Minimax optimal testing by classification
Patrik Róbert Gerber, Yanjun Han, Yury Polyanskiy
TL;DR
This work establishes classifier-accuracy testing (CAT) as a nonparametric, minimax-optimal framework for GoF, two-sample, and likelihood-free hypothesis testing across discrete, smooth, and Gaussian sequence models. Central to the results is the construction of separating sets S that maximize separation sep(S) while keeping τ(S) small, enabling test statistics based on CAT to achieve near-minimax sample complexities in high-probability regimes. The paper provides concrete separating-set constructions for the discrete case (and extensions to smooth densities via binning and to Gaussian sequences via truncation), recovering known optimal rates for GoF/TS and closing high-probability LFHT gaps up to polylog factors. Practically, these results justify training binary or probabilistic classifiers on simulated data and using their downstream accuracy as a powerful, theoretically justified test statistic in simulator-based scientific experiments.
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 $ε$ and the probability of error $δ$ 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.