Testing convexity of functions over finite domains
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Bibliographic record
Abstract
We establish new upper and lower bounds on the number of queries required to test convexity of functions over various discrete domains. We provide a simplified version of the non-adaptive convexity tester on the line. We re-prove the upper bound in the usual uniform model, and prove an upper bound in the distribution-free setting. We show a tight lower bound of queries for testing convexity of functions f: [n] → ℝ on the line. This lower bound applies to both adaptive and non-adaptive algorithms, and matches the upper bound from item 1, showing that adaptivity does not help in this setting. Moving to higher dimensions, we consider the case of a stripe [3] × [n]. We construct an adaptive tester for convexity of functions f: [3] × [n] → ℝ with query complexity O(log2 n). We also show that any non-adaptive tester must use queries in this setting. Thus, adaptivity yields an exponential improvement for this problem. For functions f: [n]d → ℝ over domains of dimension d ≥ 2, we show a non-adaptive query lower bound .
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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