The Eigenstep Method: An Iterative Method for Unconstrained Quadratic Optimization
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Bibliographic record
Abstract
Th is paper presents a method for the unconstrained minimizat ion of convex quadratic programming problems. The method is a line search method, an iterative nonmonotone gradient method that is a modification of the classical steepest descent method. The two methods are the same in the choice of the negative gradient as the search direction, but differ in the choice of step size. The steepest descent method uses the optimal step size introduced by Cauchy in the n ineteenth century and the proposed method uses the reciprocal of the eigenvalues of the Hessian matrix as step sizes. Thus, the proposed method is referred to as the eigenstep method. We introduce and study three more recent developments, also modifications of the steepest descent method that alter the optimal Cauchy choice of steplength with nonmonotone steplength choices. Nu merical examp les with encouraging results are given to illustrate our new algorith m and a co mparison is made to two standard optimizat ion methods as well as to the three more recent developments in line search methods presented in this paper.
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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 | 0.012 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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