BEYOND NEWTON: ROBUST METHODS FOR SOLVING LARGE NONLINEAR MODELS IN TROLL
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Bibliographic record
Abstract
Newton's method is an important algorithm for solving nonlinear systems of equations. For any solution algorithm, the principle concerns are robustness (finding a solution reliably) and efficiency (finding a solution quickly). Newton is simple in principle, but a useful implementation must deal with a variety of practical and theoretical obstacles. By using partial derivatives, Newton's method can model the shape of the residual surface to provide quadratic convergence near the solution: the number of correct digits doubles each iteration. But the full step may be illegal, leading to economic nonsense like negative prices and numerical problems like taking the log of a negative number. Automatic backtracking — taking shorter steps along the Newton direction — can improve global convergence in such cases. This paper describes enhancements to Newton's method used in the TROLL modeling system and illustrates them with a variety of contemporary models. Acknowledgements I would like to thank Hope Pioro of the Bank of Canada, who provided the stochastic simulation program and model that I used for some of the experiments. I would also like to thank Sarma Jayanthi and Doug Laxton of the International Monetary Fund for providing other test models. All errors are mine. Beyond Newton: Robust Methods for Solving Large Nonlinear Models in TROLL 1. The problem Given a system of equations: F ( x) = 0 (1) where F is a vector of equations and x is a vector of variables and: x (0) , an initial guess for x, find a “solution ” x * such that F(x * ) ≈ 0. F is assumed to be differentiable, so the Jacobian matrix,
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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 | 0.000 | 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