Apple researchers show separation between testing and verification for location-invariant properties
A new paper from Apple’s Machine Learning Research team demonstrates that while testing complexity for location-invariant properties closely tracks distribution properties, verification complexity does not, introducing new bounds for interactive proofs of proximity.
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- A new Apple Machine Learning Research paper distinguishes testing and verification complexity for location-invariant properties of functions.
- The work shows that relationships proven for testing do not extend to verification, particularly for doubly-sublinear interactive proofs of proximity (IPPs).
- The paper provides explicit query complexity bounds for two natural location-invariant function properties, contrasting them with known limitations for corresponding distribution properties.
- Findings include new doubly-sublinear IPPs for balanced frequency functions and functions with restricted value occurrences.
Apple’s Machine Learning Research group published a paper titled “Location-Invariant Properties of Functions Versus Properties of Distributions: United in Testing but Separated in Verification,” which examines how location-invariant properties behave under testing versus verification. A location-invariant property is one that depends only on the frequency of values in a function, not their positions.
The authors—Oded Goldreich and Guy N. Rothblum—note that prior work established close relationships between the query complexity of testing location-invariant properties of functions and the sample complexity of testing corresponding properties of distributions. Their new result shows that this relationship breaks down in the context of verification, especially for interactive proofs of proximity (IPPs).
The paper introduces new bounds for doubly-sublinear IPPs, a class of verification protocols where both the verifier’s query complexity and the honest prover’s complexity are sublinear relative to baseline testing and learning complexities. Specifically, the authors present two constructions:
First, for the set of functions from [m] to [n] where each value occurs exactly m/n times, they give a doubly-sublinear IPP with verifier query complexity O(n^{0.5−α}) and prover query complexity e^{O(n^{0.5+α}/ε^2)} for any α ∈ (0, 0.5).
Second, for functions from [m] to [n] where each value occurs either m/k times or not at all, they provide a doubly-sublinear IPP with verifier query complexity poly(1/ε) · k^{(2/3)−2α} and prover query complexity poly(1/ε) · e^{O(k^{(2/3)+α})} for any α ∈ (0, 1/3).
The authors contrast these results with known limitations for the corresponding distribution properties, where doubly-efficient IPPs are not possible. For instance, the uniformity property over [n] has no IPP where the verifier uses o(n^{1/2}) samples, regardless of other efficiency measures.
The findings highlight a structural separation between testing and verification for location-invariant properties and contribute to the broader study of interactive proof systems for statistical verification.
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