Efficient Testing Implies Structured Symmetry
This work provides a theoretical characterization for efficient property testing in machine learning and computer science, building incrementally on prior results to address computational constraints.
The paper tackles the problem of characterizing which properties of Boolean functions can be tested efficiently from few random samples, showing an equivalence between efficiently testable properties and those with structured symmetry based on low-complexity partitions. The result extends to broader sample sizes and generalizes to high-entropy distribution testing on arbitrary domains.
Given a small random sample of $n$-bit strings labeled by an unknown Boolean function, which properties of this function can be tested computationally efficiently? We show an equivalence between properties that are efficiently testable from few samples and properties with structured symmetry, which depend only on the function's average values on parts of a low-complexity partition of the domain. Without the efficiency constraint, a similar characterization in terms of unstructured symmetry was obtained by Blais and Yoshida (2019). Our main technical tool is supersimulation, which builds on methods from the algorithmic fairness literature to approximate arbitrarily complex functions by small-circuit simulators that fool significantly larger distinguishers. We extend the characterization along other axes as well. We show that allowing parts to overlap exponentially reduces their required number, broadening the scope of the construction from properties testable with $O(\log n)$ samples to properties testable with $O(n)$ samples. For larger sample sizes, we show that any efficient tester is essentially checking for indistinguishability from a bounded collection of small circuits, in the spirit of a characterization of testable graph properties. Finally, we show that our results for Boolean function testing generalize to high-entropy distribution testing on arbitrary domains.