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.

翻译：我们提出了一个框架，用于从$f$-散度族中构建实用的基于核的双样本检验。检验统计量通过散度的正则化变分表示形式的见证函数计算得出，我们使用核方法对该函数进行估计。所提出的检验对超参数（如核带宽和正则化参数）具有自适应性。我们为$f$-散度估计族提供了统计检验功效的理论保证。虽然我们的检验涵盖了多种$f$-散度，但我们特别关注曲棍球棒散度，其动机在于其在差分隐私审计和机器遗忘评估中的应用。对于双样本检验，实验表明不同的$f$-散度对不同的局部差异具有敏感性，这说明了利用多样化统计量的重要性。对于机器遗忘，我们提出了一种相对检验，用于区分真实的遗忘失败与安全的分布变化。