/) (2014)
- [Understanding the Sparse Vector Technique for Differential Privacy](https://mlsanity.com/paper/1603.01699/) (2016)
- [Efficient Aggregated Kernel Tests using Incomplete U-statistics](https://mlsanity.com/paper/2206.09194/) (2022)
- [A Test of Relative Similarity For Model Selection in Generative Models](https://mlsanity.com/paper/1511.04581/) (2015)
- [Learning new physics efficiently with nonparametric methods](https://mlsanity.com/paper/2204.02317/) (2022)
- [AutoML Two-Sample Test](https://mlsanity.com/paper/2206.08843/) (2022)
- [Property Testing for Differential Privacy](https://mlsanity.com/paper/1806.06427/) (2018)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [A Practical Introduction to Kernel Discrepancies: MMD, HSIC & KSD](https://mlsanity.com/paper/2503.04820/) (2025)
- [DP-Auditorium: a Large Scale Library for Auditing Differential Privacy](https://mlsanity.com/paper/2307.05608/) (2023)
- [Goodness of fit by Neyman-Pearson testing](https://mlsanity.com/paper/2305.14137/) (2023)
- [Differentially Private Permutation Tests](https://mlsanity.com/paper/2310.19043/) (2023)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [Multiple testing for signal-agnostic searches of new physics with machine learning](https://mlsanity.com/paper/2408.12296/) (2024)
- [Analysis of k-Nearest Neighbor Distances with Application to Entropy Estimation](https://mlsanity.com/paper/1603.08578/) (2016)
- [DUAL: Learning Diverse Kernels for Aggregated Two-sample and Independence Testing](https://mlsanity.com/paper/2510.11140/) (2025)

## Built on by (most cited first, 2 of 2)

- [Zero-Flow Two-Sample Tests](https://mlsanity.com/paper/2607.21542/) (2026)
- [Behavioral Guarantees for Proxy-Based Unlearning](https://mlsanity.com/paper/2605.10680/) (2026)

## Abstract (arXiv)

We propose a framework to construct practical kernel-based two-sample tests from the family of f-divergences. The test statistic is computed from the witness function of a regularized variational representation of the divergence, which we estimate using kernel methods. The proposed test is adaptive over hyperparameters such as the kernel bandwidth and the regularization parameter. We provide theoretical guarantees for statistical test power across our family of f-divergence estimates. While our test covers a variety of f-divergences, we bring particular focus to the Hockey-Stick divergence, motivated by its applications to differential privacy auditing and machine unlearning evaluation. For two-sample testing, experiments demonstrate that different f-divergences are sensitive to different localized differences, illustrating the importance of leveraging diverse statistics. For machine unlearning, we propose a relative test that distinguishes true unlearning failures from safe distributional variations.