Error bounds for overdetermined and underdetermined generalized centred simplex gradients
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Bibliographic record
Abstract
Abstract Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient. We develop error bounds and, under a full-rank condition, show that the error bounds have ${\mathcal O}(\varDelta ^2)$, where $\varDelta $ is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also ${\mathcal O}(\varDelta ^2)$. We provide several examples to illustrate the results and some benefits of these new methods.
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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 | 0.000 | 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 | 0.000 | 0.000 |
| 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