Observations on greedy composite Newton methods
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Bibliographic record
Abstract
The only robust general-purpose numerical methods for approximating the solution to systems of nonlinear algebraic equations (NAEs) are based on Newton's method. Many variants of Newton's method exist in order to take advantage of problem structure; it is often computationally infeasible to solve a given problem without taking some advantage of this structure. It is generally impossible to know a priori which variant of Newton's method will be optimal for a given problem. In this paper, we describe an algorithm for automatically selecting a composite Newton method, i.e., a sequential combination of Newton variants, for solving NAEs. The algorithm is based on a greedy principle that updates the current state at regular intervals according to the best performing Newton variant. Preliminary results show that it is possible for composite Newton methods to outperform optimal classical implementations of Newton's method, i.e., ones that only use one Newton variant on a given problem.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| 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