Testing Distributions Against Bounded Distinguishers

2026-07-17

AI Summary

This work addresses the challenge of distribution identity testing in high-dimensional or continuous domains, where traditional total variation distance becomes ineffective. The authors propose a fooling distance framework based on bounded discriminator classes, integrating integral probability metrics with Boolean function classes to achieve sample-efficient testing algorithms. By unifying three directions—testable learning, verification of learning, and structured distribution testing—the study extends the Ak-testing framework and establishes theoretical connections among them. Technically, the approach combines Rademacher complexity analysis, membership query mechanisms, and polynomial density modeling over the hypercube, yielding testable proper learners for halfspaces and decision trees, deriving verification lower bounds, and designing efficient identity testers for decision tree distributions and low-degree polynomial densities.

Abstract

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.

Problem

• distribution testing
• bounded distinguishers
• identity testing
• fooling distance
• high-dimensional domains

Innovation

• fooling distance
• distribution testing
• integral probability metrics
• testable learning
• bounded distinguishers