Adversarial Attack and Defense for Non-Parametric Two-Sample Tests

Non-parametric two-sample tests (TSTs) that judge whether two sets of samples are drawn from the same distribution, have been widely used in the analysis of critical data. People tend to employ TSTs as trusted basic tools and rarely have any doubt about their reliability. This paper systematically uncovers the failure mode of non-parametric TSTs through adversarial attacks and then proposes corresponding defense strategies. First, we theoretically show that an adversary can upperbound the distributional shift which guarantees the attack's invisibility. Furthermore, we theoretically find that the adversary can also degrade the lower bound of a TST's test power, which enables us to iteratively minimize the test criterion in order to search for adversarial pairs. To enable TST-agnostic attacks, we propose an ensemble attack (EA) framework that jointly minimizes the different types of test criteria. Second, to robustify TSTs, we propose a max-min optimization that iteratively generates adversarial pairs to train the deep kernels. Extensive experiments on both simulated and real-world datasets validate the adversarial vulnerabilities of non-parametric TSTs and the effectiveness of our proposed defense. Source code is available at https://github.com/GodXuxilie/Robust-TST.git . where x i , x j ∈ S P and y i , y j ∈ S Q . In this paper, we investigate six types of non-parametric TSTs as follows since Liu et al. (2020a; 2021) have shown they are powerful on complex data. • ) for tests based on MMD with Gaussian kernels (MMD-G) (Sutherland et al., 2017) with the learnable lengthscale σ φ , in which ) for tests based on MMD with deep kernels (MMD-D) (Liu et al., 2020a). Note that k (D) (x, y) = [(1γ) exp(-1 2σ φ φ(x)φ(y) 2 ) + γ] exp(-1 2σq xy 2 ) where γ, σ φ , σ q are the learnable parameters and φ(•) is a parameterized deep network to extract the features. • D (S) (•, •) (Eq. ( 10 )) for C2ST based on Sign (C2ST-S) (Lopez-Paz & Oquab, 2016). A classifier f : X → R that outputs the classification probabilities is utilized by C2ST. Liu et al. (2020a) pointed out that the test statistic of C2ST-S is equivalent to MMD with kernel k (S) , i.e., D (S) (•, •) = MMD 2 (•, •; k (S) ) where k (S) (x, y) = 1 4 1(f(x) > 0)1(f (y) > 0).