Optimal Prediction-Augmented Algorithms for Testing Independence of Distributions

Motivated by the challenge of testing distributions over high-dimensional or continuous domains, we study distribution testing with respect to bounded classes of distinguishers. A representative task is to use samples from an unknown distribution $P$ over a very large domain to decide between two cases: $P = P_{\mathsf{ref}}$ for a fixed reference distribution $P_{\mathsf{ref}}$, or there exists a distinguisher $f$ in a bounded class $\mathcal{F}$ which witnesses the separation $|\mathbf{E}_P[f] - \mathbf{E}_{P_{\mathsf{ref}}}[f]| > ε$. This is the task of identity testing with respect to fooling distance, a name inspired by the conceptual connection with pseudorandomness. (Formally, our model instantiates integral probability metrics from Boolean classes of bounded expressivity.) We show that testing with respect to fooling distance is not only a natural computational problem that admits sample-efficient algorithms even in high-dimensional settings, but also one that reveals and underlies connections between three seemingly unrelated areas of study: testable learning, verification of learning algorithms, and testing of structured distributions (whose "$\mathcal{A}_k$-testing" model our framework extends). These connections yield new results for all of these models, including: 1. Testable proper learners using membership queries for halfspaces and decision trees. 2. A lower bound for testable PAC verification in terms of Rademacher complexity, and a distribution-free verification protocol for disjoint unions of $k$ multidimensional rectangles. 3. Identity testers (with respect to total variation distance) for decision tree distributions and distributions with low-degree polynomial densities, over Boolean and continuous hypercube domains.

Deep learning methods have proved highly effective for classification and image recognition problems. In this paper, we ask whether this success can be transferred to hypothesis testing: if a neural network can distinguish, for example, an image of a handwritten digit from another, can it also distinguish an "image of a sample" (such as a scatter plot) generated under a given statistical model from one generated outside that model? Motivated by this idea, we propose a novel procedure called deep-testing, which approaches the classical inferential problem of hypothesis testing through deep learning. More specifically, the test statistic is a classification map learned by a deep neural network from simulated data satisfying the null and alternative hypotheses, leveraging its strong discriminating power to construct a highly powerful test. As a proof of concept, we apply deep-testing to the problem of independence testing, arguably one of the most important problems in statistics. In a large-scale simulation study, deep-testing achieves the highest overall power against nineteen competing methods across a broad range of complex dependence structures, confirming the viability of the proposed approach.

Testing conditional independence is fundamental yet intrinsically difficult: without additional assumptions, Type I error control is impossible in general. The "Model-X'' paradigm addresses this difficulty by assuming exact knowledge of a relevant conditional distribution. While small deviations from this assumption can sometimes be tolerated in classical one-shot testing, existing sequential conditional independence tests typically require the Model-X conditional to be known exactly, making them fragile when it must instead be estimated. We propose a new approach that is substantially more robust to such estimation error. Our method applies testing-by-betting to an adaptively optimized Kernel Conditional Independence statistic, together with a normalization scheme and a truncate-and-shift calibration strategy. These modifications greatly reduce Type I error inflation while preserving high power across high-dimensional synthetic benchmarks and real-world fairness tasks, outperforming existing sequential Model-X approaches. Code is available at https://github.com/he-zh/SKCI.