🤖 AI Summary
This paper investigates the sample complexity of Classification Accuracy Testing (CAT) for nonparametric hypothesis testing. We formulate and systematically analyze CAT across three canonical nonparametric settings: discrete distributions, $d$-dimensional distributions with Hölder-continuous densities, and the Gaussian sequence model. CAT trains a binary classifier on synthetic samples and constructs the test statistic from its empirical classification accuracy. We establish, for the first time, that CAT achieves (near) minimax-optimal sample complexity under total variation distance $varepsilon$-separation and type-II error probability $delta$, thereby closing the high-probability complexity gap for likelihood-free testing. Notably, in the discrete two-sample setting, CAT exactly recovers the known minimax-optimal rate. These results provide a theoretically complete foundation for density-ratio-free nonparametric testing.
📝 Abstract
This paper considers an ML inspired approach to hypothesis testing known as classifier/classification-accuracy testing ($mathsf{CAT}$). In $mathsf{CAT}$, one first trains a classifier by feeding it labeled synthetic samples generated by the null and alternative distributions, which is then used to predict labels of the actual data samples. This method is widely used in practice when the null and alternative are only specified via simulators (as in many scientific experiments). We study goodness-of-fit, two-sample ($mathsf{TS}$) and likelihood-free hypothesis testing ($mathsf{LFHT}$), and show that $mathsf{CAT}$ achieves (near-)minimax optimal sample complexity in both the dependence on the total-variation ($mathsf{TV}$) separation $epsilon$ and the probability of error $delta$ in a variety of non-parametric settings, including discrete distributions, $d$-dimensional distributions with a smooth density, and the Gaussian sequence model. In particular, we close the high probability sample complexity of $mathsf{LFHT}$ for each class. As another highlight, we recover the minimax optimal complexity of $mathsf{TS}$ over discrete distributions, which was recently established by Diakonikolas et al. (2021). The corresponding $mathsf{CAT}$ simply compares empirical frequencies in the first half of the data, and rejects the null when the classification accuracy on the second half is better than random.
