Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism
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Bibliographic record
Abstract
A function f :{ −1,1} n → { −1,1} is a k -junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k -juntas, where the testing algorithm must accept any function that is ε- close to some k -junta and reject any function that is ε′-far from every k ′-junta for some ε′ = O (ε) and k ′ = O ( k ). Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ε. This result is obtained via a new polynomial-time approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function. Our second result considers the case where k ′ = k . We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that, given ρ ∈ (0,1), accepts any function that is ε ρ/16-close to some k -junta and rejects any function that is ε-far from every k -junta. The query complexity of the algorithm is O ( k log k /ε ρ (1-ρ) k . Finally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g . We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k -junta.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it