Concentration and robustness of discrepancy-based ABC via Rademacher complexity

Classical implementations of approximate Bayesian computation (ABC) employ summary statistics to measure the discrepancy among the observed data and the synthetic samples generated from each proposed value of the parameter of interest. However, finding effective summaries is challenging for most of the complex models for which ABC is required. This issue has motivated a growing literature on summary-free versions of ABC that leverage the discrepancy among the empirical distributions of the observed and synthetic data, rather than focusing on selected summaries. The effectiveness of these solutions has led to an increasing interest in the properties of the corresponding ABC posteriors, with a focus on concentration and robustness in asymptotic regimes. Although recent contributions have made key advancements, current theory mostly relies on existence arguments which are not immediate to verify and often yield bounds that are not readily interpretable, thus limiting the methodological implications of theoretical results. In this article we address such aspects by developing a novel unified and constructive framework, based on the concept of Rademacher complexity, to study concentration and robustness of ABC posteriors within the general class of integral probability semimetrics (IPS), that includes routinely-implemented discrepancies such as Wasserstein distance and MMD, and naturally extends classical summary-based ABC. For rejection ABC based on the IPS class, we prove that the theoretical properties of the ABC posterior in terms of concentration and robustness directly relate to the asymptotic behavior of the Rademacher complexity of the class of functions associated to each discrepancy. This result yields a novel understanding of the practical performance of ABC with specific discrepancies, as shown also in empirical studies, and allows to develop new theory guiding ABC calibration.